A Comprehensive Dictionary of Mathematics

_cP!_ COMPREHENIIVE vvvvOfvvvv [)ICTI()~Al?"

MATHEMATICS Chief Editor & Compiler: -~

Roger Thompson

~ ABHISHEK

All rights reserved. No part of this book may be reproduced in any form, electronically or otherwise, in print, photoprint, micro film or by any other means without written permission from the publisher.

ISBN Copyright

Revised Edition

978-81-8247-341-6 Publisher

2010

Published by ABHISHEK PUBLICATIONS, S.C.O. 57-59, Sector 17-C, CHANDIGARH-1600 17 (India) Ph.-2707562,Fax-OI72-2704668 Email: [email protected]

Preface Mathematicians, scientists, engineers, and students will look this dictionary for defmitive coverage of all branches of mathematics, both pure and applied. Featuring more than 1500 terms, the book defmes terms and expressions in algebra, number theory, operator theory, logic, complex numbers, fmite mathematics, topology, and other areas----each with definition along with pictorial representations of many terms. This dictionary is authoritative, comprehensive comprising latest terms and is carefully reviewed to ensure its accuracy, clarity, and completeness. This Dictionary of Mathematics puts a wealth of essential information at your fingertips. Whether you're a professional, a student, a writer, or a general reader with science curiosity, this comprehensive resource defines the current language of pure and applied Mathematics and gives you a better understanding of the ideas and concepts you need to know. It has, for the first time, brought together in one easily accessible form the best-expressed thoughts that are especially illuminating and pertinent to the discipline of Mathematics. This dictionary will be a handy reference for the mathematician or scientific reader and the wider public interested in who has said what on mathematics.

· The overall aim of the dictionary is to provide an accessible description of what one judges to be the core material for damn good dictionary. The subject has a reputation for being disagreeably difficult. I have tried to alter that perception by showing that key ideas can be presented simply judiciously but not overwhelmingly deployed, clarifies and provides power.

11 2-by-2 table I abeliangroup

5

.2-by-2 table 1. this is a two-way table where the numbers of levels of the ro·.v and column-classifications are each 2. If the row and column classifications each divide the observational units into subsets, then it is likely that it will be useful to analyse the data using the Fisher Test.

. ; .

• 3-D figure a set of points in space; examples: box, cone, cylinder, parallelpiped, prism, pyramid, regular pyramid, right cone, right cylinder, right prism, sphere.

; 'prime'; designates an image : corresponding to the preimage ~ using the same variable.

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.• aRb I a is an element in b

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abacus ~ a Japanese counting device and : calculator.

~ • abelian group I

a group in which the binary

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• 45-45-90 triangle an isoscoles right triangle

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is commutative, that for all elements a abd b m the group. • abscissa the x-coordinate of a point in a 2-dimensional coordinate systern . • absolute value the positive value for a real n~ber, disregarding the sign. Wntten 1x I. For example, 131 =3, 1-41 =4, and 101 =0. • abundant number a positive integer that is smaller than the sum of its proper divisors.

~ • additive identity property ; the sum .pf any number and : zero is the original number· ~ zero is the identity element of ~ addition. ; • adjacent angles . .g ht and nonzero ; two non-straJ. : angles that have a common side ~ in the interior of the angle ; f?rmed by the non-s:ommon : SIdes

• acceleration ~e rate of change of velocity With respect to time. • "'......-...hl I els f --1'~ e eVi 0 accuracy ~e p~eClSlon determined by the sItuation or the given numbers stude?ts should help develo~ what IS acceptable according to the situation.

. ; • ad)a~t interior angle : the mtenor angle that forms a ~ linear pair with a given exterior I angle of a triangle. : . ." adjacent SIde ~ (of an angle.in a triangle~, one : of the two SIdes of the mangle ~ that form the sides of the angle.

IS,. ab= ba

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• accuracy ; • affine superimposition the closeness of a measurement ; a su~rimposition for which the or estimate to its true value. : assocIated transformations are I all affine. • acute angle : an angle whose measure is ~ • affine transforma,tion greater than 0 but less than 90 ; a transformation for which degrees. : parallel lines remain parallel.· I

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Mfine transformations of the plane take squares into parallelograms and take circles into ellipses of the same shape. Mfine transformations of a 3-dimensional space take cubes into parallelopipeds (sheared bricks) and spheres into ellipsoids all of the same shape. Similar results are produced in higher dimensional spaces . Equivalent to "uniform transformation" . As far as form is concerned' (that is, ignoring translation and rotation), any affine transformation can be diagrammed as a pure strain taking a square to a rectangle on the same axes. In studies of shape, where scale is ignored as well, the picture is the same but now the sum of the squares of the axes is unchanging. Still ignoring scale (that is, as far as shape is concerned), any affine transformation can be also diagrammed as a pure shear tak ing a square into a parallelogram of unchanged base segment and height. This diagram of shear came into morphometrics via an application to principal components analysis somewhat before it

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: _ algebraic number ~ a number that is the root of ; an algebraic polynomial. For : example, (sqrt 2) is an alge~ braic number because it is a ~ solution of the equation x2 =2.

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terion level (often 0.05). The

.'1· ... {-~. -Ti.~. -rz}' 11, ; outcome is classified as showU .... U1C(.-21J,.1.'1· •• {-~, IT'~' -Ti}; 11, 1 ing statistical signifi·lCance if the 1 AlqebCnct, · 2 '1 . 11' to, -~. 0, -,}. 11 . u""0;cl.-2I1'.'I'., {o, i, 0, ~}, 11, I actual alpha (probability of the

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outcome under the null hypothesis) is no greater than this _ algorithm nominal alpha criterion level. 1. a formal statement, clear . ~ This reasoning is applicable for all types of statistical testing, complete and unambiguous, of I how a certain process needs to : including re-randomisation stabe undertaken. I tistics which are the concern of 2. an algorithm expressed in a I this present glossary. programming language for a - alphametic computer . a cryptarithm in which the let_ allometry I ters, which represent distinct any change of shape with size. I digits, form related words or It describes any deviation of the meaningful phrases. bivariate relation from the - alternate exterior angles simple functional form yIx = c, I if two parallel lines are cut by a where c is a constant ~d x and I transversal, alternate exterior y are size measures in units of : angles are outside the parallel I lines and on opposite sides of the same dimension. I the transversal. - alpha also known as size or type-l er- I - alternate interior angles if two parallel lines are cut by a ror. This is the probability that, according to some null hypothesis, a statistical test will gener- I y ate a false-positive error : affirming a non-null pattern by chance. Conventional method- I ;"';';'---z ology for statistical testing is, in advance of undertaking the test, to set a nominal alpha cri- I .1"""<1> -211'. 11' •.

I

9

*================= transversal, alternate interior angles are inside the parallel lines and on opposite sides of the transversal. • alternative hypothesis in hypothesis testing, a null hypothesis (typically that there is no effect) is compared with an alternative hypothesis (typically that there is an effect, or that there is an effect of a particular sign). For example, in evaluating whether a new cancer remedy works, the null hypothesis typically would be that the remedy does not work, while the alternative hypothesis would be that the remedy does work. When the data are sufficiently improbable under the assumption that the null hypothesis is true, the null hypothesis is rejected in favour of the alternative hypothesis. (This does not imply that the data are probable under the assumption that the alternative hypothesis is true, nor that the null hypothesis is false, nor that the alternative hypothesis is true.

~

; :

~ ~

; : ~ ~

or the line containing the opposite side, or the length of the altitude segment. • altitude of a conic solid the length of a segment whose endpoints are the vertex and a point on the plane of the base that is perpendicular to the plane of the base

altitude of a cylindric solid the distance be,tween the planes : of the bases I •

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~ • altitude of a trapezoid I

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the distance between the bases of a trapeziod • altitude of a triangle. the perpendicular segment from a vertex to the line containing the opposite side of a triangle

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: • ambiguous ~ not stable; changing I • amicable numbers ; two numbers are said to be : amicable if each is equal to the ~ sum of the proper divisors of ; the other.

; • analyse • altitude : to break down into parts and (of a triangle) A line segment ~ explain or demonstrate the drawn from a vertu that is per- I logic of a situation or a process. pendicular to the opposite side

10

angle I anisotropy

• angle 1. (of a polygon) an angle having its vertex at one of the polygon's vertices, and having two of the polygon's sides as its sides. 2. two noncollinear rays (the sides of the angle) having a common endpoint (the angle's vertex). • angle bisector a ray that has the vertex of the angle as its endpoint, and that divides the angle into two congruent angles. LI!OH'''' --..\.ooe •• _

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tex at the viewer's eye, one side horizontal, and the viewer's line of sight to the object as the other side. angle of rotation the angle between a point and its image under a rotation, with its vertex at the center of the rotation. and sides that go through the point and its image. Also, the measure of the angle.

I •

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angle ruler a hinged ruler with a protractor attached for reading the measure of an angle in degrees.

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angle side one of the two rays forming an angle

I •

• angular velocity I (of an object moving around a circle) the rate of change, with respect to time, of the measure • angle of depression I of the central angle that inter(of an object lower than the I cepts the arc between the obviewer) The angle with its ver- : ject and a ftxed point. tex at the viewer's eye, one side I horizontal, and the viewer's line I • anisotropy of sight to the object as the anisotropy is a descriptor of one other side. I aspect of an affine transformation. In two dimensions, this is • angle of elevation the ratio of the axes of the el(of an object highet than a I lipse into which a circle is transviewer) The angle with its ver- formed by an affine transfor-

IllInnulus I lire length

11

mation. In general, it is the ~ _ antecedent maximum ratio of extension of ; the 'if' part of a conditional' l~n~ in one direc~on to exten- ; represented by p; aka hypoth: s~on m a perpendlCular direc- : esis, given, problem I •• tlon. : - antlpnsm I a polyhedron resulting from - annulus the region between two concen- : rotating one base of a prism tric circles of unequal radius. ~ and connecting the vertices so I that the lateral faces are tri~ angles. ~ - apothem : (of a regular polygon) a line ~ segment between the center of ; the polygon's circumscribed : circle to a side of the polygon ~ that is also perpendicular to ; that side. Also, the length of : that line segment. I

_ ANSI acronym for the American Nationa! Standards Institute. This body publishes specifications for a number of standard programming languages. The specifications are generally arranged to concur with those of ISO, _ ante the up-front cost of a bet: the money you must pay to play the game. From Latin for "before."

: - arc ~ (of a circle) Two points on the ; circle (the endpoints of the arc) : and the points of the circle be~ tween them. An angle interI cepts an arc if the sides of the ~ angle ~tersect the circle at the : ~ndpomts of the arc. Th~ arc is ~ mcluded by the chord wlth the ; same endpoints. ~ - arc length

: the portion of the circumference ~ of the circle described by an arc, ; measured in units of length.

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12 • arc measure the measure of the central angle that intercepts an arc, measured in degrees.

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• area the amount of space taken up in a plane by a figure • arithmetic mean the arithmetic mean of n numbers is the sum of the numbers divided by n.

• association two variables are associated if some of the variability of one can be accounted for by the other. In a scatterplot of the two variables, if the scatter in the values of the variable plotted on the vertical axis is smaller in narrow ranges of the variable plotted on the horizontal axis (i.e., in vertical "slices") than it is overall, the two variables are associated. The correlation coefficient is a measure of lillear

I

association, which is a special case of association in which large values of one variable tend to occur with large values of the other, and small values of one tend to occur with small values of the other (positive association), or in which large values of one tend to occur with small values of the other, and vice versa (negative association). associative property property about grouping of numbers; of addition, the c formula (a + b) + c = a + (b + c).

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• assume to accept as true without facts or proof. asymptote a straight line always approaching but never intersecting a curve.

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automatic drawer a computer program that lets you build constructions

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• automorphism an isomorphism from a set onto itself. average a sometimes vague term. It usually denotes the arithmetic

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13 ~

ther lands heads or lands tails ; is the sum of the chance that the : coin lands heads and the chance ~ that the coin lands tails, because I both cannot occur in the same : coin toss. All other mathemati~ cal facts about probability can I be derived from these three axi~ oms. For example, it is true that : the chance that an event does ~ not occur is (100% the chance ; that the event occurs). This is a : consequence of the second and • axioms of probability ~ third axioms. there are three axioms of prob- I • axis ) the line segment Iability: Chances Th are always at .I (ofa cYlinder, east z~ro. e c~ance that : connecting the centers of the something happen! IS 100%. If I bases two events cannot both occur at : . the same time (if they are dis- ~ • base joint or mutually exclusive), the ; I. the side of an isoscoles trichance that either one occurs is : angle whose endpoints are the the sum of the chances that ~ vertices of the base angles each occurs. For example, con- I 2. a side of a polygon or face of sider an experiment that con- : a solid used for reference when sists of tossing a coin once. The ~ drawing an altitude or other first axiom says that the chance I feature. ~at the coin lands heads, for ~ 3. a face of a solid used for refmstance, must be at least zero. : erence when drawing an altiThe second axiom says that the I tude or other feature. chance that the coin either lands ; 4. the congruent parallel polyheads or lands tails or lands on : gons of a prism. If the faces are its edge or doesn't land at all is ~ all rectangular, any parallel pair 100%. The third axiom says; can be considered the bases. that the chance that the coin eimean, but it can also denote the median, the· mode, the geometric mean, and weighted means, among other things.

14 ~~~~~~~~.

Otherwise, the nonrectangular ~ P(AIB) = P(B IA) xP(A)/( pair forms the bases. i P(B IA) xP(A) + P(B lAc) 5. in the expression xY, x is called : xP(Ac». the base and y is the exponent. ~ • bending energy • base angles ~ bending energy is a metaphor 1. (of an isosceles triangle) The i borrowed for use in two angles opposite the two : morphometrics from the mecongruent sides. ~ chanics of thin metal plates. 2. (of an isosceles triangle) The I Imagine a configuration of two angles opposite the two : landmarks that has been printed ~ on an infinite, infinitely thin, congruent sides. 3. (of a trapezoid) A pair of I flat metal plate, and suppose angles with a base of the trap- ~ that the differences in coordi: nates ~ these same ¥dmarks ezoid as a common side. • b Ii I in ano er picture are taken as ase ne : . A: I f tho for a system of two-point shape I vernc 'tf'~ a~ements o. IS r. I dm ks· : plate perpendicular to Itself, dina coordinates lor an ar m a I e · . th lin one artesIan coor te at a I th b lin pane, ease elS e e : . Th be din f connecting the pair of land- I nme. e n g energy 0 . d 1::._ d : one of these out-of-plane madcs that are asslgne to llAe I " h h )). th (·deal '. (0 0) d th . s ape c anges IS e); 1ocanons , an (1 ,0)·me· ized) th . I I I energy at would be reconstructIon. n genera,. . ed be d th tal I baselines work better if they are i qwr to n e me pate closely aligned with the long : s.o that the landmarks .were . f I lifted or lowered appropnately. aXIS 0 the mean landmark . Whil· h· bending m p YSI~ shape and pass near the centroid i. eeal eneder~ f th t h . IS a r quannty, measur m o a mean s ape ; appropriate units (g cm2 sec2), ~ there i~ an alternate formula • Bayes' rule Bayes' rule expresses the con- : that remains meaningful in ditional probability of the event ~ morphometries: bending enA given the event B in terms of i ergy is proportional to the inthe conditional probability of : tegral of the summed squared the event B given the event A: ~ second derivatives of the "ver-

II hentlinamewmlltriJ&lhettl

.~~~~~~~~~15

tical" displacement the extent to ~ energy of a general transformawhich it varies from a uniform; tion is the sum xtLk-lX +y'Lk:ly tilt. The bending energy of a ~ of ~e bending energy of Its shape change is the sum of the : honzontal:-co~~nent, modbending energies that apply to I eled as ~ vemcal pl~te, pl~ any two perpendicular coordi- ~ the bending energy of Its v~~­ nates in which the metaphor is : cal y-component, modeled Slffilevaluated. The bending energy I !arly as a "vertical" plate. of an affme transformation is ~ • Bernouilli process zero since it corresponds to a : this is the simplest probability tilting of the plate with?ut any ~ model a single trial between bending. The value obtamed for I two possible outcomes such as the bending energy correspond- : a coin toss. The distribution . ing to a given ~splacement is ~ depends upon a . single inversely pr?pomonal to scale. ~ parameter,'p', representmg the Sl~ch quantities should not probability attributed to one interpreted as measures o~ dis- : defmed outcome out of the two similarity (e.g:, taxonOIll1C or ~ possible outcomes. evolutionary distance) between . ~ • beta two forms. . ' 2 be I also known as type- error, ta • bending energy ~atrix : is the complement to power : the formula for bending ~nergy ~ beta = (I-power). This is the the formula whose ~alue IS pro- ; probability that a statistical t~t portional to that mtegral of : will generate a false-negative those summed squared. second ~ error: failing to assert a defmed derivatives is. a quadratic form ; pattern of deviation from a null (usually wntten Lk·l) deter- : pattern in circumstances where mined by the coordinates of the ~ the defmed pattern exists. Conlandmarks of the reference I ventional methodology for staform. That is, if h is a vector : tistical testing is to set in addescribing the heights of a plate ~ vance a nominal alpha criterion above a set of landmarks, then I level the corresponding level for bending energy is htLk-lh. In : BETA will depend upon the morphometrics, the bending ~ nominal alpha criterion level

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bias I biMr,ynumber

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and upon further considerations J swer that differs from the truth. including the strength of the J The bias is the average (expattern in the data and the pected) difference between the sample size. Interest is gener- J measurement and the truth. For ally in the relative power of dif- J example, if you get on the scale ferent tests rather than in an with clothes on, that biases the absolute value. It is question- : measurement to be larger than able whether the concept of J your true weight (this would be BETA error is properly appli- : a positive bias). The design of cable without considering the an experiment or of a survey can concept of sampling from a J also lead to bias. Bias can be population, which is separate deliberate, but it is not necesfrom the concerns of this Glos- sarily so. sary. Applicability of this rea- I • biconditional soning is also closely bound up I Q conditional and its converse with the choice of test statistic. ~ where the converse is also true; r

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• bimodal having two modes. binary number a number written to base 2.

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• bias a measurement procedure or estimator is said to be biased if, on the average, it gives an an-

bilateral symmetry reflectional symmetry with only one line of symmetry.

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r. . . - -"~~;;:c!W~");'~~5~ =.-1 ~ r.1'IWJ'>(.~-s!:i'"

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uses the words if and only if. • bijection a one-to-one onto function.

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• binary operation ~ a binary operation is an opera- I tion that involves two operands. ; For example, addition and sub-. traction are binary operations. ; •. . : bmolD1~.

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an expression that IS the sum of two terms. • binomial coefficient the coefficients of x in the expansion of (x+ 1)n.

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• binomial distribution a random variable has a binomial distrib,:!:tion (with parameters n -and p) if it is the number of "successes" in a fixed number n of independent random trials, all of which have the same probability p of resulting in "success." Under these assumptions, the probability of k successes (and n _k failures) is nC pk(l-p)n-k, where nC is the k k number of combinations of n objects taken k at a time: nC = k n!j(k!(n-k)!). The expected value ~f a r~do~ v~iabl~ wi~ the Bmomial dIstnbutIon IS nxp, and the.stanruu:d error of a random vanable With the Binomial distribution is (nxpx(l p» V2. This page shows the

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probability histogram of the binomial distribution. • binomial test .. . . . this IS a stansnc~ test refemng to a repeated bmary process such as would be expected to generate outcomes with a binomial distribution. A value for the parameter 'p' is hypothesised (null hypothesis) and the difference of the actual value from this is assessed as a value of alpha.

~

• biplot a single diagram that represents two separate scatterplots on the same pair of axes. One scatter is of some pair of columns of ~ the matrix U of the singular I value decomposition of a ma: trix S, and the other scatter is ~ of the matching pair of columns I of V. When S is a centered data ~ ~atrix, the effect is to plot prin: Clpal component loadings and ~ scores on the same diagram. ~ • biquadratic equation ; a polynomial equation of the : 4th degree. : ~ ; :

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•

~. bl~e~t

. ; to diVide mto two congruent I parts.

18

bisecWrpflJ~t I bootstmp

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• bisector pf a segment ~ human subjects, it is usually any plane, point or two-dimen- ; necessary to administer a plasional figure containing the the cebo to the control group. midpoint of the segment and no I • bootstrap other points on that segment I this is a form of randomisation • bit I test which is one of the alternaa binary digit. tives to exhaustive re• bivariate randomisation. Th~ bootstrap having or having to do with two I scheme involves generating variables. For example, bivari- I subsets of the data on the basis ate data are data where we have of random sampling with retwo measurements of each "in- I placements as the data are dividual." These measurements I sampled. Such resampling promight be the heights and vides that each datum is equally in the weights of a group of people I represented ". di ·dual"· ) th I randomisation scheme; how(~ghID VI IS a person, e: ever, the bootstrap procedure hel ts 0 f fathers and sons (an I h Co hi h di· .h ".IIId··d . asf leatures IVI ua1"·IS a f at h er-son .. h w c d songws f .) th d I It rom t e proce ure 0 a pair, e pressure an tem-.. M earl0 test. The distm. onte.. perature of a fIXed volume of I ". di .dual"· th 1· gwshing features of the boot( gas an ID VId IS ~ vo - ; strap procedure are concerned lin th . ume 0 f gas un er a certam set. .th of experimental conditions), ; WI o~er-samp thg erebels nO · · constramt upon e num r 0 f etc. Scatterp1ots, the corre1anon·. tha d be coefficient, and regression ~ nmes d ~ a atum ~ay inglrepmake sense for bivariate data ; resente lingID genberaong a ~ e .. d . resamp su set; the SIZe 0 f but not uruvariate ata. ; the resampling subsets may be • blind experiment in a blind experiment, the subjects do not know whether they are in the treatment group or the control group. In order to have a blind experiment with

fixed arbitrarily independently : of the parameter values of the ~ experimental design and may ; even exceed the total number of : data. The positive motive for ~ bootstrap resampling is the gen-

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II bootstrapestimateofstandarderror I: eral relative ease of devising an appropriate resampling algorithm when the experimental design is novel or complex. A negative aspect of the bootstrap is that the form of the resampling distribution with prolonged resampling converges to a form which depends not only upon the data and the test statistic, but also upon the bootstrap resampling subset size thus the resampling distribution should not be expected to converge to the gold standard form of the exact test as is the for Monte-Carlo case resampling. An effective necessity for the bootstrap procedure is a source of random codes or an effective pseudo-random generator.

model

19

~ the sample to estimate the SE ; of sampling from the popula: tion. For sampling from a box ~ of numbers, the SD of the ; sample is the bootstrap estimate : of the SD of the box from ~ which the sample is drawn. For I sample percentages, this takes ~ a particularly simple form: the : SE of the sample percentage of I n draws from a box, with re~ placement, is SD(box)/nY2, : where for a box that contains ~ only zeros and ones, SD(box) ; - ( (fraction of ones in : box) x (fraction of zeros in box) ~ )Y2. The bootstrap estimate of ; the SE of the sample percent: age consists of estimating ~ SD(box) by ((fraction of ones I in sample) x (fraction of zeros : in sample»'h. When the ~ sample size is large, this apI proximation is likely to be ~ good. : - box ~ a surface made up of rect~ angles; a rectangular parallel; epiped

- bootstrap estimate of standard error the name for this idea comes from the idiom "to pull oneself up by one's bootstraps," which connotes getting out of a hole without anything to stand on. The idea of the bootstrap is to ~ • box model assume, for the purposes of es: an analogy between an experitimating uncenainties, that the ~ ment and drawing numbered sample is the population, then ; tickets "at random" from a box use the SE for sampling from : with replacement. For example, I

Msthemsties=========~ II

20 =~~~~~~=*

, to evalu- I, suppose we are trymg M ate a cold remedy by giving it ; R or a placebo to a group of n in- : dividuals, randomly choosing ~ half the individuals to receive I Q p the remedy and half to receive : the placebo. Consider the me- ~ dian time to recovery for all the I individuals (we assume every- ~ N one recovers from the cold : eventually; to simplify things, ~ • box plot we also ~sume that no one ,re- ; a representation of data above covered m exactly the median: a numbered scale where the time, and that n is' even). By ~ "box" encloses all data between defmition, half the individuals ;~ the median of the lower half got better in less than, the me- : (quartile 1) and the median of dian time, and half 10 more ~ the upper half (quartile 3), with than the median time. The in- I a vertical line inside the box to dividuals who received the: indicate the median of the data; treatment are a random sample ~ a dot represents each of the high of size n/2 from the set of n I and low values of the data, and subjects, half of whom got bet- ~ a horizontal line called a whister in less than median time, and : ker connects each dot to the half in longer than median ~ box. time, If the remedy is ineffec-; b h d b d . • ranc tIve, the number 0 f sub'Jects : , -an - oun . , ' d the remedy and I exploratIon w h0 recelve " , 'of a randOlrusanon 1 th : distrlbunon m such a way as to ' ' , w ho recovered mess an me- I 'lik th f nl2 . anOclpate the effect' of the next ' dian orne lS e e sum 0 , "I th ' replacement firom a I, rannOffilSatIOn to ale draws Wlth d 're atIve . This box with two tickets in it: one ; present r~ OffilSanon. ,with a "1" on it and one with a : lows selecove search of ~aro~"0"" I lar zones of a randoffilsatIon on It. . the context 0 f a .. distrl'b' uoon; m ~ randomisation test such selec-

II =======MsthtmUJms

II hreaItdownpoint I canonicalcorreUJ";"nalysis

21

tive search may be concerned ~ relations. Each score (linear with the tail of the i combination) from either list is randomisation distribution : correlated with no other com~ bination from its list and with • breakdown point . the breakdown point of an esti- ~ ~nly one score from the other mator is the smallest fraction of i st. observations one must corrupt to make the estimator take any '------------value one wants. • byte the amount of memory needed to represent one character on a computer, typically 8 bits. • calculator notation ~ • canonical correlation the symbols used by a calcula- i analysis tor for scientific notation. : a multivariate method for as; sessing the associations be• caliban puzzle a logic puzzle in which one is I tween two sets of variables asked to infer one or more facts : within a data set. The analysis ~ focuses on pairs of linear comfrom a set of given facts. I binations of variables (one for • canonical ~ each set) ordered by the maga canonical description of any : nitude of their correlations with statistical situation is a descrip~ each other. The fIrst such pair tion in terms of extracted vec; is determined so as to have the tors that have especially simple : maximal correlation of any ordered relationships. For in~ such linear combinations. Substance, a canonical correlations ; sequent pairs have maximal coranalysis describes the relation : relation subject to the constraint between two lists of variables ~ of being orthogonal to those in terms of two lists of linear I previously determined. combinations that show a remarkable pattern of zero cor- I

Msthmulties======= II

22

canonical 11am; analysis I causation, causal relation

II

• canonical variates analysis ~. capacity a method of multivariate analy- ; the amount of liquid that can fill sis in which the variation among : an object. • I groups is expressed relative to I the pooled within-group cova- I • cartesian p ane . . . C . I . a rectangular coordinate svstem . . . J' nance matrIX. anoruca varIal . finds lin I COnSIStIng of a honzontal numates an. YSIS ear tr~s- ber line (x-axis) and a vertical fiormatlons of the data which I b lin ( .). maximise the among group ~um ethr e . ~-axIS , mtersect. . , I mg at e ongm (zero on each varIatIOn relatIve to the pooled . be lin) within-group variation. The ca- ; num r e. nonica! variates then may be I • categorical variable displayed as an ordination to : a variable whose value ranges show the group centroids and ~ over categories, such as {red, scatter within groups. This may ; green, blue}, {male, female}, be thought of as a "data reduc- : {Delhi, Calcutta, Mumabai}, tion" method in the sense that ~ {short, tall}, {Asian, Mricanone wants to describe among ; American, Caucasian, Hispanic, group differences in few dimen- : Native American, Polynesian}, sions. The canonical variates are ~ {straight, curly}, etc. Some cat'uncorrelated, however the vec- ; egorical variables are ordinal. tors of coefficients are not or- : The distinction between catthogonal as in Principal Com- ~ egorical variables and qualitaponent Analysis. The method is I tive variables is a bit blurry. closely related to multivariate .I • catenary analysis of variance : a curve whose equation is y = (MANOVA), multiple discrimi- ~ (a/2)(e +eA chain susnant analysis, and canonical cor- I pended from two points forms relation analysis. A critical as- : this curve sumption is that the within- ~ •. group variance-covariance;. causation, causal relation structure is similar otherwise : two variables are causally rethe pooling of th~ data over ~ lated if changes in the value of groups is not very sensible. i one cause the other to change. For example, if one heats a rigid X/ 3

X/ 3 ) .

II =======.M.thelulies

II

23

ceilingfunction I centroid

*================= container filled with a gas, that I • center of gravity causes the pressure of the gas I the mean of the coordinates of in the container to increase. points in a figure, whether one, Two variables can be associated I two, or three-dimensional without having any causal rela- I • central angle tion, and even if two variables I (of a chord or arc) An angle have a causal relation, their cor- : whose vertex is the center of a relation can be small or zero. I circle and whose sides pass I through the endpoints of a • ceiling function the ceiling function of x is the chord or arc. smallest integer greater than or • central angle of a circle equal to x. an angle whose vertex is the • center (of a circle or sphere) The point from which all points on the figure are the same distance. • center of a circle the point that all points in the circle are equidistant from .

I

I

I

I I

\

... \

''''

• center of a rotation the point where the two intersecting lines of a rotation meet

central limit theorem the central limit theorem states that the probability histograms of the sample mean and sample sum of n draws with replacement from a box of labeled tickets converge to a normal curve as the sample size n grows, in the following sense: As n grows, the area of the probability histogram for any range of values approaches the area tmder the normal curve for the same range of values, converted to standard units.

I •

I

--..

center of the circle

I

• centroid the point of concurrency of a triangle's three medians.

MlJthemR.tiu=======

II

",,24=========== ..centroid size I Chebychev'si1UlJUR1ity

II

_ centroid size - cevian centroid size is the square root I a line segment extending from of the sum of squared distances a vertex of a triangle to the opof a set of landmarks from their I posite side. centroid, or, equivalently, the I square r'?Ot of the sum of the ~ variances of the landmarks about that centroid in xand y- I directions. Centroid Size is used in geometric morphometries _ chance variation, chance because it is approximately I error uncorrelated with every shape a random variable can be devariable when landmarks are composed into a sum of its exdistributed around lTiean posi- ~ pected value and chance variations by independent noise of ; tion around its expected value. the same small variance at ev- The expected value of the ery landmark and in every di- I chance variation is zero; the rection. Centroid Size is the size I standard error of the chance measure used to scale a con- variation is the same as the stanfiguration of landmarks so they dard error of the random varican be plotted as a point in I able-the size of a "typical" difKendall"s shape space. The de- ference between the random nominator of the formula for variable and its expected value the Procrustes distance between I - Chebychev's inequality two sets of landmark configuI for lists: For every number rations is the product of their k>O, the fraction of elements Centroid Sizes. in a list that are k SD's or furI ther from the arithmetic mean - certain event an event is certain if ItS prob- of the list is at most Ijk2. For ability is 100%. Eve~ if an event random variables: For every is certain, it might not occur. I number k>O, the probability However, by the complement I that a random variable X is k rule, the chance that it does not SEs or further from its expected occur is 0%. I value is at most l/k2.

II chi-square curve I chi-squared d~tion • chi-square curve the chi-square curve is a family of curves that depend on a parameter called degrees of freedom (d.f.). The chisquare curve is an approximation to the probability histogram of the chi-square statistic for multinomial model if the expected number of outcomes in each category is large. The chi-square curve is positive, and its total area is 100%, so we can think of it as the probability histogram of a random variable. The balance point of the curve is d.f., so the expected value of the corresponding random variable would equal d.f.. The standard error of the corresponding random variable would be (2xd.f.)V2. As d.f. grows, the shape of the chi-square curve approaches the shape of the normal curve. • chi-square statistic the chi-square statistic is used to measure the agreement between categorical data and a multinomial model that prediets the relative frequency of outcomes in each possible category. Suppose there are n in-

25

~ dependent trials, each of ; which can result in one of k : possible outcomes. Suppose ~ that in each trial, the probabil; itythatoutcome i occurs is pi, : for i = 1, 2, ... , k, and that ~ these probabilities are the I same in every trial. The ex: pected number of times out~ come 1 occurs in the n trials I is n.xp1; more generally, the ~ expected number of times out: come i occurs is ~ • chi-squared distribution ~ where expected frequencies are ; sufficiently high, hypothesised : distributions of counts may be ~ approximated by a normal dis; tribution rather than an exact : binomial distribution. The cor~ responding distribution of the ; chi-squared statistic can be de: rived algebraically this is the ~ chi-squared distribution which I has been computed and pub~ lis~ed historically as extensive : prmted tables. Use of the tables ~ is notably simple, as the chi; squared distribution depends : upon only one parameter, the ~ degrees of freedom, defined as ; one less than the number of cat: egories. I

Mslthema.ties======= II

chi-SlJ.tl4red statistic I circumfoYence II

26

~============*~===========

• chi-squared statistic I this is a long-established test statistic for measuring the extent to which a set of categori- : cal outcomes depart from a. I hypothesised set of probabili- ~ ties. It is calculated as a sum of : terms over the available catego- ~ ries, where each term is of the i form: ((0-E)2)fE; '0' represents the observed frequency for I the category and 'E' represents the corresponding expec~ed ~e- ~ quency based upon multIplymg : the sample size by the I h thesised probability for the : being considered (therefore 'E' will generally not I be an integer value). In situa- : tions where the numbe~ of cat- ~ egories is 2 an alternatIve pro- ~ cedure is to use an exact i biniomial test.

c~gory

~

Q Chord

c

D

"A

E

F

A line segment that connects 2 points on a circle.

CD

and

EF

are chords of circle A.

• circle the set of points on a plane at a certain distance (radius) from a certain point (ceI1ter); a polygon with infinite sides •

• cl1'cular cone . . a cone whose base IS a crrcle. • circularity when on a search, circling back to a previous place visited (definition, web site, etc.), ustially unhelpful or redundant

i • circumcenter • chord : the circumcenter of a triangle 1. a line segment whose end- ~ is the center of the circum; scribed circle. points lie on a circle.. 2. the line joining two pomts on i • circumcircle a curve is called a chord. : "the circle circumscribed about I • chord of a circle : a figure. a segment whose end~oints ~ • circumference are on a circle i the distance around a circle, ~ given by the formula C = 2m;

II circumscribed I cluster analysis

27

*=============== where r is the radius of the circle. • circumscribed passing through each vertex of a figure, usually referring to circles circumscri bed around polygons or spheres circum _ scribed around polyhedrons. The figure inside is inscribed in the circumscribed figure. • cissoid a curve with equation y2(a_ X)=X3.

P2

~

: ~

• class interval in plotting a histogram, one starts by dividing the range of v~ue~ into a set of non-ov~rlap­ pmg ~tervals, called class mtervals, m such a way that every ~atum is contained in some class mterval. • class of functions family of functions such as linear, quadratic, power (polynomial), exponential, or logarith-

;

mic.

; :

~

~ .

~

I

I

~

~ • classes of numbers : family of numbers or number ~ systems such as natural, inte; ger, rational, irrational, real, or : complex. I

: • classify ~ to categorise something accord; ing to some chosen character: istics. I

: • clockwise ~ in orientation, the direction in ; which the points are named : when, if traveling along the line, ~ the interior of the polygon is on ~ the right. • class boundary a point that is the left endpoint I • cluster analysis of one class interval, and the ~ a method of analysis that repright endpoint of another class : resents multivariate variation in I data as a series of sets. In biolinterval. ~ ogy, the sets are often con-

II

corJJicient I combinatitms

28

structed in a hierarchical manner and shown in the form of a tree-like diagram called a dendrogram. • coefficient a coefficient, in general, is a number multiplying a function. In multivariate data analysis, usually the "function" is a variable measured over the cases of the analysis, and the coefficients multiply these variable values before we add them up to form a score. A coefficient is not the same as a loading. • coincide lying exactly on top of each other. Line segments that coincide are identical; they have all the same points.

~

; :

~ ~ ~

I

: I I

I

I

I I

: I : I

I

~

;

II

• coincidental lines lines that are identical (one and the same) • collinear lying on the same line. • combinations the number of combinations of n things taken k at a time is the number of ways of picking a subset ofk of the n things, without replacement, and without regard to the order in which the elements of the subset are pickc:d. The number of such combinations is nCk = n!j(k!(nk)!), where k! (pronounced "k factorial") is kx(k-l)x(k-2)x ... x 1. The numbers nCk are also called the Binomial coefficients. From a set that has n elements one can form a total of 2n subsets of all sizes. For example, from the set {a, b, c}, which has 3 elements, one can form the 23 = 8 subsets U, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}. Because the number of subsets with k elements one can form from a set with n elements is nck, and the total number of subsets of a set is the sum of the numbers of possible subsets of each size, it follows that nco +nC1 +nC2 + ... +nCn = 2n.

II ~m#n~~I~~n~~~~~~~~~~~2==9 The calculator has a button ~ - complementary angles (ncm ) that lets you compute the ; two angles whose measures number of combinations of m : have the sum 90°. things chosen from a set of n I things. To use the button, first I ArSfc. type the value of n, then push the nCm button, then type the value of m, then press the "=" I 8 () button. mLABC + mLCBD =90"

~

LABC and LCBD are complementary angles.

- commutative properties I properties about order of addi- : - complex numbers tion' a + b = b + a; of multi- ~ complex numbers are an algeplication, a x b = b x a" ; braic way of coding points in : the ordinary Euclidean plane - compass ~ so that translation (shift of a drawing tool used to draw ; position) corresponds to the circles at different radii :I addition of complex numbers . - compatible numbers : and both rescaling (enlargenumbers that can be easily ma- I ment or shrinking) and rotanipulated and operated on men- ~ tion correspond to multiplicatally. : tion of complex numbers. In _ complement rule ~ this system of notation, inthe probability of the comple- ; vented by Gauss, the x-axis is ment of an event is 100% mi- : identified with the "real numnus the probability of the event: ~ bers" (ordinary decimals P(Ac) = 100% peA). ; numbers) and the y-axis is : identified with "imaginary -thcompIement I f b f .I num b ers" (t h e square roots 0 f e.comp e~ent 0 a su .set 0 ; negative numbers). When you a gIven set IS the collecnon of" ul" I " thi" b all elements of the set that are ; ~ tip pomts on " s axiS hY not elements of the subset. : t emse ves accord mg to t e I rules, you get negative points : on the "real" axis just defined. ~ Many operations on data in

r

MJJ.th_tics========== II

~30~~~~~~~~~~~~~~mmW~l~i~p70b~i~ two dimensions can be proved valid more directly if they are written out as operations on complex numbers. • compose numbers put a set of numbers together to form a new number using addition or multiplication. • composite transformation the composite of a first transformation S and a second transformation T is the transformation mapping a point P onto T(S(P».

/I

concentric circles circles that share the same center, but have different radii

I •

I

I

I

I

• concrete materials objects to be manipulated (e.g., • composition (of transformations) The trans- I pattern blocks, snap cubes, formation that results when geoboards, tangrams, color one transformation is applied ~ tiles, base ten blocks). after another transformation. • concurrent I intersecting at a single point • compound eventS the point of two or more events in a prob- ( called I concurrency). ability situation such as flipping a coin and spinning a spinner. I • conditional I a statement that tells if one • concave thing happens, another will folcurved from the inside. low. • concave polygon a polygon having at least one • conditional probability diagonal lying outside the poly- I suppose we are interested in the probability that some event A gon; not convex. occurs, and we learn that the I event B occurred. How should I

II =======MRthem4ries

31

*================= we update the probability of A to reflect this new knowledge? This is what the conditional probability does: it says how the additional knowledge that B occurred should affect the probability that A occurred quantitatively. For example, suppose that A and B are mutually exclusive. Then if B occurred, A did not, so the conditional probability that A occurred given that B occurred is zero. At the other extreme, suppose that B is a subset of A, so that A must occur whenever B does. Then if we learn that B occurred, A must have occurred too, so the conditional probability that A occurred given that B occurred is 100%. For in-between cases, where A and B intersect, but B is not a subset of A, the conditional probability of A given B is a number between zero and 100%. Basically, one "restricts" the outcome space S to consider only the part of S that is in B, because we know that B occurred. For A to have happened given that B happened requires that AB happened, so we are interested in the event AB. To have a legitimate probability requires that P(S) = 100%, so

~

if we are restricting the out; come space to B, we need to : divide by the probability of B ~ to make the probability of this ; new S be 100%. On this scale, : the probability that AB hap~ pened is P(AB)fP(B). This is , the deftnition of the conditional : probability of A given B, pro~ vided P(B) is not zero (division ~ by zero is undefined). Note ; that the special cases AB = {} : (A and B are mutually exclu~ sive) and AB = B (B is a subset ; of A) agree with our intuition : as described at the top of this ~ paragraph. Conditional prob; abilities satisfy the axioms of : probability, just as ordinary ~ probabilities do. ~ - conditional proof ; a proof of a conditional state: ment.

,

: ~ , :

~ ~

; : ~

; :

- cone a solid whose surface consists of a circle and its interior, and all points on line segments that connect points on the circle to a single point (the cone's vertex) that is not coplanar with the circle. The circle and its interior form the base of the cone. The radius of a cone is the radius of the base. The altitude of a cone

Msthemtllti&s==========

II

~32~~~~~~~~~~~Mlmn~m~M is the line segment from the vertex to the plane of the base an~ perpendic~ar to it. The ?elgh~ of a cone 18 t.he length of Its aln~de. If the line segment c~nnecnng the vertex of a cone with th~ center of its base is perpendi~ar ~o the base, then the cone 18 a nght cone; otherwise it is oblique. • confidence interval for a given re-randomisation distribution, a family of related distributions may be defined according to a range of hypothe~cal values of the pattern whIch the test statistic measures. For instance, for the pitman permutation test to test for a scale shift between .two groups, a related distribution may be formed by shifting all the observations in one group by a common amount where this common shift is r~garded as a continuous variable, With fmite numbers of data the number of related distributions will be fmite, and typically considerably smaller than the number of points of the randomisation distribution. The likelihood of the outcome value may be calculated for each distribution in

II

~ the family, and these likelihoods ; may be then used to defme a ~ contiguous set of values which : occupy a certain proportion of I the total unit weight of the like: lihoods integrated over all val~ ues of the test statistic, the con~ fidence interval is defmed by ' the minimum and maximum ~ values of the range of values so ~ defined. The proportion of the ; total weight within the range of : values is regarded as an alpha ~ probabili~ ~at, the v~ue of the I test stanstic hes WIthin this ~ range, Generall~ the defmition : of a confidence mterval cannot ~ be unique wi~out imposing ' further constramts. Approaches ~ to providing suitable con~ ~traints, s~ch that a confidence ; mtet;al will be unique, include : d~g the confidence interval ~ : to mclude the whole of one tail ; of the ~istribution; or to be : centred m some sense upon the ~ outcome value; or to be centred ; bet,ween TAILS of equal ~ weIght" In, the ,cas,e of re: randomIsanon distributions, I these are discrete distributions : so there will generally be no ~ range of,values with weight corI responding exactly to an arbi: trary nominal alpha criterion

II ===:========Mstht:'l'/Ul.tU:s

II con.fideme level I congruentfiDU1'eS.

33

level, and the problem of non- ~ the treatment (if any). For exuniqueness is therefore not gen- ; ample, prominent statisticians eratly solvable. : questioned whether differences ~ between individuals that led ; some to smoke and others not : to (rather than the act of smok~ ing itself) were responsible for I the observed difference in the ~ frequencies with which smok: ers and non-smokers contract I various illnesses. If that were ~ the case, those factors would be - confidence level : confounded with the effect of the confidence level of a confi- ~ smoking. Confounding is quite dence interval is the chance that ; likely to affect observational the interval that will ~esult on~e : studies and experiments that data are collected wIll contam I are not randomised Confoundthe corresponding parame~er. If ~ ing tends to be d~creased by one computes confidence mter- : randomisation. vals again and again from inde- ~ pendent tiata the long-term . - congruent limit of the fr;ction of intervals ~ equilateral, equal, exactly the that contain the parameter is I same (size, shape, etc.) the confidence level. ; - congruent angles : two or more angles that have • confounding when the differences between ~ the same measure. • I the treatment and control I: - congruent clrc es groups other than the treat- I two or more circles that with the ment produce differences in re- : same radius. I sponse that are not distinguishable from the effect of the treat- : - congruent figures ~ two figures where one is the ment, those differences beI image of the other under a retween the groups are said to be ~ flection or composite of reflecconfounded with the effect of : tions.

MIIthemtlnes=======

\I

~34~~~~~~~~~~C~ongruen=.tpolygOllS I constantofRnetpmtUm • congruent polygons two or more polygons with the exact same size and shape. • congruent segments two or more segments that have the same measure or length. • conic section the cross section of a right circular cone cut by a plane. An ellipse, parabola, and hyperbola are conic sections. 1\

I

I

I

I I I

~ --

...... =:. :\ ,, .......

....... - ...

,~ - ............

I

=-'= I

;

\

-

:.,-l_tt.. I \

) '/ i

"

.'----- - - -- /

I

1\

l '

f,I .-------\\ (

~

I ..-

\

~ / \\ -

-

I \

'/

• conic solid the set of points between a point (the venex) and a non-coplan;:lI region (the base), including the point and the region. • conjecture a guess, usually made as a result of inductive reasoning. • consecutive angles (of a polygon), two angles that have a side of the polygon as a common side. • consecutive sides (of a polygon), two sides that have a common vertex.

• consecutive vertices (of a polygon or polyhedron), two vertices that are connected by a side or edge. consensus configuration a single set of landmarks intended to represent the central tendency of an observed sample for the production of superimpositions, of a weight matrix, or some other morphometric purpose. Often a consensus configuration is ·computed to optimize some measure of fit to the full sample : in particular, the Procrustes mean shape is computed to minimise the sum of squared Procrustes distances from the the consensus landmarks to those of the sample.

I •

/ \ !

•......:

II

I

I I I

• consequent the second or "then" part of a conditional statement. constant of an equation the term that has no variable in an equation; example: "0

I •

I I

1 (v - va) 2 x - Xo = vot + - - - t 2 t 1

1

x - Xo = Vat + -tit - -vat 2 2 1 1 2 2 1 x - Xo = -(tJo + tI)t 2

x - Xa = .,..-vat + -vt I

II =====.===MsJthmuJnes

II ~consts~~n~tffl,~te~oJi~'ch~'Q;~~~e~Iconti~~·n~Uf,~·ty=~ • constant rate of change set .of data or table of values in which ~e amount of the dependent vanable changes by a constant (fixed) value as the value of the independent variable changes by a constant value. • construct create a figure using only a straight edge and compass. • construction a precise way of drawing which allows only 2 tools: the straightedge and the compass • continuity correction in using the normal approximation to the binomial probability histogram, one can get more accurate answers by fmding the area under the normal curve corresponding to half-integers, transformed to standard units. This is clearest if we are seeking the chance of a particular number of successes. For example, suppose we seek to approximate the chance of 10 successes in 25 independent trials, each with probability p = 40% of success. The number of successes in this scenario has a binomial distribution with parameters n = 25 and p = 40%.

35

~ The expected number of suc-

; cesses is np = 10, and the stan~ dard error is (np( I-p» lh = 6lh . = 2.45. If we consider the area ; under the normal curve at the : point 10 successes transformed ~ to standard units,' we get zero: ~ the area under a point is always ; zero. We get a better approxi: mation by considering 10 suc~ cesses to be the range from 9 ; 1/2 to 101/2 successes. The only : possible number of successes ~ between 91/2 and 10 1/2 is 10, ; so this is exactly right for the : binomial distribution. Because ~ the normal curve is continuous ; and a binomial random variable : is discrete, we need to "smear ~ out" the binomial probability ' over an appropriate range. The ; lower endpoint of the range, 9 : 1/2 successes, is (9.5 10)/2.45 '. = -0.20 standard units. The ~ upper endpoint of the range, 10 : 1/2 successes, is (10.5 10)/2.45 ~ = +0.20 standard units. The ; area under the normal curve : between -0.20 and +0.20 is ~ about 15.8%. The true bino; mial probability IS : 25CI0x (0.4)10x (0.6)15 = ~ 16%. In a similar way, if we ' seek the normal approximation : to the probability that a bino-

,

36

cuntinUIIIIS tlistriburion I cuntmaitm I

II

..

mial random variable is • the ~ for any possible ~a!ue of ~e range from i successes to k suc- ; cumulative probab~ty there IS cesses, inclusive, we should find ~ an exact; c~n:espon~ value of the area under the normal curve : the statIStiC m question. from i-I/2 to k+ 1/2 successes, I • continuous variable transformed to standar~ .units. ~ a quantitative variable. is conIf we seek the probability of : tinuous if its set of pOSSible valmore than i successes and fewer ~ ues is uncountable. Examples than k successes, we should find ; inclUde temperature, exact the area under ¢e normal curve : height exact age (including corresponding to the range i + 1/ ~ parts of a second). In practice, 2 to k-I/2 successes, trans- lone can never measure a conformed to standard units. If we : tinuous variable to infinite preseek the probability of more ~ cision so continuous variables than i but no more than k suc- ; are so~etimes approximated by cesses, we should find the area: discrete variables. A random , I . unde.r the normal curve corre- : variable X is also called conttnusponding to the range i + 1/2 to I ous if its set of possible values k+ 1/2 successes, transformed ~ is uncountable, and the chance to standard units. If we se~~ the : that it takes any particular value probability of at least I but ~ is zero (in symbols, ifP(X = x) fewer than k successes, we ; = 0 for every real number x). should fmd the area under the : A random variable is continunormal curve corresponding to ~ ous if and only if its cumulative the range i-I/2 to k-I/2 suc- ; probability distributi?n function cesses, transformed to standard : is a continuous funCtIon (a funcunits. Including or excluding the ~ tion ,with no jumps). • half-integer ranges at the ends I of the interval in this manner is : • contrac~on . . called the continuity correction. ~ the oppo~lte of dilatlo~ a ~. ure resulttng from multlplymg • continuous distribution ~ all dimensions of a given figure a probability distribution of a ; by a number betwec:n zero and continuous statistic, based upon ~ one. an algebraic formula, such that : I

37 II contm,positi1le I cunwnien&e sample ================*================

- contrapositive if p and q are two logical propositions, then the contrapositive of the proposition (p IMPLIES q) is the proposition «NOT q) IMPLIES (NOT p) ). The contrapositive is logically equivalent to the original proposition. _ control for a variable to control for a variable is to try to separate its effect from the treatment effect, so it will not confound with the treatment. There are many methods that try to control for variables. Some are based on matching individuals between treatment and control; others use assumptions about the nature of the effects of the variables to ~ to model the effect mathematically, for example, using regressl0n. - control group the subjects in a controlled experiment who do not receive the treatment. -control there are at least three senses of "control" in statistics: a member of the control group, to whom no treatment is given;

~

a controlled experiment, and to ; control for a possible confound: ing variable. I

: - controlled experiment ~ an experiment that uses the ; method of comparison to evalu: ate the effect of a treatment by ~ comparing treated subjects I with a control group, who do ; not receive the treatment. :I - controlled, randomised • : expenment I a controlled experiment in ~ which the assignment of sub: jects to the treatment group or ~ control group is done at ran; dom, for example, by tossing a : com. I

: - convenience sample ~ a sample drawn because of its ; convenience; not a probability : sample. For example, I might ~ take a sample of opinions in ; Delhi (where I live) by just ask: ing my 10 nearest neighbors. I : That would be a sample of conI venience, and would be unlikely : to be representative of all of ~ Delhi. Samples of convenience ~ are not typically representative, ; and it is not typically possible : to quantify how unrepresentaI

MMIIenIaeics==--====____.11

38

..

~~~~WD~e,!!!!!·c~O'1¥I1i~m!!!!!~!!!!!mee~!!!!!1coo!!!!!~"~rJt~~na~te II.

tive results based on samples of ~ 2. if P and q are two logical i propositions, then the converse convenience will be. of the proposition (p IMPLIES -~~-=.:=.:~ q) is the proposition (q IM~ i i PLIES p). v.,...,

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i-conversion factor : relationship between two umts ~ from different systems of meai surement used to convert from ; . one system ~o the other (e.g., : 2.~4 centimeters corresponds to ~ 1 mch). i_convex polygon : a polygon having no diagonal ~ lying outside the polygon.

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. ':" ~ : : : - ::

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_ converge convergence a sequence ~f numbers xl, x2, x3 . . . converges if there is a number x such that for any number E>O, there is a number k (which can depend on E) such that Ixj xl < E whenever j > k. I . . . . - convex If such a number x eXISts, It 18 f set hich all egcalled the limit of the sequence ~ a set 0 pomts om w fSth 1 x2 3 i ments connectmg pomts 0 e x, , x set lie entirely in the set; There - convergence in probability ~ are three things one can do to a sequence of random variables i see if a figure is convex look for Xl, X2, X3 converges in : "dents", extend the segments probability if there is a random ~ (they shouldn't enter the figvariable X such that for any jure), and connect any two number E>O, the sequence of : points within the figure with a numbers P( IXI XI < e), P( 1X2 ~ segment (if any part of the segXI < e), P( IX3 XI < e), . I ment lies outside the figure, it's converges to 100%. : concave). 0

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- converse 1. (of a conditional statement), the statement formed by exchanging the "if" and "then" parts 0 f an "if.th" - en statement

: _ coordinate ~ a number that identifies (or I helps to identify) a point on a : number line (or on a plane, or . space) :I m I

39 ~ vector may be thought of as - coordinate geometry the study of geometrically rep- ; coordinates in a geometric resenting ordered pairs of num- : sense. I bers : - coordinatised. line ~ a line on which every point is ; identified with exactly 1 num: her and vice versa; a one-di~ mensional graph. The distance I between 2 parts on a : coordinatised line is the abso~ lute value of the difference of I their coordinates. - coordinate plane a plane in which every point is I _ coplanar identified with exactly 1 num- ~ lying in the same plane. ber and vice versa; a two-di- ; - coprime mensional graph : integers m and n are coprime if _ coordinate proof ~ gcd(m,n) =1. a proof using coordinate geom- ~ _ corollary to a theorem ; a theorem that is easily proved etry. _ coordinate system : from the first set of ordered pairs used to 10- I p cate an object or point on the I two-dimensional plane.

- coordinates a set of parameters that locate a point in some geometrical I space. Cartesian coordinates, for instance, locate a point on a plane or in physical space by I projection onto perpendicular . lines through one single point, :I - correI abon the origin. The elements of any I relation between two or more ; variables. Frequently the word

M/lthemR,riu==--======

II

correlation moment correlation which is I a measure of linear association the covariance divided by the between two (ordered) lists. product of the standard ~ Two variables can be strongly deviations,rxy=Sxy / Sx.Sy- This ; correlated without having any correlation coefficient IS + 1 or causal relationship, and two -1 when all values fall on a variables can have a causal restraight line, not parallel to ei- I lations hip and yet be ther axis. However, there are uncorrelated. also Kendall, Spearman, • corresponding angles tetrachoric, etc. correlations I if two parallel lines are cut by a which measure other aspects of ~ transversal, corresponding die relation between two vari- angles are translations of each ables. , other along the transversal . is used for Pearson's producr-

I •

• correlation coefficient • cosine the correlation coefficif!nt r is I (of an acute angle) The ratio of a measure of how nearly a the length of the adjacent side scatterplot falls on a straight I to the length of the hypotenuse line. The correlation coeffi- I in any right triangle containing cient is always between -1 and I the angle. + 1. To compute the correla- . tion coefficient of a list of ; • coterminal angles pairs of measurements (X,Y), : two angles that have the same first transform X and Y indi- .~" terminal side vidually into standard units. "I • countable set Multiply correspopding ele- ; a set is countable if its elements ments of the transformed : can be,put in one~to-one correpairs to get a single list of ~ spondence with a subset of the numbers. The correlation co- ; integers. For eXaniple the sets efficient is the mean of. that {O~ 7 -3} {red, gre~n, blue} list of , , "'-2 :1" 0" 1 2 ... }', . products. This page .~. { ... contams a .too~ tha~ lets y?U I {straiWtt, curly}, and the set of generate" bIVarIate data WIth : all fractions are countable. If a . YQU .I set is not countable ' · coeffiICIent any correIanon . , it is unwant.

II counterclockwise I criticalveUue

41

~~~======~*==~===========

countable. The set of all real ~ given transformation has only numbers is uncountable. lone direction of covariants, but : a full plane (four landmarks) or • counterclockwise in orientation, the direction in ~ hyperplane (five or more landwhich points are named when I marks) of invariants.

· travelling on the line, the in-' if terior of the figure is on the left side.

• counterexample a situation in a conditional for which the antecedent is true, but the conditional is false; aka contradiction • counting techniques a variety of methods used to determine the total possible outcomes, typically in a probability situation, including the multiplication principle, trees and lists. • covariant a covariant of a particular shape change is a shape variable whose gradient vector as a function of changes in any complete set of shape coordinates lies precisely along the change in question. For transformations of triangles, the relation between invariants and covariants is a rotation by 90 degrees in the shape-coordinate plane. For more than three landmarks, a

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• cover a confidence interval is said to cover if the interval contains the true value ofl the parameter. Before the data are collected, the chance that the confidence interval will contain the parameter value is the coverage probability, which equals the confidence level after the data are collected and the confidence interval is actually computed.

; • coverage probability : the coverage probability of a ~ procedure for making confi; dence intervals is the chance : that the procedure produces an ~ interval that covers the truth. ~ • critical value ; the critical value in an hypoth: esis test is the value of the test I

Msthernades,=======-II

42

cross-seaional study I Ciunulati; Probability Distribution Pmu:tUm...

statistic beyond which we would reject the null hypothesis. The critical value is set so that the probability that the test statistic is beyond the critical value is at most equal to the significance level if the null hypothesis be true.

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• cross-sectional study I . • a cross-sectlonal study com- . pares different individuals to ~ each other at the same time- ; it looks at a cross-section of a ; population. The differences between those individuals can cQn- ~ found with the effect being ex- ; plored. For example, in trying : to determine the effect of age ~ on sexual promiscuity, a cross- ; sectional study would be likely : to confound the effect of age ~ with the effect of the mores the ; subjects were taught as children:: . di 'duals b theoIder m VI were...1:.a:' pro -. . ed·th ably r~s WI a very ~ler- ~ . cu- ; ~ntthanatt1tuthde towards proIDlS Ity e younger sub~ects .. · uld be . rud ; Thus It Imp• ent . to: 'b wo ...1:.a: attn ute wuerences m proIDlS- I . th' C f. CUlty to e agmg process. ..: 1 .rudinal tud I ong! s y. : • cryptarithm ~ a number puzzle in which an ; indicated arithmetical operation :

has some or all of its digits replaced by letters or symbols and where the restoration of the original digits is required. Each letter represents a unique digit. • cube a solid figure bounded by 6 congruent squares. ub'IC equa0'on • C I 'al ti f d a po oml equa on 0 egree .

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u1 u' 'D-. b bili'ty um a ve .ero a D'IS m'bU u'on F unc0'on . (CDF) the cumulative distribution fun' f d . bl etlon 0 a ran om varia e .IS the ch ance that the rand om . bl . 1 than ual varia e IS ess or eq to fun' f In bols X, as a etlon 0 x, sym , if F is the cdf of the random variable X, then F(x) = P( X < = x). The cumulative distribution function must tend to zero as x

C

II :........... 1~tur,fue approaches minus inftnity, and must tend to unity as x approaches inftnity. It is a positive function, and increases monotonically: ify > x, thenF(y) >= F(x). The cumulative distribution function completely characterises the probability distribution of a random variable. . - curved ~pace. a space wlth coordinates and a distance f~ction such that the area of Circles, volume of spheres, etc. are not pro~ortional to the ap.propnate power of the radIUs, e. g., Kendall's shape space. In curved spaces, the usual iI?tui tions about what "straight to do lines" can be expected . will be fa~lty. For mstaI?ce, corresponding to every trlangular shape i? Kendall's sha~e space, there IS. another t?at I! "as far fronl ~t as 1?osslble, just like there IS a pomt on the surface of the earth as far as possible from where you now sit.

*=========!!!!!!4!!!!!!3 ~ - cyclic polygon . . ; a polygon whose vernces lie on ~ a circle. : _ cyclic quadrilateral . ~ a quadrilateral that can be m; scribed in a circle.

; - cylinder . : a solid whose surface COnsISts of ~ all points on two circl~ in ~o I parallel planes, along WIth pomts : in their interiors (the bases of the ~ cylinder), and all points o~ line I segments joining the two arcles. : The axis of the cylinder is the line ~ segment that joins the centers of I the bases. The radius of the cyl: inder is the radius of a base. An ~ altitude of a cylinder is a line segI ment between. and perpendicu:I lar to the planes of the bases. The : height of a cylinder is ~e ~ength ~ of an altitude. If the axIS IS per; pendicular to the basc:s, then the : cylinder is a right cylinder; oth~ erwise, it's oblique. I

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: - cylindric s?lid ~ the set o~ pomts ~~n a re; gion ~d I~ trans~nondm.s~ce, : including llle region an Its .lffi-

- cycle ~ age (of a periodic curve) One se~tion of a curve that, when laid ; - cylindric surface out repeatedly end-to-end, :I the union of the bases and the : lateral surface forms the entire curve.

~lhemsti&s======= II

44 =================*

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~ data . inform~tlon

; • decompose numbers used as a basis for ; break up numbers into addends reasonmg. . or factors.

• decagon ~ • deductive reasoning ~ reasoning accepted as logical a ten-sided polygon • decimal number ; from agreedupon assumptions a number written to the base 10. : and proven facts. I

• decision rule : a rule for comparing the out- ~ come value of alpha with a ; nominal alpha criterion level: (such as 0.05). An outcome ~ value smaller (more extreme) ; than the nominal alpha criterion : level leads to a decision of sta- ~ tistical significance of the find- ; ing that the test statistic has a : value other than its (null-) ~ hypothesised value. ~ • deck of cards ; a standard deck of playing cards : contains 52 cards, 13 each of ~ four suits: spades, hearts, dia- ~ monds, and clubs. The thirteen ; cards of each suit are {ace 2 3 : 4? 5,6,7,8,9, 10, jack, q~n: ~ king}. The face cards are {jack, I queen, king}. It is typically as- : sumed that if a deck of cards is ~ shuflled well, it is equally likely ; to be in each possible ordering. ;

- deductive system an arrangement of premises and theorems, in which each theorem can be proved by deducti~e reasoning using only the premIses ~nd ~revious theorems, and m which each definition uses only terms that have been defmed previously in the system. • deficient coordinate in addition to landmark locations, a digitiser can be used to supply information of other sorts. For example, a point can used .to encode part of the inf~rma~o~ about a curving arc by Identifying the spot at which the ar~ lies farthest from some other unage structure (perhaps another such curving arc). The null model of independent Gauss~ noise does not apply to posItion along the tangent

?e

II--==---==---==---==~~Mstherntmes

45· II MJkimtnumbwl derWed~~~~~~~~~~

direction of the curve that is digitised in this ~ay, an~ s~ that Cartesian coordinate IS defi cient." The usual model of independent G~ussi~ ~oise is inapplicable m pnnclple for such points.

~ missing observation can be ; computed as. ; _ denominator : in the fraction xly, x is called the ~ numerator and y is called the ; denominator.

; - dense line - deficient number : the line that contains the shona positive integer that is lar~~r ~ est path between two points than the sum of its proper diVlsors. ~ - density ; the ratio of the mass of an ob- definition : ject to its volume. a statement that clarifies or ex- I plains the meaning of a word : - density scale ~ the vertical axis of a histogram or phrase. ; has units of percent per unit of - degree : the horizontal axis. This is called a unit of m,easure for angles and ~ a density scale; it measures how arcs equivalent to 1/360 of ro- I "dense" the observations are in tation around a circle. A right ~ each bin .. angle measures 90 degrees, . de . bl s bolized b 90°. ; - depen nt vana e . ym y : in regression, the vanable - degrees of freedom . ~ whose 'values are supposed to given a set of parameters ~stt- ; be explained by chang~ in the mated from the data, the de- : other variable (the the mdepengrees of freedom" of s?me sta- ~ dent or explanatory variable). tistic is the n~ber of m~epen- ; Usually one regresses th~ dedent observanon~ ~equ'tred to : pendent variable on the mdecompute the stattsttc. For ex- ~ pendent variable. ample the variance has n-1 de- ; grees ~f freedom because only : - derived measurem~nt n-1 of the observations are ~ a. me~urement that IS a comneeded for its computation ; bmatton of two other measuregiven the sample mean. The : I

MMhemsnes===--==--==--====

II

46

=========.

describe I dijfermtUJte

II

ments such as speed in miles per ~ • diagonal hour. I. a line segment connecting two • describe :I nonconsecutive vertices of a to explain orally or in writing. : polygon or polyhedron. I di • determine .• ameter to know, or make it possible ~ the. longest ~ord of ~ figure. In to know, all the characteristics ; a CIrcle, a diameter IS a chord of a fi?ure. For example, : that passes through the center three Sides determine a tri _ .~ of the circle. angle; three angles do not de- ~ • diameter of a circle (or ; sphere) termine a triangle. • develop : the segment whose endpoints to be ~volved in reasoning, ex- ~ are points <;>n a circle (or sphere) ploratlon, conjecturing, using I ~t co~tams. the. center of the manipulatives or sketches to gain : CIrcle as Its mldpomt; the length understanding of concepts or re- ~ of that segment I • difference of means lationships. • develop fluency ; a test statistic of intuitive appeal to. become skillful in working ; for. measuring difference in 10With numbers both in accu- . cation between two samples ~ wi~ inte~al-scale data. Emracy and speed. . ploymg this test statistic in an • deviation :I exact test defmes the pitman a deviation is the difference I permutation tests(l or 2). b etween a d atum and some : reference value, typically the ~ • differential calculus mean of the data. In comput- ; th.at part of calculus that deals ing the SD, one fmds the rms : ~I~ the opeation of differenof the deviations from the ~ tIatIon of functions. mean, the differences be- ; • differentiate tween the individual data and ; to distinguish from other mem: bers of a class, based upon the mean of the data. ~ some chosen properties or cri; teria.

II tIi,gimetie I discrete line

47

• digimetic ~ - direction a cryptarithm in which digits ; the way a number goes positive represent other digits. : or negative I

• digit in the decimal system, one of the numbers 0, I, 2, 3, 4, 5, 6, 7, 8, 9.

: - direction of a translation ~ the compass direction in which ; a translation goes. ~ _ disc

- dihedral angle : a circle together with its intethe angle formed by two planes ~ rior. meeting in space. ~ - dis.crete distribution ; a probability distribution of : some statistic, based upon ~ algebraic formula or upon reI randomisation or upon actual ~ data, in which the cumulative c : probability increases in non- dilation ~ infmitesmal steps corresponda nonrigid transformation that ; ing to non-infmitesmal weight enlarges or reduces a geomet- : associated with possible values ric figure by a scale factor rela- ~ of the statistic in question. This tive to a point (the center of the ; situation is characteristic of dilation). : randomisation distributions, • dimensions ~ and also of test statistics which the width, length, and height of ; are essentially discrete. a plane or space figure ~ _ discrete line /, _ diophantine equation : a line made of dots with space an equation that is to be solved ~ in between their centers. in integers.

an

• direct variation a variation of the type y = kx; the graph is a straight line through the origin.

E

48 ~~======. ~ a

- discrete variable a qUantitative variable whose set of possible values is countable. Typical examples of discrete variables are variables whose possible values are a subset of . . the mtegers, such as SOCial Se. be th cunty num rs, e number of · f:~_:1. d d peopIe m a aJ.IlllY, ages roun . e to the nearest year, etc. D15crete " hunky." C f . bl vana es are. blc A di. . . con. ttnuous vana e. screte random variable is one whose set l' bi of POSdSI·ble va .ubes 1~ cdi~unta eif· A ran om vana le 15 screte · ula . b and 0 nly if Its cum nve pro ability distribution function is a stair-step function; i.e., if it is piecewise constant and only increases by jumps.

• discriminant analysis a broad class of methods concemed with the development of rules for assigning unclassified objects/specimens to previously defmed groups. _ disctiminant function a discriminant function is used to assign an.observation to one of a set of groups. Linear discriminant functions take a vector of observations from a specimen and multiplies it by a vector of coefficients to produce

score which can be used to i classify the specimen as belonging to one or another predefined ~ group. i di·· tuall I : - . sJOlnt or mu Y exc uI sive events . dis· . . two events are ~omt or muI tuall . I . ifth . yexc USlve e occurrence .I 0 f one IS ..mcompan·bI·th e WI the f th th th . occurrence 0 eo er; at 15, I ifth ey can' t both happen at once I (ifth ha . ey) Eq~ nal° outltcome m comI mon . wv en y, two events : are disjoint if their intersection I · th IS e empty set. ~ • displacement . the volume of fluid that rises above the original fluid line ~ when a solid object is placed i. into the fluid . i-dissection : the result of dividing a figure ~ into pieces. I • distance ; 1. the distance between points : A and B is written as AB ~ 2. this term has several meanI ings in morphometrics; it : should never be used without a ~ prefixed adjective to qualify it, ~ e.g., Euclidean distance, i Mahalanobis distance,

Procrustes distance, taxonomic distance. 3. (of a translation), the length of the translation vector between a figure and its image.

~

; : ~

tion is zero for small enough (negative) values of x, and is unity for large enough values of x. It increases monotonically: if y > x, the empirical distribution function evaluated at y is at least as large as the empirical distribution function evaluated at x.

; : - distance between 2 parallel ~ lines I the length of a perpendicular ; segment between them : - distribution ~ the distribu~ion of a set of nuI merical data is how their val; ues are distributed over the : real numbers. It is completely ~ characterised by the empiriAF02.165_ ; cal distribution function. : Similarly; the probability dis~ tribution of a random vari; able is completely - distribution function, : characterised by its probabil~ ity distribution function. empirical the empirical (cumulative) ; Sometimes the word "distridistribution function of a set . . : bution" is used as a synonym of numencal data IS, for. eachf'I tior the empirIc .. al distn'butIon . h b bT re al vaI ue 0 f x, t he firactIon 0 . f ' observations that are less ~ d~nc~blOn. or fit e pro a I tty ; Istn utIon unctIon. t han or equaI to x. A pIot 0 f the empirical distribution ~ - distributive law function is an uneven set of : the formula a(x+y)=ax+ay. ~tairs. The ~idth of the sta~rs ~ _ distributive property IS the sp~cmg be~een adJa- ~ thepropertywhichrelatesmulcent data, the height of the . tiplication and addition' the stairs depends on how many ~ formula, a(b + c) = a x + a data have e~ac~ly ~he same I -x c. value. The distnbutlon func- :

b

50

. diPitlend I drawing

II

~~~~~~~~=*

• dividend ~ in the expression "a divided by ; b", a is the divident and b is the : . I diVIsor. : • divisor ~ in the expression "a divided by ; b", .a is the divident and b is the :I di VIsor. : • dodecagon. ~ a twelve-sided polygon I • dodecahedron ; a solid figure with 12 faces. A : regular dodecahedron is a ~ regular polyhedron with 12 I faces. Each face is a regular; pentagon. : I

; : ~

; : ~

;

• domino two congruent squares joined along an edge. • dot a description of a point in which the point has a definite size • d ouble I"me graph s graphs in which two sets of data are graphed at the same time, connecting each set with line segments. • double-blind, doubleblind experiment in a double-blind experiment, neither the subjects nor the people evaluating the subjects knows who is in the treatment group and who is in the control group. This mitigates the placeho effect and guards against conscious and unconscious prejudice for or against the treatment on the part of the evaluators.

; .draw : to create a figure using num" ~ bered scales on tools such as . • d omam . rul~ . f a fun' ers and protractors. mon fi()' x IS I• the domamo the set of x values for which the ; • drawing function is defined. ; a freehand picture using any : tool.

II dual I eigenshapes

51 *~=============

~ - edge ; (of a solid) The intersection of : two faces. I

: - edgel I an extension of the notion of ~ landmark to include partial in: formation about a curve ~ through the landmark. An ; edgel specifies rotation of a di: rection through a landmark, ~ extension along a direction ; through a landmark, or both. : The formula for thin-plate ~ splines on landmarks can be -dual I extended to encompass data (of a tessellation), the new ~es­ : about edgels as well. They are , . sellation formed by connectmg :I intended eventually to circum(with line segments) the centers ~ vent any need for deficient coof polygons with a cOI?mon ; ordinates in multivariate moredge in another tessellanon. : phometric analysis. I - duodecimal number : - Egyptian fraction system . ~ a number of the form l/x where the system of numeration with ; x is an integer is called an Egypbase 12. : tian fraction. I - ecological correlation : - eigenshapes the correlation between aver- ~ principal components for outages of groups of individual~, ; line data. An eigenshape instead of individuals. EcologI- : analysis begins with the seleccal correlation can be mislead- ~ tion of a distance function being about the association of in- ; tween pairs of outlines. At the dividuals. : end one gets "eigenshapes," I . f : which have the propernes 0 ~ principal component vectors

52

:enPeCtors I empirietdlmPof.."tII'lIJIes II

7

(uncorrelated, describing the· ~ sample in decreasing order of ; variance) and also are outline: shapes themselves, so .that the ~ scores for each speclme~ of ~ the sample can be combll~ed ; to produce a new outhne : shape that approximates it in I some possibly useful way. : Eigenshapes apply to curves ~ as relative warps apply to ~ landmark shape. . • eigenvectors in the equation given to define eigenvalues, E contains the eigenvectors. In the common data analysis case, E is an orthonormal. matrix (i. e., EtE=I and EEt~I) .. When sorted by descen?IDg elgenv~ues, t~e first elge.nve~tor IS that bnear combmatl0n of var~ables that has the .greatest vanance. The second elgenvector is t~e linear combination of vanables that has the greatest variance of such combinations orthogonal to tl}e first, and so on. • elementary function one of the functions: rational functions, trigonometric functions, exponential functions, and logarithmic functions.

I

• ellipse a plane figure whose equation is x2ja2+y2jb2=1. • ellipsoid a solid figure whose equation is x2ja2+y2jb2+z2jc2=1. elliptic Fourier analysis a type of outline analysis in which differences in x and y (and p~ssibly z) coordinates of an outline are fit separately as a function of arc length by Fourier analysis.

•

~ • elliptic geometry ; a geometry in which there are : no parallel lines. ~ • empirical law of averages ~ the Empirical Law of Aver; ages lies at the base of the fre: quency theory of probability. I This law: which is in fact an : assu'llp;ion abo~t how'the world works rather than a I mathematical 'or physical law, I states that if one repeats a random experiment over and lover, independently and under I "identical" conditions, the fraction of trials that result in a given outcome converges to I a limit as the number of trials grows without bound.

II empty set I equidisttmt

53

• empty set ~ • enumerable set the empty set, denoted {} or 0, ; a countable set. is the set that has no members. ; • epicyc10id : the locus or path of a point on a • endpoint convention in plotting a histogram, one ~ circle as it rolls around another "must decide whether to in- ; circle. clude a datum that lies at a Eptcycloid class boundary with the class interval to the left or the right of the boundary. The rule for making this assignment is called an endpoint convention. I The two standard endpoint conventions are to include the left endpoint of all ~lass inter- I vals and exclude the right, exCoustle Cuns (wrt horizontal reus) cept for the rightmost class interval, which includes both I of its endpoints, and to in- ~ • equianglular clude the right endpoint of all : having angles of the same meaclass intervals and exclude the I sure left, excep~ fo~ the leftmost in- ~ • equiangular polygon ~erval, w~ch mcludes both of ; a polygon all of whose interior ItS endpomts. : angles are equal. • end~oihts . the pomts at the ends of a line segment or arc. • ends of a kite the common vertices of the equilateral sides of a kite .enneagon a nine-sided polygon

~. equichordal point

~ a point inside a closed convex

; : ~ ; ; : I

curve in the plane is called an equichordal point if all chords through that point have the same length. • equidistant equally distant.

54

• equilateral equal in length • equilateral polygon a polygon all of whose sides are equal. • equilateral triangle a triangle whose sides are equal in length

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in place of a test upon the descriptively more valid one, with corresponding savings in amount of computation required. An example of such equivalent test statistics occurs for the situation of comparison of levels of a single intervalscale variable between two groups. In this situation, the descriptively valid statistic, as defined for the Pitman permutation test, is the difference of means, but simpler equivalent test statistics include the mean for one designated group, or (most simply) the total of scores in one designated group. error tolerance the value allowable above and

I •

• equivalent test statistic within a randomisation set, it is possible that two different sta. tistics may be inter-related in a manner which is provably monotonic irrespective of the data. In such a situation a randomisation test performed on either of these test statistics will necessarily have the same outcome in terms of alpha. If one of the statistics is of good descriptive validity whereas the other is simpler to compute, then a raridomisation test upon the simpler statistic may be used

I

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below a number or its approximation. escribed circle an escribed circle of a triangle is a circle tangent to one side of the triangle and to the extensions of the other sides. • estimatot: an estimator is a rule for "guessing" the value of a population parameter based on a random sample from the population. An estimator is a random variable, because its value depends on

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II euclitleandistancema;t'l'ixanalysis I ;"",t==========5=5 which particular sample is obtained, which is random. A canonical example of an estimator is the sample mean, which is an estimator of the population mean.

~ - Euler segment ; the line segment containing the : centroid of a triangle, whose ~ endpoints are the orthocenter ; and the circumcenter of the tri: angle.

- euclidean distance matrix analysis EDMA. A method for the statistical analysis of full matrices of all interlandmark distances, averaging elementwise within samples, and then comparing those averages between samples by computing the ratios of corresponding mean distances. • euclidean space a space where distances between two points are defmed as Euclidean distances in some system of coordinates.

: - Euler's formula for poly· ~ hedrons octahedron I an eight-sided polyhedron. The : regular octahedron is one of the ~ Platonic solids. ~ _ Euler's constant I the limit of the series 1/1 + 1/ ~ 2+1/3+ ... +1/n-ln n as n goes : to infInity. Its value is approxi~ mately 0.577216 .

• euler line the line through a triangle'S circumcenter, orthocenter, and centroid. Named after Swiss mathematician and physicist Leonhard Euler. A Euler or nine.point circIc of ABC.

I

I _ even function ; a function f(x) is called an even : function if f(x) =f( -x) for all x. I

: - even node ~ a node that has an even num; ber of arcs ; : I : ~ ;

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- even nwnber an integer that is divisible by 2. _ event an event is a subset of outcome space. An event determined by a random variable is an event of the form A=(X is in A). When the random variable X is observed, that determines whether or not A occurs: if the

56 ~~~~~~~=.

value of X happens to be in A, ~ group for the development and A occurs; if not, A does not oc- ; promulgation of the ideas of re: randomisation statistics. cur. I

• exact binomial test a statistical test referring to the binomial distribution in its exact algebraic form, rather than through continuous approximations which are used especially where sample sizes are substantial. • exact number a numerical result that has not been rounded or estimated. • exact test - the characteristic of a rerandomisation test based upon exhaustive re-randomisation, that the value of alpha will be fixed irrespective of any random sampling of randomisations or uponanydistributionalassumptions. Notable examples are the exact binomial test, fisher test, the Pitman permutation tests ( 1 and 2), and various non-parametric tests based upon ranked data.

: .excenter ~ the center of an excircle. I •

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~ an escribed circle of a triangle. I

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one or the other, but not both

exhaustive rerandomisation : a series of samples from a ~ randomisation set which is ; known to generate every randomisation. In particular, I sampling which generates every I randomisation exactly once. I •

~

; • exact-stats : this is the name of the academic ~ initiative which produced this ; present glossary. exact-stats is : a closed e-mail based discussion ~

• exhaustive a collection of events {AI, A2, A3, ... } is exhaustive if at least one of them must occur; that is, if S = Al U A2 U A3 U ... where S is the outcome space.

II existentUdstll:tement I e:cpectedi =nxpi

57

*~~==~~~ ~ pected value of a constant a • existential statement ; a conditional that uses the word times a random variable X is the ~ constant times the expected 'same' : value of X (E(axX ) = aX • expansion I E(X». a size change where k is greater t • expected i = nxpi than I • expectation, expected ~ if the model be correct, we value : would expect the n trials to rethe expected value of a random ~ s~t in o.utcome i about nxpi ~ariable is the long-term limit- ; tunes, gIve or take a bit. Let mg average of its values in in- : observed i denote the number ' dependent repeated experi- I : 0f tIm~S an outcome of type i ments. The expected value of t occurs m the n trials, for i = I, the random variable X is de- :t 2,• •... , k. The chi-squared stanoted EX or E(X). For a discrete : tlStl~ summarises the discreprandom variable (one that has I anCles between the expected a countable number of possible : number of times each outcome values) the expected value is the ~ occurs (assuming that the weighted average of its possible ~ model is trU~) and the observed ~alues, where the weight as- ; number of tImes.each outcome sIgned to each possible value is : occurs, by summmg the squares the chance that the random vari- ~ of t~e discrepancies, able takes that value. One can ; normalIsed by the expected think of the expected value of a : ~umbe~s, over all the categorandom variable as the point at ~ nes: chi-squared = (observed1 which its probability histogram ; expected1)2/expected1 + (obwould balance, if it were cut out : served2 expected2)2/expected2 of a uniform material. Taking ~ + . . . + (observed k expected the expected value is a linear t ~)2/expectedk. As the sample operation: if X and Y are two : SIZe n mcreases, if the model is random variables, the expected ~ c.OITecr, the s~pling distribuvalue of their sum is the sum of t ~on of the chi-squared statistic their expected values (E(X +Y) : IS approximated increasingly = E(X) + E(Y», and the ex- ~ well by the chi-squared curve

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58

txperiment I ~tment

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with (#categories 1) = k 1 de- ~ grees of freedom (d.f.), in the ; sense that the chance that the : chi-squared statistic is in any ~ given range grows closer and; closer to the area under the ChiSquared curve over the same range. This page illustrates the I sampling distribution of the chisquare statistic. _ experiment I what distinguishes an experi- I ment from an observational I study is that in an experiment, the experimenter decides who I receives the treatment.

_ explanatory variable in regression, the explanatory or independent variable is the one that is supposed to "explain" the other. For example, in examining crop yield versus quantity of fertilizer applied, the quantity of fertilizer would be the explanatory or independent variable, and the crop yield would be the dependent variable. In experiments, the explanatory variable is the one that is manipulated; the one that is observed is the dependent variable.

- experimental design I this term overtly refers to the planning of a pJrocess of data I collection. The term is also used I to refer to the information necessary to describe the interre- I lationships within a set of data. I Such a description involves considerations such as number of ~ cases, sampling methods, iden- I tification of variables and their I scale-types, identification of repeated measures and replica- I tions. These considerations are I essential to guide the choice of : TEST STATISTIC and the pro- ~ cess of RE-RANDOMI-· SATION.

- explicit form a formula for any term of a sequence given the number of the term. - exploration - indirect measurement clinometer a tool for measuring angle of elevation or depression, consisting of an edge to sight along, a plumb line, and a protractor. _ expoential function to base a the function f(x)=ax• - exponent in the expression xl', x is called the base and y is called the exponent.

II '-"Punmtislfunaiun Ifoetorial • exponential function the function f(x) =cX. Future Value as an Exponential FlIDction Amction

59 ~ • extrapolate ; to use given information to pre: dict values beyond the set of ~ given values using either a for; mula or a reasonable estimate.

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; • face . : (of a polyhedron) one of the ~ polygons and its interior form~ ing the surface of a polyhedron.

• expression I • face angle combination of numerals or : the plane angle formed by adnumerals and variables that in~ jacent edges of a polygonal dicate a ftnite number of opera~ angle in space. tions, not an equation. I • factor (noun) • exradius ~ an exact divisor of a number. an exradius of a triangle is the : This 7 is a factor of 28. radius of an escribed circle. I : • factor (verb) • exterior angle I to fmd the factors of a number. 1. (of a triangle), an angle that forms a linear pair with one of ~ • factor analysis the interior angles of a triangle. ; factor analysis is a multivariate 2. (of a polygon), an angle that : technique for describing a set of forms a linear pair with one of ~ measured variables in terms of the interior angles of a polygon. ; a set of causal or underlying : variables. A factor model can be • exterior of an angle ~ characterised in terms of path the nonconvex set formed by an ; diagrams to show relations beangle that measures less than : tween measured variables and 180 degrees. ~ factors. • externally tangent ~ • factorial (circles) intersecting at exactly ; the factorial operator is applione point, with neither circle : cable to a non-negative integer inside the other. ~ quantity. It is notated as the

60

foirbet

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postfixed symbol 'I'. The result- ~ ing value is the product of the ; increasing integer values from: I up to the value of the argu- ~ ment quantity. For instance : 3! ; is lx2x3 = 6. By convention O! : is taken as producing the value ~ 1 f: .al val . . ~~on. h u~ mcreas.e v~ry ~ rapt y Wlty 1mcre~~ m . ~ ; argwthume~t va ue; t diS .raP l : gro IS represente 10 the ~ similarly rapid growth in num- . bers of combinations.. ~ • fair bet ~

neously (the nwnber of Type I errors divided by the nwnber of rejected null hypotheses), with the convention that if no hypothesis is rejected, the false discovery rate is zero. .....__ :1: fr cti" J.CUlllllar a ons commonly used fractions such as halves, thirds, fourths, fifths, sixths, eighths and tenths.

a fair bet is one for which the expected value of the payoff is zero, after accounting for the cost of the bet. For example, suppose I offer to pay you Rs.2 if a fair coin lands heads, but you must ante up Rs.l to play. Your expected payoff is - Rs.I + Rs.OxP(tails) + Rs.2xP(heads) = - Rs.l + Rs.2x50% = Rs.O. This is a fair bet-in the long run, if you ma~e this bet over and over agam, you would expect to break even. • false discovery rate in testing a collection ofhypotheses, the false discovery rate is the fraction of rejected null hypotheses that are rejected erro-

farey sequence the sequence obtained by arranging in nwnerical order all the proper fractions having denominators not greater than a given integer. • Fermat number a nwnber of the form (22n+ I)

• family tree hierarchy; tower or pyramid of power or importance

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II Fibonacci numb",. Ifinite element ~'t!1=uurJ,=:y.""'SU,,,,'=======",,6=1 configuration under all possible rotations. • Fibonacci number a member of the sequence 0, 1, 1, 2, 3, 5, ... where each number is the sum of the previous two numbers.

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• field properties closure for addition and multiplication, commutative for addition and multiplication, associative for addition and multiplication, identity for addition and multiplication, inverse for addition and multiplication, distributive for multiplication over addition. • figurate numbers polygonal numbers

~ • figure ; a representation of an object by : the coordinates of a specified set ~ of points, the landmarks. ~ • figure space ; the 2por 3p-space of figures, i. : e., the original coordinate data I . vectors.

~ • finite element scaling I analysis : without the word "scaling," fi~ nite element analysis is a comI putational system for con~ tinuum mechanics that esti: mates the deformation (fully ~ detailed changes of position of ; :ill component particles) that are : expected to result from a speci~ fied pattern of stresses (forces) ; upon a mechanical system. As : applied in morphometrics, ~ FESA solves the inverse prob; lem of estimating the strains : representing the hypothetical ~ forces that deformed one speciI men into another. These results : are a function of the "finite ele~ ments" into which the space I between the landmarks is sub; divided. FESA can be compared : with the thin-plate spline, which ~ interpolates a set of landmark ; coordinates under an entirely : different set of assumptions. I

62

Jinitegroup I Fisher test

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• finite group ~ exactly once), and the SE a group containing a finite num- ; should be zero. This is indeed : what the finite population corber of elements. ~ rection gives (the numerator • finite population correc; vanishes). tion

when sampling without replace- ; ment, as in a simple random : sample, the SE of sample sums ~ and sample means depends on I the fraction of the population: that is in the sample: the ~ greater the fraction, the smaller ~ the SE. Sampling with replace- ; ment is like sampling from an : infinitely large population. The ~ adjustment to the SE for sam- ; pling without replacement is : called the ftnite population cor- ~ rection. The SE for sampling ; without replacement is smaller : than the SE for sampling with ~ replacement by the fmite popu- ; Iation correction factor ((N -n)/ : (N 1»'12. Note that for sample ~ size n=l, there is no difference I between sampling with and without replacement; the fmite population correction is then I unity. If the sample size is the entire population of N units, there is no variability in the re- ~ sult of sampling without re- ; placement (every member of : the population is in the sample ~

• Fisher test named after the statistician RA Fisher. This is an exact test to examine whether the pattern of counts in a 2x2 cross classification departs from expectations based upon the marginal totals for the rows and columns. Such a test is useful to examine difference in rate between two binomial outcomes. The randomisation set consists of those reassignments of the units which produce tables with the same row and column totals as the outcome. The randomisation set will thus consist of a number of tables with different respective patterns of counts; each such table will have a number of possible randomisations which may be a very large number. For this test there are several reasonable test statistics, including : the count in anyone of the 4 cells, chi-squared, or the number of randomisations for each 2x2 table with the given row and

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II Fisher'sl'&ll&ttest Iflexibly

63

column totals; these are equiva- ~ pothcsis be true, the two lent test statistics. The calcula- ; sampl~s arc like one larger tion for the Fisher test is rela- : sample frum a single population tively undemanding ~ of zeros and ones. The allocacomputationally, making refer- ; tion of ones between the two ence to the algebra of the hy- : samples would be expected to pergeometric distribution, and ~ be proportional to the relative the test was widely used before , sizes of the samples, but would the appearance of computers. : have some chance variability. This test has historically been ~ Conditional on G and the two regarded as superior to the use ~ sample sizes, under the null of chi-squared where sample ; hypothesis, the tickets in the sizes are small. Statistical tables : first sample are like a random have been published for the ~ sample of size ni without reFisher test for a number of ; placement from a collection of small 2x2 tables defined in : N = ni + n2 units of which G terms of row and column totals. ~ are labelled with ones. Thus, _ Fisher's exact test ; under the null hypothesis, the for the equality of two percent- ~ number of tickets labeled with ages, Consider two populations . ones in the first sample has of zeros and ones. Let pI be the ' (conditional on G) an hyper of ones in the frrst : geometric distribution with paProportion ~ rameters N, G, and nl. Fisher's population, and let p2 be the , exact test uses this distribution proportion of ones in the second population. We would like ~ to set the ranges of observed to test the null hypothesis that : values' of the number of ones in pI = p2 on the basis of a simple ~ the first sample for which we random sample from each; would reject the null hypothpopulation. Let ni be the size ~ eSlS. of the sample from population : - flexibly 1, and let n2 be the size of the ~ usually applied to computation, sample from population 2. Let ; where students should be able G be the total number of ones : to mentally manipulate numin both samples. If the null hy- ~ bers and components of num-

64

bers to create a solution to a ~ • foot of altitude problem. i the intersection of an altitude of • floor function a triangle with the base to which I it is drawn. the floor function of x is the greatest integer in x, i.e. the ~ • foot of line largest integer less than or i the point of intersection of a line equal to x. : with a line or plane. • flowchart I a concept map that shows a: step-by-step process. Boxes I represent the steps, and arrows connect the boxes to order the : I process. • flowchart proof a logical argument presented in the form of a flowchart. I • focal chord a chord of a conic that 'passes through a focus.

i • form

• focal radius a line segment from the focus

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of an ellipse to a point on the perimeter of the ellipse.

• football-shaped scatterplot in a football-shaped scanerplot, most of the points lie within a tilted oval, shaped more-or-Iess like a football. A footballshaped scatterplot is one in which the data are homoscedastically scattered about a straight line.

I

in morphometries, we represent the form of an object by a point in a space of form variables, which are measurements of a geometric object that are tIDchanged by translations and rotations. If you allow for reflections, forms stand for all the figures that have all the same interlandmark distances. A form is usually represented by one of its figures at some specified location and in some specified orientation. When represented in this wa~~ location and

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II form space Ifranu. mmpli"9franu *========6=5 orientation are said to have been "removed." _ form space th~ space of figures with differences due to location and orientation removed. It is of 2p-3 dimensions for two-dimensional coordinate data and 3p-6 dimensions for three-dimensional coordinate data. _ formula a concise statement expressing the symbolic relationship between two or more qu;mtities. - Fourier analysis in morphometries, the decomposition of an outline into a weighted sum of sine and cosine functions. The chapter by Rohlf in the Blue Book provides an overview of this and other methods of analysing outline data. - fourier series a periodic function with period 2 pi. _ fractal a self-similar geometric figure. _ fractal dimension a measure of the complexity of a structure assuming a consistent pattern of self-similarity

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: sample .will be drawn. Ideally, I the frame is identical to the ; population we want to learn : about; more typicaHy, the ~ frame is only a subset of the ; population of interest. The : difference between the frame ~ and the population can be a ; source of bias in sampling de. sign, if the parameter of I terest has a different value for I the frame than it does for the population. For example, one

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might desire to estimate the ~ (fraction or percentage) of obcurrent annual average in- ; servations in different ranges, come of 1998 graduates of : called class intervals. the University of Delhi. I pro- I pose to use the sample mean income of a sample of graduates drawn at random . To fa- I cilitate taking the sample and I contacting the graduates to obtain income information I from them, I might draw I names at random from the list of 1998 graduates for whom the alumni association has an I accurate current address. The I population is the collection of : 1998 graduates; the frame is I those graduates who have cur- - frustum rent addresses on file with the for a given solid figure, a realumni association. If there is I lated figure formed by two para tendency for graduates with I allel planes meeting the given higher incomes to have up-to- solid. In particular, for a cone date addresses on file with the I or pyramid, a frustum is deteralumni association, that would I mined by the plane of the base introduce a positive bias into and a plane parallel to the base. the annual average income es- I NOTE: this word is frequently timated from the sample by I incorrectly misspelled as the sample mean. frustrum. _ function rule _ frequency the number of times a value I the set of operations that deoccurs in some time interval. I scribes the process that takes the independent variable and _ frequency table transforms it into the dependent a table listing the frequency I variable in a consistent way. (number) or relative frequency

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• fundamental region a region used in a tesselation • fundamental rule of counting if a sequence of experiments or trials TI, T2, T3, ... , Tk could result, respectively, in nl, n2 n3, · .. , nk possible outcomes, and the numbers nl, n2 n3, ... ,nk do not depend on which outcomes actually occurred, the entire sequence of k experiments has nl x n2 x n3 x x nk possible outcomes. • game theory a field of study that bridges mathematics, statistics, economics, and psychology. It is used to study economic behaviour, and to model conflict between nations, for example, "nuclear stalemate" during the Cold War.

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variance and correlation of the variables in measuring distances between points, i. e., differences in directions in which there is less variation within groups are given greater weight than are differences in directions in which there is more varia-

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• generalised superimposition the superimposition .ofa set of configurations. onto their consensus configuration. The fitring may involve least-squares, resistant-fit, or other algorithms and may be strictiy orthogonal or allow affine transformations.

• generalised distance I d. A ,synonym for Mahalanobis distance. Defined by the equa- ; • geoboard tion for two row vectors x. and a flat board into which nails • Xj for two individuals, and p I have been driven in a regular variables as: , where S is the pxp I rectangular pattern. These nails variance-covariance matrix. It represent the · lattice points in takes into consideration the the plane.

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~ The expected value of the geo• geodesic the arc on a surface of shortest . metric distribution is lip, and length joining two given points. its SE is (l-p)V2/p.

• geometric mean • geodesic distance the length of the shortest path I the non-negative number whose between two points in a suitable , square is the product of two geometric space (one for which given non-negative numbers; curving paths have lengths). On I the side of a square having the a sphere, it is the distance be- I same area as a rectangle whose tween two points as measured length and width are given. ,. ________________. __, along a great circle. • geodesy a oranch of mathematics dealing with the shape, size, and curvature of the Earth.

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• geometric distribution ... the geometric distribution de- I • geometric morphometrics scribes the number of trials ·up , geometric morphometrics is a to and including the first suc- collection of approaches for the cess, in independent trials with I multivariate statistical analysis the same probability of success. I of Cartesian coordinate data, The geometric distribution de- usually (but not always) limited pends only on the single param- to landmark point locations. eter p, the probability of success I The "geometry" referred to by in each trial. For example, the the word "geometric" is the number of times one must toss geometry of Kendall's shape a fair coin until the first time I space: the estimation of mean the coin lands heads has a geo- shapes and the description of metric distribution with param- sample variation of shape using eter p = 50%. The geometric I the geometry of Procrustes disdistribution assigns probability tance. The multivariate part of px(l p)k-lto the event that it geometric morphometrics .is' takes k trials to the first success. ' usually carried out in a linear

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IIBeometrie probllhility 1 probllhility IB~~1 symmetry . tangent space to the non-Euclidean shape space in the vicinity of the mean shape. More generally, it is the class of morphometric methods that preserve complete information about the relative spatial arrangements of the data throughout an analysis. As such, these methods allow for the visualisation of group and individual differences, sample variation, and other results in the space of the original specimens. - geometric probability 1 probability (of an event) Its likely outcome, expressed as the ratio of the number of successful outcomes of the event to the number of possible outcomes.

69

~ - geometry ; the branch of mathematics that : deals with the nature of space ~ and the size, shape, and other . ; properties of figures as well as : the transformations that pre. :I serve these propernes. I

•

: - gergonne pomt I in a triangle, the lines from the : vertices to the points of contact ~ of the opposite sides with the I inscribed circle meet in a point ~ called the Gergonne point. ; - given : information assumed to be true ~ in a proof ~ - glide reflection ; an isometry that is a composi: tion of a translation (glide) and ~ a reflection over a line that is ; parallel to the translation vec: tor.

- geometric progression I a sequence in which the ratio of : each term to the preceding ~ term is a given constant.

- geometric series I a series in which the ratio of : each term to the preceding I term is a given constant. ; - glide-reflectional symme: try - geometric solid ~ the property of a geometric the bounding surface of a 3-di~ figure that it coincides with its mensional portion of space.

70 =======~.

image under some glide reflection. - gnomon magic square a 3 X 3 array in which the elements in each 2 X 2 corner have the same sum. - gold standard the gold standard is the form of test which ~s ~ost ~ait?ful t? the randOI~llsatl0n distnbunon, for a gIven test statistic ~nd experimental design. This Involves exhaustive randomisation. Other randomisation ~ests may reasonably be Judged by comparison with this form.

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- goldback's conjecture I if n is an even number greater : than 2, then there are always I : 2 prime numbers whose sum ~ is n I _ golden ratio . the ratio of two numbers ; (larger number : smaller; number) whose ratio to each : other equals the ratio of their ~ sum to the larger number. ; _ golden rectangle ; a rectangle in which the ratio : of the length to the width is ~ the golden ratio.

_ golden spiral a spiral through vertices of nested golden rectangles . - grace~ g~aph a graph IS Said to be graceful if you can number the n vertices with the integers from 1 to n and ~en label each edge with the dIfference between the numbers at the vertices, in such a way that each edge receives a different label. _ grad (0" grade) 1/100th df a right angle - grade the tilt of a real-life object in relation to the horizontal, often ~e~ to determine how steep a hill IS - graph a gr~ph is a set of points (called vertices) and a set of lines (cal~ed edges) joinging these vernces.

IloraPhofaveraoes Igrowingpattem _ graph of averages for bivariate data, a graph of averages is a plot of the average values of one variable (say y) for small ran~es of values of the. other van able (say x), ag~st the value of ~e s.econd vanable (x) at the ffildpomts of the ranges . - graph theory the mathematics of complicated networks

• great circle a circle on a sphere with a diameter equal to that of the sphere. The shortest path connecting two points on the surface of a sphere lies along the great circle passing through the points.

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:

- greatest common factor the greatest number that is a factor of each of the given numbers. _ greatest lower bound the greatest lower bound of a set of real numbers, is the largest real number that is smaller than each of the numbers in the set. _ grid ' f congruent 0 a tesse IatlOn

:I , squares sometimes used ~ measure distance

to

'. . - group ~ a mathematical system consist~ ing of elements from a set G ; and a binary operation * such : that ~ x*y is a member of G whenever ; x and yare : (x*y) *z=x* (y*z) for all X, Y, and ~ z ; there is an identity element e : such that e*x=x*e=e for all x ~ each member x in G has an in; verse element y such that ; x*y=y*x=e

• greatest common divisor : . growing pattern the greatest common divisor of ~ a pattern where the number a sequence of integers, is the I of objects in the pattern inlargest integer that divides each ~ creases from term to term. of them exactly.

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• growth formula either a linear ·or exponential equation that describes the growth over time. • growth pattern a set of values usually visualised by plotting points on a grid and fitting either a linear or an expOnential equation to the scattel'" plot.

harmonic mean the harmonic mean of two numbers a and b is 2abj(a + b).

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Hannoni c Mean I

1 n HM=---=--n 1 1 n 1

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• half-line a ray.

HM

• half-plane the part of a plane that lies on one side of a given line.

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• Hankel matrix a matrix in which all the elements are the same along any diagonal that slopes from northeast to southwest.

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• harmonic analysis the study of the r.epresentation of functions by means of linear operations on characteristic sets of functions. • harmonic division a line segment is divided harmonically by two points when it is divided externally and internally int he same ratio.

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-1

L

=1 Xi

L -

i = 1 Xi

1 n 1 =- L n i =1 Xi

• hectare a unit of measurement in the metric system equal to 10,000 square meters (approximately 2.47 acres). height l.the length of an altirude. 2. (of a prism), the length of an altitude.

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helix the path followed by a point moving on the surface of a right circular cylinder that moves along the cylinder at a constant ratio as it moves around the cylinder. The parameteric equation for a helix is x=a cos t y=a sin t z=bt

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1/ hemisphere I ~mm

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_ hemisphere ~ - hexahedron half of a sphere including a ; a polyhedron having 6 faces. : The cube is a regular hexahegreat circle as its base. ~ dron. - heptagon a polygon with 7 sides. ~ - hexomino ; a six-square polyomino. _ heronian triangle a triangle with integer sides and ; - hidden lines : broken lines used to signify integer area. ~ lines that normally wouldn't be - heteroscedasticity I seen in a drawing "Mixed scatter." A scatterplot or residual plot shows ; - hierarchy heteroscedasticity if the scatter : a chart that shows varying levin vertical slices through the ~ els of importance plot depends on where you take I h" : - Istogram the slice. Linear regression is I a histogram is a kind of plot that not usually a good idea ·if the ~ summarises how data are disdata are heteroscedastic. : tributed. Starting with a set of ~ class intervals, the histogram is - hexagon a polygon with 6 sides. ; a set of rectangles ("bins") sit: ting on the horizontal axis. The - hexagonal number a number of the form n(2n-l) . ~ bases of the rectangles are the ; class intervals, and their heights - hexagonal prism : are such that their areas are proa prism with a hexagonal base. ~ portional to the fraction of ob; servations in the corresponding : class intervals. That is, the ~ height of a given rectangle is I the fraction of observations in : the corresponding class interval, ~ divided by the length of the corI responding class interval. A his~ togram does not need a verti: cal scale, because the total area I

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historical controls .' homology

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=========* of the histogram must equal I 100%. The units of the vertical I axis are percent per unit of the horiwntal axis. This is called I the density scale. The horiwn- I tal axis of a histogram needs a scale. If any observations coincide with the endpoints of class I intervals, the endpoint convenI tion is important. This page contains a histogram tool, with I controls to highlight ranges of ~ values and read their areas. - historical controls sometimes, the a treatment group is compared with individuals from another epoch who did not receive the treatment; for example, in studying the possible effect of fluoridated water on childhood cancer, we might compare cancer rates in a community before and after fluorine was added to the water supply. Those individuals who were children before fluoridation started would comprise an historical control group. Experiments and studies with historical controls tend to be more susceptible to confounding than .those with contemporary controls, beca.use many factors that might affect

the outcome other than the treatment tend to change over time as well. (In this example, the level of other potential carcinogens in the environment also could have changed.) - homeomorphism a one-to-one continuous transformation that preserves open and closed sets.

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- homology the notion of homology bridges the language of geometric morphometrics and the language of its biological or biomathematical applications. In theoretical biology, only the explicit entities of evolution or development, such as molecules, organs or tissues, can be "homologous." Following D'Arcy Thompson, morphometricians often apply the concept instead to discrete geometric structures, such as points or

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htmumwrpmsm I hypergeometric

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curves, and, by a further exten- ~ in vertical slices through the sion, to the multivariate de- ; plot does not depend much on scriptors (e.g., partial warp : where you take the slice. C.f. scores) that arise as part of ~ heteroscedasticity. most multivariate analyses. In ~ • horizontal line this context, the term "homolo- ; a line whose slope is zero gous" has no meaning other than that the same name is used ; • hyperbola for corresponding parts in dif- : a curve with equation x2/a2_y2/ ferent species or developmen- •. b2 =1. tal stages. To declare something ~ • hyperbolic spiral "homologous" is simply to as- • the curve whose equation in sert that we want to talk about : polar coordinates is r*theta=a. processes affecting such struc- • . tures as if they had a consistent : • hyperboloid ' biological or biomechanical ~ a geometric solid whose equa2 2 2 2 2 . S"l mearung. 1m1 ar1y, to d ec Iare • tion is x /a + y2/b _Z /C = 1 0[, 2 2 2 . lanon ' (such as a thin.: an rnterpo - x /a + y2/b _z,2/C,2= ,_ 1. plate spline) a "homology map" ~ • hypergeometric ~stribumeans that one intends to refer tion , to its features as if they had ~ the hypergeometricdi~tribu­ something to do with valid bio- ; cion with parameters,N, G and logical explanations pertaining : n is the distribution of the to the regions between the ~ number of "good~ obje'ct~ in landmarks, about which we ; a simple random sample of have no data. : size n (i.e., a random sample • homomorphism ~ without replacement in which a function that preserve the op- ; every subset of size n has the erators associated with the : same chance of occurring) specified structure. ~ from a population of N ob• jects of which G are "good." • homoscedasticity : The chance of getting exactly "Same scatter." A scatterplot or ~ g good objects in such a residual plot shows • sample is GCg x N-GCn-g! homoscedasticity if the scatter ; NCn, provided g < = n, g < = MIJ. . .n u = = = = = = = =___ 11

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hyperplllne I hypothesis tutino

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~~~~~~~~~.

G, and n g < = N G. (The probability is zero otherwise.) The expected value of the hypergeometric distribution is n x GjN, and its standard error is ((N-n)j(N-I»V2 x (n X GjN x (I-GjN) ) V2 .

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• hypersphere a generalisation of the idea of a sphere to a space of greater than three dimensions.

hypothesis testing statistical hypothesis testing is formalised as making a decision between rejecting or not rejecting a null hypothesis, on the basis of a set of observations. Two types of errors can result from any decision rule (test): rejecting the null hypothesis when it is true (a Type I error), and failing to reject the null hypothesis when it is false (a Type II error). For any hypothesis, it is possible to develop many different decision rules (tests). Typically, one specifies ahead of time the chance of a Type I error one is willing to allow. That chance is called the significance level of the test or decision rule. For a given significance level, one way of deciding which decision rule is best is to pick the one that has the smallest chance of a Type IT error when a given alternative hypothesis is true. The chance of correctly rejecting the null hypothesis when a given alternative hypothesis is

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• hyperplane : a k-I dimensional subspace of I a k-dimensional space. A hy- : perplane is typically characterised by the vector to I which it is orthogonal. • hyperspace a space of more than three dimensions.

hypotenuse the side opposite the right angle in a right triangle. The other two sides are called legs.

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• hyperv~lu.me . ~ a generalisanon of the ldea of ; volume to a space of more than three dimensions.

II icoSahedron I implies, ~ical imPlica;

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true is called the power of the ~ tion ( (p AND q) OR ((NOT test against that alternative. ; p) AND (NOT q» ). • icosahedron a polyhedron with 20 faces. Usually refers to a regular icosahedron, one of the Platonic solids . .d .1 empotent

; • image : the reflection of the preimage I

: • imaginary axis ~ the y-axis of an Argand diagram.

.~ I III some algebraJ.c. the element x . . . . structure IS called Idempotent if ~ x*x=x. . .

.

• identify to choose from a set or to name cases in which the desired resuIt is present or true.

• imaginary number a complex number of the form . h . al and'l=sqrt( Xl were x IS re 1).

imaginary Simplify:

a)./=4 .

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• identitiy transformation a size change where k equals 1 I

• identity reflection : • imaginary part a reflection where the preimage ~ the imaginary part of a complex and the image are the same. I munber x+iy where x and y are : real is y. • iff I if and only if. : • implies, logical implication • iff, if and only if if p and q are two logical propositions, then(p iff q) is a proposition that is true when both p and q are true, and when both p and q are false. If is logically equivalent to the proposition ( (p IMPLIES q) AND (q IMPLIES p) ) and to the proposi-

~ logical implication is an opera-

tion on two logical proposi~ tions. If p and q are two logical : propositions,(p IMPLIES q) is ~ a logical proposition that is true ; if P is false, or if both P and q : are true. The proposition (p ~ IMPLIES q) is logicallyequiva; lent to the proposition ((NOT : p) ORq). I

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,,;,,78==========im~7 subset I irulepnulmt, irukperulm&e II • improper subset a subset that includes the entire parent set. • incenter the point of concurrency of a triangle's three angle bisectors. • incircle the circle inscribed in a given figure.

independent and identically distributed a collection of two or more random variables {Xl, X2, . . . , } is independent and identically distributed if the variables have the same probability distribution, and are independent .

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independent variable in regression, the independent • included angle ~ variable is the one that is supan angle formed between two I posed to explain the other; the given sides of a triangle. term is a synonym for "explana• inclusive OR tory variable." Usually, one reone or the other, or both; and/ I gresses the "dependent variable" on the "independent varior able." There is not alwavs a • incoming angle I clear choice of the independent the angle formed between the variable. The independent varipath of an approaching object able is usually plotted on the (a billiard ball, a light ray) and I horizontal axis. Independent in the surface it rebounds against I this context does not mea.., the (a cushion, a mirror) . same thing as statistically independent . I •

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• independent, independence two events A and B are (statisti~ally) independent if the chance that they both happen simultaneously is the product of tl:ie chances that each occurs individually; i.e., if P(AB) = P(A)P(B). This is essentially equivalent to saying that learn-

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iruJ,icaturrandmn variable I indirta;s"=rem&n==t=========

ing that one event occurs does not give any information about whether the other event occurred too: the condi tional probability of A given B is the same as the unconditional probability of A, i.e., P(AIB) = P(A). Two random variables X and Yare independent if all events they determine are independent, for example~ i~ the event {a < X < = b} IS mdependent of the event {c < Y < = d} for all choices of a, b, c, and d. A collection of more than two random variables is independent if for every proper subset of the variables, every event determined by that subset of the variables is independent of every event determined by the variables in the complement of the subset. For example, the three random variables X, Y, and Z are independent if every event determined by X is independent of every event determined by Y an~ ~very event determined by X IS mdependent of every event determined by Y and Z and every event determined by Y is independent of every event determined by X and Z and every event deter~ mined by Z is independent of

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every event determined by X and Y : • indicator random variable ~ the indicator [random variable] ~ of the event A, often written ; lA, is the random variable that : equals unity if A occurs, and ~ zero if A does not occur. The ; expected value of the indicator : of A is the probability of A, ~ P(A), and the standard error of I the indicator of A is (P(A) x (1: P(A)) V2. The sum lA + IB + ~ 1C + ... of the indicators of a ; collection of events {A, B, C, .. ~ . } counts how many of t~e : events {A, B, C, ... } occur m ~ a given trial. The product of the ; indicators of a collection of : events is the indicator of the ~ intersection of the events (the ; product equals one if and only : if all of indicators equal one). ~ The maximum of the indicators ; of a collection of events is the : indicator of the union of the ~ events (the maximum equals lone if any of the indicators : equals one). ~ . . ; • mdirect measure~en.t : a meas~ement .that IS lffipOS~ sible o~ lffipraCtlcal t? be me~­ I sured directly or phYSICallY, us I

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80

indirect proof I inst:ribetl1l1llJk "

~~~~=~~~=.

ally calculated using a formula or a known relationship.

infinitesimal a variable that approaches 0 as a limit.

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• indirect proof a proof that begins by assum- • inflection ing the conclusion is not true ~ a point of inflection of a plane and leads to a contradiction of ; curve is a point where the either the assumption or a pre- curve has a stationary tangent, at which the tangent is changviously proved theorem. I ing from rotating in one di• indirect technique the method used to determine I rection to rotating in the oppostie direction. an indirect measurement. • initial side • inductive reasoning I the side that the measurement the process of observing data, of an angle starts from .. recognizing patterns, and male-I ing conjectures about • injection a one-to-one mapping. generalisations. • inscribed • inequality the statement that one quantity I (in a polygon or polyhedron) is less than (or greater than) Intersecting each side or face of a figure exactly once. Usuanother. I ally referring to circles in• infinite becoming large beyond bound. I scribed in polygons or spheres inscribed in polyhedrons. The I figure outside is circumscribed around the inscribed figure. • inscribed angle an angle formed by two chords I of the circle with a common endpoint (the vertex of the angle).

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instance ofa sentence I intersecting p:es===========""8,,,1

• instance of a sentence ~ circle IS less than that of the a situation where the statement ; radius. is true ; • interior of an angle the convex set formed by an • integer any whole number or its oppo- I angle that measures less than Site. 180 degrees. • integral coefficient in the expression, 3x, 3 is the coefficient. Integral coefficients are coefficients that are integers. • intercepted arc an arc of a circle whose endpoints are marked by the sides of a central angle or an lllscribed angle.

Angle A

= Angle B

• interior angles angles between two lines cut by a transversal.

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interpolate determine a value within a set of given values using a formula, rule, or reasonable estimation.

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interpolation given a set of bivariate data (x, y), to impute a value ofy corre~ sponding to some value of x at ; which there is no measurement of y is called interpolation, if I the value of x is within the range of the measured values of x. If the value of x is outside the I range of measured values, imI puting a corresponding value of y is called extrapolation. I •

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• interior of a circle the set of points whose distance from the center of the

internally tangent (circles) Intersecting at exactly one point, with one circle inside the other.

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• inter-quartile range the inter-quartile range of a list of nwnbers is the upper quartile minus the lower quartile.

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intersecting planes planes that share a line

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MR.thematics=================== II

"",8"",2========'"",'n"",tersec==~oftwosetSAllndB I i~verseoperation I

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fully be added or subtracted and that the mean is a representative measure of central tendency. Such data are common in the domain of physical sciences or engineering e.g. lengths or weights .

• invar.iant an invariant, generally speaking, is a quantity that is unchanged (even though its formula may I have changed) when one A.B or A.,-,B The above operation is cOIIum.alive. associative, changes some inessential aspect anddistribttive of a measurement. For inAB=BA (AB)C=A.(BC) A(B+C)=AB+AC I stance, Euclidean distance is an We note that if A. c B, thenAB =A.. Hence invariant under translation or AA =.'1. {$} A = {$} AF= A rotation of one's coordinate sys• intersection I tern, and ratio of distances in the intersection of two or more I the same direction is an invarisets is the set of elements that ant under affine transformaall the sets have in common; the I tions. In the morphometrics of elements contained in every one I triangles, the invariants of a of the sets. The intersection of : particular transformation are I the events A and B is written the shape variables that do not '1\ and B" and '1\B." C.f. union. I change under that transformation. • interval scale a characteristic of data such that • inverse the difference between two val- a form of conditional; if not p, ues measured on the scale has I then not q. the san1e substantive meaning! I • inverse operation significance irrespective of the examples of inverse operations common level of the two val- I are addition and subtraction, ues being compared. This im- I multiplication and division, explies that scores may meaning• intersection of two sets A andB the set of elements which are in both A and B.

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II in17eNe sine, cosine, or tangent I isome; tracting a root and raising to a power.

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combinations of shapes; shapes that may contain curved portions rather than straightline segments .

• inverse sine, cosine, or I tangent (of a number) The acute angle '.ISO whose sine, cosine, or tangent I acronym for the International is the given number. Standards Organisation, based in Geneva, Switzerland. This • inverse variation I body publishes specifications a variation of the type xy = k; the graph is a hyperbola with I for a number of standard programming languages. The axes as asymptotes. I specifications are arranged gen• irrational number erally to concur with those of decimal number that never ANSI. I ends, never repeats (Ex: pi) : • isogonal conjugate • irrational number ~ isogonal lines of a triangle are a number that is not rational. ; cevians that are symmetric with : respect to the angle bisector. • irregular region region whose boundary is not the ~ Two points are isogonal conjuunion of circular arcs or seg- ; gates if the corresponding lines to the vertices are isogonal. ments • isometric drawing • irregular shapes shapes that are not one of the I a type of drawing that shows named geometric shapes or I three faces of a three-dimensional object in one view. The isometric drawing of a cube ~. I shows all the edges equal, but each square face is represented as a 60°-120°-60°-120° rhomI bus.

••

I • isometry : an isometry is a transformation of a geometric space that leaves

Mllthemntics==================== II

isosceles tetrahedron I isotropic

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=======* distances between points unchanged. If the space is the Euclidean space of a picture or an organism, and the distances are distances between landmarks,

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ean translations, rotations, and • isosceles triangle reflections. If the distances are I a triangle with at least two conProcrustes distanc~s between I gruent sides. If a triangle has shapes, the isometnes (~or the I exactly two congruent sides, simplest case, landmarks ~n two : they are called the legs and the dimensions) are the rotatlons ~f I ano-Ie between them is called the Kendall's shape spat~ .. For. tn- ~ ve~ex angle. The ~ide opposite angles, these can be ~lsuahsed : the vertex angle IS called the as ordinary rotatIOns of I base. The nonvertex angles are Kendall's "spherical black- called the base angles. board." • isoscoles trapezoid • isosceles tetrahedron a trapezoid that has a pair of a tetrahedron in which each pair I equiangular base angles of opposite sides have the same I • isotomic conjugate length. two points on the side of a triangle are isotomic if t~ey ~re I equidistaru: from the. rru~po.mt of that side. 1\vo pomts mSlde a triangle are isotomic conjuI o-ates if the correspon d..lllg b I cevians through these pomts meet the opposite sides in isotornic points . • isosceles trapezoid a trapezoid whose two nonpar- • isotropic allel sides are congruent. I invariant with respect to direction. Isotropic errors have the same statistical distribu-

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II jointprobability distribution 'I Kenda;'S""S""hR""ifJ""esp=ac""e=~=====8~5 tion in all directions implying equal variance and zero correlation between the original variables (e.g., axis coordinates).

~ - Jordan matrix

; a matrix whose diagonal ele: ments are all equal (and non~ zero) and whose elements ; above the principal diagonal are : equal to 1, but all other eleI . ments are O.

_ joint probability distribution 0 0 0 -1 0 0 0 if Xl, X2, ... , Xk are random variables, their joint I 0 0 0 0 0 0 probability distribution gives 0 0 2 0 0 0 0 the probability of events de- .. 0 0 2 0 0 0 termined by the collection of I A:= 0 0 -1 0 2 0 0 random variables: for any collection of sets of numbers -3 2 2 0 -1 {AI, ... ,Ale}, the joint prob- I 0 -1 0 0 0 0 ability distribution deter- I • mines P( (Xl is in AI) and: - Joule (X2 is in A2) and ... and (Xk ~ a unit of energy or ~ork. is in Ak) ). ~ _ jump discontinuity _ joint probability function ; a discontinuity in a function a function that gives the prob- : where the left and righ-hand ability that each of two or ~ limits exist but are not equal to more random variables takes ; each other. at a particular value. ; - justify : give a logical explanation or in- Jordan curve ~ formal proof of a mathematia simple closed curve. ; cal situation, computation or : property. I

: - Kendall's shape space ~ the fundamental geometric conI strUction, due to David Kendall, : underlying geometric ~ morphometrics. Kendall's

MRthmuJri&s======= II

86

kilonIem-l I

~~~~~~~~=*

shape space provides a complete geometric setting for analyses of Procrustes distances among arbitrary sets of landmarks. Each point in this shape space represents the shape of a configuration of points in some Euclidean space, irrespective of size, position, and orientation. In shape space, scatters of points correspond to scatters of entire landmark configurations, not merely scatters of single landmarks. Most multivariate methods of geometric morphometrics are linearisations of statistical analyses of distances and directions in this underlying space. • kilometer a unit of length equal to 1,000 meters . • kinematics a branch of mechanics dealing with the motion of rigid bodies without reference to their masses or the forces acting on the bodies. • kite a quadrilateral with exactly two distinct pairs of congruent consecutive sides. The angles between the pairs of congruent

II

sides are called the vertex angles. The angles between the pairs of non-congruent sides are I called the non-vertex angles. I • knight's tour I a knight's tour of a chessboard : is a sequence of moves by a I knight such that each square of ~ the board is visited exactly once. : ~ • knot . a ·curve in space formed by interlacing a piece of string and I then joining the ends together. I

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the crucial matrix for computing the thin-plate spline interpolant between two landmark configurations. In this entry, k stand,s"for the number of landmarks, for historical

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reasons . The equation of the thin-plate spline has coefficients Vlh, where h is. a vector of the xor y-coordinates of the landmarks in a target fo;m, followed by three D's (for two dimensional data, four D's for three-dimensional data). The entries in the matrix L are wholly functions of the starting or reference form for the spline. Bending energy is the upper k-by-k square of

~ • lateral faces ; the faces of the lateral surface : of a prism, or a face of a pyra~ mid that is not a base

~ • lateral surface ; the surface not included in the : base(s) I

: • latin square ~ an n X n array of numbers in I which only n numbers appear. : No number appears more than . :I once In any row or coIumn.

VI.

~ • latitude

.la lateral area

the angular distance of a point ~ on the Earth from the equator, : measured along the meridian I through that point.

• latera recta plural of lattice rectum. • lateral area the area of the lateral surface of a solid • lateral edge a segment whose endpoints are corresponding points of a cylindric solid's bases, or whose endpoints are the vertex of a conic solid and a vertex of its base

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• lattice points ; points in the coordinate plane : with integer coordinates

• lateral edge (of a prism) The intersection of I two lateral faces. : • latus rectum I a chord of an ellipse passing through a focus and perpen-

M..themiJtUs======= II

~awofaverages I law ofla1lJenumhers II

88

dicular to the major axis of the ellipse.Plural: latera recta.

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• law of averages the Law of Averages says that the average of independent observations of random variables that have the same probability distribution is increasino-Iy likely to be close to the expec~ed value of the random variables as the number of observations grows. More precisely, if X!, X2 , X3 , . . . , are IIId ependent random variables with the same probability distribution, and E(X) is their common expected value, then for every number E > 0, P{ I (Xl + X2 + ... + Xn)j n E(X) I < E} converges to 100% as n grows. This is equivalent to saying that the sequence of sample means Xl, (XI+X2)j2, (Xl+X2+X3)j3,. .. converges in probability to E(X).

law of contrapositive the type of valid reasoning that concludes the truth of a statement from the truth of its contrapositive .

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• laW' of cosines the theorem that, for any triangle with angles of measure A, ~ o B, and C, and sides of lengths ; a, b~ and c (a opposite A, b oppoSIte B, and c opposite C), c2 I = a2 + b2 - lab cos C. o

law of large numbers the Law of Large Numbers says that in repeated, independent trials with the same probability P of success in each trial, the percentage of successes is increasingly likely to be close to the chance of success as the number of trials lOncreases. More precisely, the chance that the percentage of successes differs from the probability p by more than a fixed positive amount, E > 0, converges to zero as the number of trials n goes to infInity, for every number e > O. Note that in contrast to the difference between the percentage of successes and the probability of success, the difference between the number of successes and the expected number

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law ofsines I likelihood ratio test

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*================= of successes, n X p, tends to grow as n grows. The following tool illustrates the law of large numbers; the button toggles between displaying the difference between the number of successes and the expected number of successes, and the difference between the percentage of successes and the expected percentage of successes. The tool on this page illustrates the law of large numbers.

~ • least-squares estimates ; parameter estimates that : minimise the sum of squared ~ differences between observed ; and predicted sample values.

; • leg of a right triangle : a side of a right triangle that ~ include the 90 degree angle I

: • legs I (of an isosceles triangle), the : two congruent sides of a non~ equilateral isosceles triangle. I

.• lemata • law of sines the theorem that, for any tri- ~ plural of lemma. angles with angles of measure lemma A~ B~ and C~ and sides of lengths ~ a theorem whose importance is a~ b~ and c (a opposite A~ b op- : primarily as part of a proof of posite B~ and c opposite C), sin I another theorem. sin sin A = sin B = sin C I • likelihood ratio test abc ; a test based on the ratio of the • least common multiple : likelihood (the probability or the least common multiple of a ~ density of the data given the set of integers is the smallest ; parameters) under a general integer that is an exact multiple : model, to the likelihood when of every number in the set. ~ another, specified hypothesis is ; true. Many of the commonly • least upper bound the least upper bound of a set : used statistical tests are likeliof numbers is the smallest num- ~ hood ratio tests, e.g., the t-test ber that is larger than every I for comparisons of means, : Hotelling's T2, and the analysis member of the set. I · : 0f vanance F-test.

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limit lline~t

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• line of symmetry the line of reflection of a figure having reflectional symmetry. line perpendicular to a plane a line perpendicular to every line in the plane that it intersects (or anyone of them)

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IE • line of best fit I / .I given a collection of points, a : /1 line that passes closest to all of ~ .' them, as measured by some . Li __________.___ j I given criterion. I • line segment • line of reflection two points (the endpoints of the the line over which every point I line segment) and all the points of a figure is moved by a reflec- I between them that are on the tion. line containing them. The line segment connects the points. I The measure of a line segment I is its length. /

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~ • linear operation ; suppose f is a function or op: eration that acts on things we ~ shall denote generically by the ; lower-case Roman letters x and : y, Suppose it makes sense to ~, multiply x and y by numbers • linear association two variables are linearly asso- I (which we denote by a), and ciated if a change in one is as- I that it makes sense to add sociated with a proportional things like x and y together, We change in the other, with the I say that f is linear if for every f ' al I number a and every value of x . same constant 0 propornon d fi hi h C() d C ) · h h h f : an y or w c l' x an 1(y are lty t roug out t e range · 0 I d efimed , (') C( ).IS d efimed 1 l' a x x al £:() d ") £:( measurement, The corre lanon: d coefficient measures the degree I an equ s aX x, an (u x of linear association on a scale : + y) is defined and equals f(x) of -1 to 1. ~ + f(y) . ~ • linear pair • linear combination a sum of values each multiplied ; 2 supplementary adjacent : angles whose noncommon by some coefficient. A linear combination can be expressed ~ sides form a line.

• line symmetry a figure has line symmetry if there is a least one line that divides the figure into two parts that are mirror images of each other.

as the inner product of two vec- ~ • linear pair of angles tors, one representing the data ; two adjacent angles whose disand the other a vector of coelli- tinct sides lie on the same line. cients. • linear equation ax + By + C = 0


)

Linear Pair • linear function a function that, when applied to I • linear term of an equation consecutive whole numbers, ; the term w~th a variabl~, but no generates a sequence with a : exponent m an equauon; exconstant difference between I ample: By in a linear equation consecutive terms .

.MiJt"'nu======= II

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lineartransfOrmatUm IlocatUm

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• linear transformation I ments, usually bits of geometry in multivariate statistics, a lin- I or whole functions, that can be ear transformation is the con- added together and can be mulstruction of a new set of vari- I tiplied by real numbers in an abIes that are all linear combi- I intuitive way. The points of a nations of the original set. In plane don't form a linear vecgeometric morphometrics, one tor space (what is "five times a linear transformation takes I point"?), but lines segments Procrustes-fit coordinates to connecting all the points to the partial warp scores; another origin do form such a space. takes them to relativewarp I • lines of sight scores. A linear transformation I lines from an eye to what it sees of a matrix A can be written in that show perspective 'and what the form y = Ax, where y is the ~ size to draw it resulting linear combination of ; x, a column vector, with the • loading I the correlation or covariance of rows of A. a measured variable with a linear combination of variables. A x j I loading is not the same as a coI efficient. In general, coefficients supply formulas for the compuI tation of scores whereas loadI ings are used for the biological interpretation of the linear combination . • linear vector space • local behaviour in morphometrics, the most I a description of the values of a common k-dimensional linear function or relation within a vector space is the set of all real I small interval of the indepenk-dimensional vectors, includI dent variable. ing all sums of these vectors and their scalar multiples. More I • location generally, but informally, a lin- one of the four main descripear vector space is a set of ~le- tion of a point

II

=======MlJthenuuies

111OCRtion, measureofllongitudinaI7==========~9=3 • location, measure of a measure of location is a way of summarising what a "typical" element of a list is-it is a one-number summary of a distribution. • locus the path of a moving point; the set of all points in a plane satisfying some given condition or property. • logarithm a logarithm of a number is the exponent to which a given base must be raised to produce the given number. • logarithmic growth a set of values which are approximated by an equation of the form y = log b x. • logarithmic notation use of the symbols "log" or "in" in context. • logic the study of the formal laws of reasonmg. • logical argument a set of premises followed by statements, each of which relies on the premises or on previous statements, and ending with a final statement called a conclu-

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sion. An argument is valid if the conclusion has been arrived at through deductive reasoning. • logistic regression this relates to an experimental design for predicting a binary categorical (yes/no) olltome on the basis of predictor variables measured on interval scales. For each of a set of values of the predictor variables, the outcomes are regarded as representing a binomial process, with the binomial parameter 'p' depending upon the value of the predictor variable. The modelling accounts for the logarithm of the odds ratio as a linear function of the predictor variable. Fitting is via a weighted 1east-squares . regreSSIOn method. randomisation tests for this purpose have been developed by Mehta & Patel. • longitudinal study a study in which individuals are followed over time, and cornpared with themselves at different times, to determine, for example, the effect of aging on some measured variable. Longitudinal studies provide much more persuasive evidence about

Mathemlltics====================

II

94

loJl1er bountl

=========*

I mRBie tour

II

the effect of aging than do crosssectional studies.

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numbers. Lo =2, L} =1, Ln =Ln_

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• lower bound any number below which a function value may approach but not pass.

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• lowest common denorninator the smallest number that is exactly divisib.1e by each denominator of a set of fractions. .loxodrome on a sphere, a curve that cuts all parallels under the same angle.

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+ Ln_r

lune the portion of a sphere between two great semicircles having common endpoints (including the semicircles).

• magic square a square array of n numbers such that sum of the n numbers in any row, column, or main diagonal is a constant (known as the magic sum).

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magic tour if a chess piece visits each square of a chessboard in succession, this is called a tour of the chessboard. If the successive squares of a tour on an n X n chessboard are numbered from 1 to n A 2, in order, the tour is called a magic tour if the resulting square is a magic square.

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• l-tetromino a tetromino in the shape of the letter L.

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• lucas number a member of the sequence 2, 1, 3, 4, 7, ... where each number is the sum of the previous two

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II =======MAtlmnRtics

II magnitude IMann-Whitney test

95

*================

~ • magnitude the value of a number; its dis- ; : tance from the origin ~ • magnitude of a rotation the amount of rotation in de- ~ ; grees : • magnitude of a translation the distance between any point ~

and its image • major arc an arc whose endpoints form an angle over 180 degrees with the center of the circle; written the extra letter is used to distinguish it from a minor arc. • major axis the major axis of an ellipse is it's longest chord.

Reference srs1ltml

• malfatti circles three equal circles that are mutually tangent and each tangent to two sides of a given triangle.

• manipulatives objects that can be arranged, built, and moved around by hand,

• Mann-Whitney test this is a test of difference in location for an experimental design involving two I samples with data mea: sured on an ordinal scale or ~ better. The test statistic is ~ a measure of .
MsthmuJties======= II

!!!!!!96~~~~~~~~==.MANi~.OVA I mmeimumliltelihootlestimtlte

II

.MANOVA ~ world situation is abstracted to see multivariate analysis of vari- I a model, the related mathance. ematical problem is posed and I solved, and the mathematical .mapping I solution is interpreted back into making a transformation the real-world situation as a • margin of error solution to the real-world proba measure of the uncertainty in I lem. an estimate of a parameter; I • mathematical notation unfortunately, not everyone agrees what it should mean. correct use of labels, symbols, The margin of error of an esti- ~ and abbreviations in a mathI ematics contett. mate is typically one or two times the estimated standard I • matrix arrangement of pixels error of the estimate.

• Markov's inequality for lists: If a list contains no negative numbers, the fraction of numbers in the list at least as large as any given constant a>O is no larger than the arithmetic mean of the list, divided by a. For random variables: if a random variable X must be nonnegative, the chance that X exceeds any given constant a>O is no larger. than the expected value of X, divided by a. • mathematical model a mathematical q,bject (such as a geometric fig~re, graph, table, or equation) representing a real-world situation. In mathematical modeling the real-

• maximum the largest of a set of values. I

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• maximum likelihood estimate the maximum likelihood estimate of a parameter from data is the possible value of the parameter for which the chance of observing the data largest. That is, suppose that the parameter is p, and that we observe data x. Then the maximum likelihood estimate of p is estimate p by the value q that makes P(observing x when the value of p is q) as large as possible. For example, suppose we are trying to estimate the chance that a (possibly biased) coin

II = = = = = = = M A . t h e m s r i e s

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97

mean I meamremmttype

*==~~====~~==

lands heads when it is tossed. Our data will be the number of times x the coin lands heads in n independent tosses of the coin. The distribution of the number of times the coin lands heads is binomial with parameters n (known) and p (unknown). The chance of observing x heads in n trials if the chance of heads in a given trial is q is nex qx( l-q)nx. The maximum likelihood estimate of p would be the value of q that makes that chance largest. We can find that value of q explicitly using calculus; it turns out to be q = xjn, the fraction of times the coin is observed to land heads in the n tosses. Thus the maximum likelihood estimate of the chance of heads from the number of heads in n independent tosses of the coin is the observed fraction of tosses in which the coin lands heads. • mean average • mean squared error the mean squared error of an estimator of a parameter is the expected value of the square of the difference between the estimator and the parameter. In

~ symbols, if X is an estimator of ; the parameter t, then

; • mean, arithmetic mean : the sum of a list of numbers, ~ divided by the number of numi bers. See also average. I.meaning : a version of a conditional that I : defmes a term, where the term I is in the antecedent.

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measure the amount of openness in an : angle ~

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: • measure of an angle I the smallest amount of rotation ~ necessary to rotate from one : ray of the angle to the other, ~ usually measured in degrees. I • measure of an arc ~ the measure of minor arc or : major arc is the measure of its ~ central angle. ~ • measurement type ; this is a distinction regarding : the relationship between a phe~ nomenon being measured and i the data as recorded. The main : distinctions are concerned with ~ the meaningfulness of numeri~ cal comparisons of data (nomi. nal scale versus ordinal scale I . al : versus mterv scale versus ra-

Mathematics================= II

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tio scale this is known as Stevens' typology), whether the scale of the measurements (other than nominal scale measurements) should be regarded as essentially continuous or discrete, and whether the scale is bounded or unbounded. • medial triangle the triangle whose vertices are the midpoints of the sides of a given triangle.

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mangle rmethod ofcomptlrison

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in the list after sorting the list into increasing order. IT the list has an even number of entries, the median is the smaller of the two middle numbers after sorting. The median can be estimated from a histogram by fmding the smallest number such that the area under the histogram to the left of that number is 50%. 2. (of a triangle), a line segment connecting a vertex to the midpoint of the opposite side. member of a set something is a member (or element) of a set if it is one of the things in the set.

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mersenne number a number of the form 2p-l where p is a prime.

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• ~edian size a SIZe measure based on the re- I peated . median of ~ interlandmark distances. Used· in resistant-fit methods. ; : • median ~ 1. "Middle value" of a list. The ; smallest number suc~ that .at : least half the numbe.rs m the ~t ~ are no greater than It. IT the ~t I has an odd number of entrIes, : the median is the middle entry ~ I

• mersenne prime a Mersenne number that is prime. • method of comparison the most basic and important method of determining whether a treatment has an effect: compare what happens to individuals who are treated (the treatment group) with what happens to individuals who are not treated (the control group).

1\ metric space I minor lire

• metric space a space and a distance function defined on every pair of points that meets the requirements of the definition of "metric" above.

99 ~ • minimal path ; the path of shortest length, as : when fmding the shortest path ~ from one point to another by ; way of a fIxed line.

; • minimal-change sequence • mid-p : exploration of a randomisation proposed by H.O Lancaster, ~ distribution is such a sequence and further promoted by G.A. I that the successive Barnard. This is a tail defIni- : randomisations differ is a tion policy that the alpha value ~ simple way. In the context of a should be calculated as the I ranodmisation test this can sum o( the proportion of the ~ mean that the value of the test tail for data strictly more ex- : statistic for a particular treme than the outcome, plus ~ randomisation may be calcuone half of the proportion of ; lated by a simple adjustment to the distribution corresponding : the value for the preceding to the exact outcome value. ~ randomisation .. This gives an unbiased esti~ • minimax strategy mate of alpha. ; in game theory, a minimax : strategy is one that minimises • midpoint (of a line segment) The point ~ one's maximum loss, whatever on the line segment that is the ; the opponent might do (whatsame distance from both end- : ever strategy the opponent points. ~ might choose). I ° ° • mtOdsegment .• mtmmum 1.(of a trapewid), the line seg- I the smallest of a set of values. ment connecting the midpoints I • minor arc. of the two nonparallel sides.· h d· 2 (f nan . gl) lin . an arc w ose en pomts form . 0 a t: ~ a .de s~gmen} ~ an angle less than 180 degrees conn~dctmg t e ml pomts 0 ; with the center of the circle. : two Sl es

MathBm4tics=======

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minor IlI4ds I monic polynomi,lll

100

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~

a representation of a mathematical relationship or situ: ation.

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: • modulo I the integers a and b are said to I be congruent modulo m if a-b is divisible by m . • modus ponens the type of valid reasoning that uses "if P then Q' and the statement P to conclude that Q must be true.

• minor axis the minor axis of an ellipse is its smallest chord.

.mira a plastic device which is used to determine and complete symmetries by reflecting images and allowing the user to also see through the reflecting surface. • mixed variation variation that contains both direct and inverse variation. • mode for lists, the mode is a most common (frequent) value. A list can have more than one mode. For histograms, a mode is a relative maXImum ("bump"). • model to create, using concrete materials, drawings or symbols;

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• modus tollens the type of valid reasoning that uses "if P then Q" and statement "not Q' to conclude that "not P" must be trUe.

moment the kth moment of a list is the average value of the elements raised to the kth power; that is, if the list consists of the N elements xl, x2, ... ,xN, the kth moment of the list is ( xlk' + x2k + xNk )fN. The kth moment of a random variable X is the expected value of Xk, B(Xk).

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monic polynomial a polynomial in which the coefficient of the term of highest degree is 1.

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• monochromatic triangle . ~ randomisation set, sampled a triangle whose vertices are all ; without replacement, and using colored the same. : the values of the test statistic to ~ generate an estimate of the • monohedral tiling ; form of the full randomisation a tessellation in which all tiles : distribution. This procedure is are c;ongruent. ~ in contrast to the bootstrap pro• monomial I cedure in that the sampling of an algebraic expression consist- ~ the randomisation set is withing of just one term. : out replacement. An advantage I of the Monte-Carlo test over the • monotone a sequence is monotone if its ~ bootstrap is that with succesterms are increasing or decreas- : sive resamplings it converges to ~ the gold standard form of the mg. . . ; exact test. An effective necessity • mon~to~c funCtlO~ . : for the Monte-Carloprocedure a funct10n IS monotone if It only I is a source of random codes or ~ncreases or only de~reases:. f ~ an effective pseudo-random Increases monotOnIcally (IS : generator. . monotonic increasing) if x > y, ~ implies thatf(x) > = f(y). A ; • morphometrlcs function f decreases monotoni- . from the Greek: "morph," cally (is monotonic decreasing) I meaning "shape," and : "metron " meaning "measureif x > y, implies thatf(x) < = I ' f(y),. A function f is strictly : ment." Schools of monotonically increasing if x > I morphometrics are y, implies thatf(x) > f(y), and ~ characterised by what aspects of strictly monotonically decreas- : biological "form" they are coning if if x > y, implies thatf(x) ~ cerned with, what they choose < f(y). ; to measure, and what kinds of : biostatistical questions they ask • Monte-Carlo test . ~ of the measurements once they named after the famous SIte of . are made. The methods of this gambling. casinos. A m~nte- ~ glossary emphasise configuracarlo test mvolves generaong a ~ tions of landmarks from whole random subset of the;

Msth_ties======= II

102

mse(x)

=;(

(#&-')2) I multinomial tlistributiun

II

organs or organisms analysed ~ of the bias and SE of the estiby appropriately invariant bio- ; mator: MSE(X) = (bias (X) )2 metric methods (covariances of : + (SE(X))2. taxon, size, cause or effect with I • multimodal distribution position in Kendall's shape I a distribution with more than space) in order to answer bio- lone mode. The histogram of a logical questions. Another sort multimodal distribution has of morphometries studies tis- I more than one "bump." sue sections, measures the densities of points and curves, and .4 uses these patterns to answer I •• •• questions about the random processes that may be controlling the placement of cellular I 4 structures. A third, the method of "allometry," measures sizes of separate organs and asks I • multinomial questions about their correla- I an algebraic expression consisttions with each other and with ing of 2 or more terms. measures of total size. There • multinomial distribution are many others. consider a sequence of n in• mse(x) = e( (X-t)2 ) I dependent trials, each of the MSE measures how far which can result in an outthe estimator is off from what come in any of k categories. it is trying to estimate, on the I Let pj be the probability that average in repeated experi- each trial results in an outments. It is a summary mea- come in category j, j = 1,2, .. sure of the accuracy of the es- I ., k, so pI + p2 + ... + pk = timator. It combines any ten- 100%. The number of outdeney of the estimator to comes of each type has a mulovershoot or undershoot the I tinomial distribution. In partruth (bias), and the variabil- I ticular, the probability that the ity of the estimator (SE). The n trials result in nl outcomes MSE can be written in terms I of type 1, n2 outcomes of type

II =======MathimuJries

II multiple I multi"'ariatemultipleregr~ 2, . . . , and nk outcomes of ~ type k is n!/(nl! x n2! x . . .; X nk!) x plnl X p2n2 X .•. : X pknk, ifnI, ... ,nk are non- ~ negative integers that sum to ; n; th~ chance is zero other- : Wlse. ~ • multiple the integer b is a multiple of the integer a if there is an integer d such that b=da. _ multiple discriminant analysis discriminant analysis involving three or more a priori-defmed groups. _ multiple regression the prediction of a dependent variable by a linear combination of two or more independent variables using least-squares methods for parameter estimation.

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• multiplicity in hypothesis tests in hypothesis testing, if more than one hypothesis is tested, the actual significance level of the combined tests is not equal to the nominal significance level of the individual tests .

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- multivariate analysis of variance ~ MANOVA. An analysis ofvari~ ance of two or more dependent ; variables considered simulta~ neously. : - multivariate data I a set of measurements of two ; or more variables per indi~ vidual. : - multivariate ~ morphometries ; °a term historically used for the : application of standard multi~ variate techniques to measure; ment data for the purposes of - multiplication rule the chance that events A and : morphometric analysis. SomeB both occur (i.e., that event ~ what confusing now as any AB occurs), is the conditional I morphometric technique must : be multivariate in nature. probability that A occurs given I that B occurs, times the un- : - multivariate multiple I • conditional probability that B : regression occurs. I the prediction of two or more ~ dependent variables using two

MIIthernatics=======

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~ • nearly normal distribution ; a population of numbers (a list : of numbers) is said to have a I .. nearly normal distribution if the I histogram of its values in stan: dard units nearly follows a nor~ mal curve. More precisely, supI pose that the mean of the list is : JL and the standard deviation of ~ the list is SD. Then the list is I nearly normally distributed if, for every two numbers a < b, the fraction of numbers in the I list that are between a and b is I approximately equal to the area under the normal curve beI tween (a JL)/SD and (a JL)/SD. I • negation /' I (of a statement) A statement : that is false if the original state~ ment is true, and true if the ; original statement is false. The : negation can usually be made by ~ appropriately adding the word I not to the statement, or by pre: ceding the statement with the phrase "It is not the case that. . I .. " A double negation of a state• natural number anyone of the numbers 1,2,3, ment is a negation of the negation of that statement. 4,5, ....

or more independent variabies. • . • multivanate regression the prediction of two or more dependent variables using one independent variable.. . • nadir the point on the celestial spehere in the direction downwards of the plumb-line. • nagel point in a triangle, the lines from the vertices to the points of contact of the opposite sides with the excircles to those sides meet in a point called the Nagel point.

• navigational system compass directions or bearings in a variety of formats.

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• negative binomial distribution consider a sequence of independent trials with the same

II negllti:"enumbQ' I ninepointcircle

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probability p of success in ~ each trial. The number of tri- ; als up to and including the rth ; success has the negative Bino- : mial distribution with param- I eters n and r. If the random : variabJe N has the negative ~ binomial distribution with ; parameters nand r, then :I P(N=k) = k-1Cr-1 x pr x (1- : p)k-r, for k = r, r+ I, r+2, ... ~ , and zero for k < r, because there must be at least r trials ~ to have r successes. The nega- ; tive binomial distribution is I derived as follows: for the rth : success to occur on the kth ~ trial, there must have been r- ~ 1 successes in the first k-1 tri-. als and the kth trial must re- ~ suIt in success The chance of ; the former is cite chance of r- : 1 successes in k -1 independent ~ trials with the same probabil- I ity of success in each trial, which, according to the Binomial distribution with param- I eters n=k-1 and p, has probability k-1Cr-1 x pr-l x (1p)k-r. The chance of the lat- I ter event is p, by assumption. Because the trials are independent, we can find the I chance that both events occur I by multiplying their chances

together, which gives the expression for P(N =k) above. • negative number a number smaller than O. • net a two-dimensional pattern that you can cut out and fold to form a three-dimensional figure. • network a group of nodes and arcs • n-gon a polygon with n sides • nine point center in a triangle, the circumcenter of the medial triangle is called the nine point center. . . . • rune pomt circle in a triangle, the circle that passes through the midpoints of ~e sides is called the nine point CIrcle.

M/lthemtI,tics=======-:;;;;;;;;;:'

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~1~06~~~~~~~~~no~M~~~wi~mdm8ni~~I~mJ • no causation without manipulation a slogan attributed to Paul Holland. If the conditions were not deliberately manipulated (for example, if the situation is an observational study rather than an experiment), it is unwise to conclude that there is any causal relationship between the outcome and the conditions. See post hoc ergo propter hoc. • node a description of a point in a network where it is possible for two different segments to share the same endpoints

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alpha criterion level, other; wise not ('non-significant'). : The commonest conventional I values for the nominal alpha I criterion level are 0.05 and 0.01. ~ • nominal scale

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• nominal alpha criterion I level I a publicly agreed value for type-l error, such that the I outcome of a statistical test is I classified in terms of whether I the obtained value of alpha is : extreme as compared with I this criterion level. The fine : detail of the comparison invol ves the tail definition I policy. The outcome is classified as showing statistical significance ('significant') if the I outcome has low alpha as compared with the nominal

this is a type of measurement scale with a limited number of possible outcomes which cannot be placed in any order represenring the intrinsic properties of the measurements. Examples : Female versus Male; the collection of languages in which an international treaty is published.

• nomograph a graphical device used for computation which uses a straight edge and several scales of numbers. • nonagon a nine-sided polygon. 0cId_

II nf11'Ull9onal number 11I01II-PfJ//rametric:==========~1~O~7 ~ • nonlinear association • nonagonal number a number of the form n(7n-5)/ ; the relationship between two 2. : variables is nonlinear if a ~ change in one is associated with .nonary ; a change in the other that is associated with 9 : depends on the value of the • non-constant rate of ~ first; that is, if the change in the change ; second is not simply proporset of data or table of values : tional to the change in the: first, in which the amount of the ~ independent of the value of the dependent variable does not I first variable. change by a constant value as the value of the independent I • non-overlapping regions variable changes by a constant ~ regions that don't share interior : points value. I • : • non-parametric test • non~on~ex ~et hi h all I a number of statistical tests a set 0 POInts In ~ c ~ot ~ were devised, mostly over the segments connectIng POInts of. . d 1930-1960 , Wit . h the . I . th . perlO the set lie entlre . 0 f by-passy In e set; .I spec!°fi!C 0 b'Jectlve synonym: concave. ; ing assumptions about sam• non-euclidean geometries : pIing from populations with hyperbolic geometry ~ data supposedly conforming a geometry in which, through ; to theoretically modeled staa point not on a line, there are : tistical distributions such as infinitely many lines parallel ~ the normal distribution. Sevto the given line. ; eral of these tests were : explictly concerned with ordi• non-euclidean geometry ~ nal-scale data for which modsolid geometry I eling based upon continuous • non-included side ~ functions is clearly inapprothe side of a triangle that is not : priate. These tests are implicincluded by 2 given angles ~ itly re-randomisation tests. 0

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108

non-persp:edrtllwing I1UJrmtJl4ppyoximation

• non-perspective drawing a three-dimensional drawing that doesn't use perspective

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ciable, the survey suffer from large nonresponse bias.

I • nonresponse • non-regular shape in surveys, it is rare that everya shape that does not have all lone who is "invited" to particisides congruent and all angles I pate (everyone whose phone . congruent. number is called, everyone who is mailed a questionnaire, ev• nonresponse bias I eryone an interviewer tries to in a survey, those who respond may differ from those I stop on the street . . . ) in fact who do not, in ways that are responds. The difference berelated to the effect one is try- I tween the "invited" sample ing to measure. For example, I sought, and that obtained, is the a telephone survey of how nonresponse.

many hours people work is likely to miss people who are working late, and are therefore not at home to answer the phone. When that happens, the survey may suffer from nonresponse bias. Nonresponse bias makes the result of a survey differ systematically from the truth. • nonresponse rate the fraction of nonresponders in a survey: the number of nonresponders divided by the number of people invited to participate (the number sent questionnaires, the number of interview attempts, etc.) If the nonresponse rate is appre-

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• nonrigid transformation a transformation that does not preserve the size and shape of the original figure. • nonvertex angles (of a kite) The two angles between consecutive noncongruent sides of a kite.

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• normal perpendicular

~ • normal approximation I

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the normal approximation to data is to approximate areas under the histogram of data, transformed into standard units, by the corresponding areas under the normal curve. Many probability distributions can be approximated by

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nomudcurve I nomuddistributUm

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a normal distribution, in the ~ The tool on this page illussense that the area under the ; trates the normal approximaprobability histogram is close : tion to the binomial probabilto the area under a corre- ~ ity histogram. Note that the sponding part of the normal ; approximation gets worse curve. To fmd the correspond- : when p gets close to 0 or 1, ing part of the normal curve, ~ and that the approximation the range must be converted ~ improves as n increases. to standard units, by subtractI • normal curve ing the expected value and di- : the normal curve is the familviding by the standard error. ~ iar "bell curve:," illustrated on For example, the area under I this page. The mathematical the binomial probability his~ expression for the normal togram for n = 50 and p = : curve is y = (2xpi)-V2E-x2/2, 30% between 9.5 and 17.5 is ~ where pi is the ratio of the cir74.~%. To use the normal ap- ; cumference of a circle to its proximation, we transform : d·lameter. (3 14159265 . .. ) , the endpoints to standar d ~ and E is the base of the natuunits, by subtracting the ex- ; rallogarithm (2.71828 ... ). pected value (for the Bino- Th al . .I d . bl : e norm curve IS symmet.. d h . mla ran om vana e, n X p = .I nc aroun t e pomt x -- 0 ,and 15 fior these vaIues 0 f nan· d .. c. al f ul b I pOSItiVe lor every v ue 0 x. d di .di th p ) an Vi ng e res t .y : The area under the normal the ~tandard error (for a BI- ~ curve is unity, and the SD of nOffilal, (n x p x (l-p)) 1/2 = ; the normal curve suitably de3.24 for these values of n an~ : fined, is also unit~ Many (but p). The area normal approXI- ~ not most) histograms con. . mation is the area under the ; verte d Into stan d ar d'units, normaI curve b etween (9 .5: . I c. 11 th 15)/3.24 = -1.697 and (17.5 I approXimate y 10 ow e nor. ; mal curve. 0 2 t h at area IS I 15)/3 .24 =.77; 73.5%, slightly smaller than : • normal distribution the corresponding area under I the normal distribution is a the binomial histogram. See ; theoretical distribution applialso the continuity correction. : cable for continuous interval-

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lIO scale data. It is related mathematically to the binomial and chi-square distributions and to several named sampling distributions (including Student's t, Fisher's. F, Pearson's r); these samplmg distributions are the chara~ten~t~c t~ols of paramet.nc statlsical mfernece to WhICh re-randomisation statistics are an alternative. • normalise to normalise a geo~etric object is to transform It so th~t some function of its coordInates or other parameters has a prespecified value. For example, vectors are of~en normalised by transformation into unit vectors, which have length one. • not, negation, logical negation the negation of a logical proposition p, NOT p, is a propo~ition that is the logical opposite of p. That is, if P is true, NOT p is false, and if.p is false, NOT p is true. Negation takes precedence over other logical operations.

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• nth term . ; the number that a function rule : generates as output for a count~ ing number n. I • nuisance parameters ~ parameters of a model that : must be fit but that are not of ~ interest to the investigator. In ; morphometrics, the par~eters : for translation and rotation are ~ usually nuisance parameters. ; th . . • null hypo eslS ; in order to test whether a sup: posed interesting pattern exists ~ in a set of data, it is usual to I propose a null hypothesis that : the pattern does not exist. It is ~ the unexpectedness of the de~ gree of departure of the ob. served data, relative to the pat~ tern expected under the null ~ hypothesis, which is examined ; by the measure alpha. ReferI ence to a null hypothesis is common between re-randomisation ~ s~atistics and parametric statis; tiCS. : • null model ~ .the simplest model under conI sideration. The null model for : shape is the distribution in ~ Kendall's shape space that I arises from landmarks that are <

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null set I oblique triangle

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*================ distributed by independent cir- I cular normal noise of the same I variance in the original digitising plane or space and I drawn from a single, homoge- I neous population. It is exactly analogous to the usual assumption of "independent identically I distributed error terms" in con- I ventionallinear models (regres- : - oblique sion, ANOVA) . ~ at an angle that is not a mul; tiple of 90 degrees. - null set a set with nothing in it ; - oblique angle : an angle that is not 900 - number line I a line on which each point rep- : - oblique coordinates resents a real number. I a coordinate system in which I the axes are not perpendicular. - number theory the study of integers. f--------,p - numerator in the fraction x/y, x is called the numerator and y is called the denominator.

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• numerical analysis R the study of methods for approximation of solutions of ~ - oblique line various classes of mathematical ; a line that has a definite slope problems including error analy- : not equal to zero SIS.

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: - oblique prism or cylinder ~ a non-right prism or cylinder

- oblate spheroid an ellipsoid produced by rotat- ~ - oblique triangle ing an ellipse through 360 0 ; a triangle that is not a right triabout its minor axis. angle.

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=========* • observational study c.f. controlled experiment. • obtuse angle an angle whose measure is greater than 90 but less than 180 degrees.

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odd number an integer that is not diVisible by 2.

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• odds the odds in favour of an event I is the ratio of the probability that the event occurs to the • obtuse triangle probability that the event does a triangle that contains an obI not occur. For example, suptuse angle. pose an experiment can result in any of n possible outcomes, • octagon an eight-sided polygon. I all equally likely, and that k of the outcomes result in a "win" • octahedron and n-k result in a "loss." a polyhedron with 8 faces. I Then the chance of winning is kin; the chance of not winning is (n-k)/n; and the odds in I favour of winning are (k/n)/ «n-k)/n) = k/(n-k), which is the number of favourable outI comes divided by the number I of unfavourable outcomes. ~--.Note that odds are not syn• octant onymous with probability, but anyone of the 8 portions of ; the two can be converted back space dtermined by the 3 coor- and forth. If the odds in dinate planes. favour of an event are q, then • odd function I the probability of the event is a function f(x) is called an odd I q/(l +q). If the probability of function if f(x) =-f( -x) for all x. an event is p, the odds in • odd node I favour of the event are p/(la node with an odd number of ~ p) and the odds against the event are (l-p)/p. arcs I

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II oddsmtio I orderofopemtions

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*================ - odds ratio an alternative characterisation of the parameter 'p' for a binomial process is the ratio of . . f t h e InCIdences 0 the two alternatives : p/( I-p) ; this quantity is termed the odds ratio; the value may range from zero to infinity. This relates to a possible view of a binomial process as the com bined activity of two Poisson processes with a limit upon total count for the two processes combined.

- one to one a function f is said to be one to one if f(x) =f(y) implies that x=y. - one-dimensional having length, but no width; examples: a line, a ray, a segment

~ - open interval ; an interval that does not include : its two endpoints. ~ "t C . - °PPOSI e laces I f: th l i ' all I I : aces at e m par e panes I " . - opposIte rays ; two rays with a common end: point that form a line I : _ opposite side ~ (of an angle of a triangle) The I side that is not a side of the : angle. I

: - OR disjunction, logical ~ disjunction I an operation on two logical ~ propositions. If p and q are two propositions, (p OR) q is a ~ proposition that is true if p is ; true or if q is true (or both); : otherwise, it is false. That is, (p ~ OR) is true unless both p and q ; are false. C.f. exclusive disjunc: tion, XOR.

- one-point perspective I a method of perspective draw- : - order ing that uses one vanishing ~ to place numbers in order from ; smallest to largest or largest to point. : smallest.

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a function f is said to map A : - order of operations onto B if for every b in B, ~ the rule for using operations on there is some a in A such I numbers; first parentheses, : then exponents, then multiplif(a) =b.

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orderedpair lordinatiml

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=========* cation and division, then addition and subtraction. • ordered pair the two numbers that (called coordinates) are used to identify a point in a plane; written (x, y) • ordered pair rule a rule that uses ordered pairs to describe a transformation. For example, the ordered pair rule (x, y) ??(x + h, y + k) describes a translation horiwntally by h units and vertically by k units. • ordered triple the three numbers (called coordinates) that are used to identify a point in space; written (x, y, z) • ordinal scale a measurement type for which the relative values of data are defined solely in terms of being lesser, equal-to or greater as compared with other data on the ordinal scale. These characteristics may arise from categorical rating scales, or from converting interval scale data to become ranked data.

ordinal variable a variable whose possible values have a natural order, such as {short, medium, long}, {cold, warm, hot}, or {O, 1, 2, 3, ... }. In contrast, a variable whose possible valufs are {straight, curly} or {Arizona, California, Montana, New York} would not naturally be ordinal. Arithmetic with the possible values of an ordinal variable does not necessarily make sense, but it does make sense to say that one possible value is larger than another.

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• ordinate the y-coordinate of a point in the plane. • ordination a representation of objects with respect to one or more coordinate axes. There are many kinds of ordinations depending upon the goals of the ordination and criteria used. For example, plotting objects according to their scores on the first two principal component axes provides the two-dimensional ordination best summarising the total variability of the objects in the original sample space. Biplots combine an ordination of speci-

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orientation I orthographic drawing .. ===========1~1~5

mens and an ordination ofvariables.

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• orthic triangle the triangle whose vertices are the feet of the altitudes of a given triangle.

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• orthocenter the point of concurrency of a triangle's three altitudes (or of the lines containing the altitudes).

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• orthogonal I at right angles. In linear algebra, being "at right angles" is • orientation I defined relative to a symmetric in an image change, the direcI matrix P, such as the bendingtion in which the points energy matrix; two vectors x named go (i.e., how Ns posi- I tion relates to B's and B's re- and yare orthogonal with reI spect to P if xtpy=O. Principal lates to C's); either clockwise warps are orthogonal with reor counterclockwise for fig~ spect to bending energy, and ures ~ relative warps are orthogonal with respect to both bending • origin the point in a coordinate plane energy and the sample covariwith coordinates (0,0). I ance matrix .

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orthographic drawing a drawing of the top, front, and right side views of a solid that preserves their sizes and shapes. Orthomeans "straight;" the views of an orthographic drawing show the faces of a solid as if you were viewing them "head-on."

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orthogonal superimposition a superimposition using only transformations that are all Euclidean similarities, i. e., involve only translation, rotation, scaling, and, possibly, reflection.

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• orthonormal a set of vectors is orthonormal if each has length unity and all pairs are orthogonal with respect to some relevant matrix, P, such as the identity matrix. A matrix is orthogonal if its rows (columns) are orthonormal as a set of vectors.

..orthommnal I operlapping triangles I

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fied, and the outlier is determined to be spurious. Otherwise, discarding outliers can cause one to underestimate the true variability of the measurement process.

• overlapping angles property I the property that, if two angles have the same vertex • outcome space the outcome space is the set of ~ and overlap so that the all possible outcomes of a given I nonoverlapping parts of the random experiment. The out- angles are congruent, then the come space is often denoted by angles are congruent. the ~apital letter S. • overlapping segments • outcome value the value of the test statistic for the data as initially observed, before any rerandomisation .. • outgoing angle the angle formed between the path of a rebounding object (a billiard ball, a light ray) and the surface it collides with (a cushion, a mirror). • outlier an outlier is an observation that is many SD's from the mean. It is sometimes tempting tQ discard outliers, but this is imprudent unless the cause of the outlier can be identi-

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property the property that, if two line segments on the same line overlap so that the nonoverlapping parts are congruent, then the line segments are congruent. overlapping triangles triangles that share a side or angle

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*================ • palindrome a positive integer whose digits read the same forward and backwards.

~ • parallel ; (lines, rays, or line segments), : lying in the same plane and not I . . : mtersectmg.

• palindromic a positive integer is said to be palindromic with respect to a base b if its representation in base b reads the same from left to right as from right to left.

~ • parallel lines ; two or more coplanar lines : that have no points in com~ mon or are identical (eg, the I same line)

parallel planes • pandiagonal magic square ~ planes that have no points in a magic square in which all the : common broken diagonals as well as the ~ • parallelepiped main diagonals add up to the I a prism whose bases are magic constant. paraIleograms. • pandigital a decimal integer is called I pandigital if it contains each of the digits·from 0 to 9. I •

• paraboloid a paraboloid of revolution is a I surface of revolution produced by ; • parallelogram rotating a parabola about its axis. : a quadrilateral in which both ~ pairs of opposite sides are parI aIle!. I • parameter : in general, a parameter is a ~ number (an integer, a deciI mal) indexing a function. For : instance, the F-distribution ~ used to test decompositions I of variance has two par am-

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parameter I partial warp srores

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========* eters, both integers: the I • partial least squares counts of the degrees of free- I partial Least Squares is a muldom for the two variances tivariate statistical method for whose ratio is being tested. In I assessing relationships among morphometrics, there are I two or more sets of variables four main kinds of param- measured on the same entieters: nuisance parameters, ties. Partial Least Squares which must be estimated to I analyses the covariances beaccount for differences not of ~ tween the sets of variables particular scientific interest; rather than optimizing linear the geometric parameters, I combinations of variables in such as shape coordinates, in the various sets. Their compuwhich landmark shape is ex- tations usually do not involve pressed; statistical param- I the inversion of . eters, such as mean differ- I • partial warp scores ences or correlations, by partial warp scores are the which biological interpreta- I quantities that characterise tion is confronted with that I the location of each specimen data; and another set of geo- I in the space of the partial metric parameters, such as warps. They are a rotation of partial warp scores or I the Procrustes residuals Procrustes residuals, in which I around the Procrustes mean the findings of the statistical configuration. For the nonunianalysis are expressed. form partial .warps, the coef• parameter I ficients for the rotation are the a numerical property of a popu- principal warps, applied first to the x-coordinates of the lation, such as its mean. I Procrustes residuals, then to • parametric equations two equations which express I the y-coordinates and, for three-dimensional data, the zthe coordinates of x and y as I coordinates. Coefficients for separate functions of a common variable, called the pa- I the uniform partial warps are produced by special. rameter that is usually time.

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II partial warps I pentomino

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*========= • partial warps partial warps are an auxiliary structure for the interpretation of shape changes and shape variation in sets of landmarks. Geometrically, partial warps are an orthonormal basis for a space tangent to Kendall's shape space. Algebraically, the partial warps are eigenvectors of the bending energy matrix that describes the net local information in a deformation along each coordinate axis. Except for the very largest-scale partial warp, the one for uniform shape change, they have an approximate location and an approximate scale. • partition a partition of an event B is a collection of events {AI, A2, A3, . . . } such that the events in the collection are disjoint, and their union is B (they exhaust B). That is, AjAk = {} unless j = k, and B = Al U A2 U A3 U .. . . If the event B is not specified, it is assumed to be the entire outcome space S.

~ • patterns in fractals recur; sive rule : a rule used to fmd terms in num~ ber sequences using recursion. I ff ' . • payo matrIX ; a way of representing what each player in a game wins or loses, I as a function of his and his ~ opponent's strategies.

pedal triangle . the pedal triangle of a point P with respect to a triangle ABC I is the triangle whose vertices are the feet of the perpendiculars dropped from P to the I sides of triangle ABC. I •

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• pell number the nth term in the sequence 0, 1, 2, S, 12, ... defmed by the recurrence Po=O, P,=l, and Pn =2Pn . 1 +Pn .

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: • pentadecagon ~ a IS-sided polygon ~ • pentagon ; a five-sided polygon.

; • pentagonal number : a number of the form n(3n-l)j ~ 2.

• Pascal's triangle a triangular array of binomial ~ • pentomino coefficients. ; a five-square polyomino.

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percentile Ipennutation

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• percentile the pth percentile of a list is the smallest number such that at least p% of the numbers in the list are no larger than it. The pth percentile of a random vari able is the smallest number such that the chance that the random variable is no larger than it is at least p%. C.f. quantile.

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• • perfect square an integer is a perfect square if it is of the form m 2 where m is an integer.

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• • perimeter of a polygon • the sum of the lengths of the sides of the polygon • periodic curve • a curve that repeats in a regu• lar pattern. Closed (Periodic) Cubic S-5pline

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• perfect cube an integer is a perfect cube if it • • permutation is of the form m 3 where m is an a permutation of a set is an arrangement of the elements of integer. • the set in some order. If the • perfect number set has n things in 'it, there are a positive integer that is equal I nl different orderings of its to the sum of its proper divi• elements. For the first elesors. For example, 28 is perment in an ordering, there are fect because 28 = 1 + 2 + • n possible choices, for the sec4+7+14. • ond, there remain n-1 pos• perfect power • sible choices, for the third, an integer is a perfect power there are n-2, etc., and for the if it is of the form mn where • nth element of the ordering, m and n are integers and n> 1. • there is a single choice remammg. By the fundamental

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II perpendicular I Pitman permutation;==========~1~2~1 rule of counting, the total number of sequences is thus nx(n-1)x(n-2)x .. . xl. Similarly, the number of orderings of length k one can form from n> = k things is nx(n-1)x(n-2)x . . . x(nk+ 1) = n!j(n-k)1. This is denoted np k' the number of permutations of n things taken k at a time. C.f. combinations. • perpendicular intersecting at right angles.

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• perspective ; feeling of depth ; • perspective drawing : a technique of representing ~ three-dimensional relation; ships realistically in a draw: ing, by drawing objects ~ smaller as they recede into I the distance.

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pi : written 7t the ratio C[D where ~ C is the circumference and D is I the diameter of a circle; ~ 3.14159265359 ...

• perpendicular bisector (of a line segment) A line that ~ • piecewise function divides the line segment into : a function consisting of two or two congruent parts (bisects ~ more equations, defined for it) and is also perpendicular to ; specified intervals of the indeit. : pendent variable. I • perpendicular lines : • Pitman permutation test 2 segments, rays, or lines that ~ named after the statistician form a 90 degree angle ; E.J . Pitman who described : this test, and the PITMAN • perpendicular planes planes in which any two inter- ~ permutation test, in 1937; secting lines, one in each plane, ; this is one of the earliest in: stances of an exact test. An form a right angle. ~ exact re-randomisation test in I which the test statistic is the Q : difference of means of two OR ~ samples of univariate interP B I val-scale data. N •

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pixel I point ofaverages

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=================* • pixel small dot of color that makes up computer'and TV screens • placebo effect the belief or knowledge that one is being treated can itself have an effect that confounds with the real effect of the treatment. Subjects given a placebo as a pain-killer report statistically significant reductions in pain in randomised experiments that compare them with subjects who receive no treatment at all. This very real psychological effect of a placebo, which has no direct biochemical effect, is called the placebo effect. Administering a placebo to the control group is thus important in experiments with human subjects; this is the essence of a blind experiment.

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thickness and is therefore considered two-dimensional.

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• plane figure a set bf pO'ints that are on a plane

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plane geometry the study of two-dimensional figures in a plane

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• plane section the intersection of a figure with a plane • point an undefined term in most deductive systems. It has no size, only location and is therefore considered zero-dimensional. You can think of geometric figures as sets of points. In a Cartesian coordinate system, a point's location is represented by a pair of numbers (xJ y). point of averages in a scatterplot, the point whose coordinates are the arithmetic means of the corresponding variables. For example, if the variable X is plotted on the horizontal axis and the variable Y is plotted on the vertical axis, the point of averages has coordinates (mean of X, mean of Y).

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• placebo a "dummy" treatment that has no pharmacological effect; e.g., a sugar pill. • plane an undefined term in most deductive systems. A flat surfa~e that extends infinitely. A plane has length and width but no

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II pointofconcurrency I polyhedron

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*========= • point of concurrency the point at which more than two concurrent lines, line segments, or rays intersect. • point of tangency (of a circle) The single point where a tangent line touches a circle. • Poisson distribution the Poisson distribution is a discrete probability distribution that depends on one paranleter, m. If X is a random variable with the Poisson distribution with parameter m, then the probability that X = k is E-m x mkjk!, k = 0, 1,2, ... , where E is the base of the narurallogaritlun and ! is the factorial function. For all other values of k, the probability is zero. The expected value the Poisson distribution with parameter m is 111, and the standard error of the Poisson distribution with parameter m is m 1/2 . • Poisson process a process whereby t;vents occur independently in some COI1tinuwn (in many applications, time), such that the overall density (rate) is statistically constant but that it is impossible to

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improve any prediction of the position (time) of the next event by reference to the detail of any number of preceding observations. The corresponding distribution of intervals between events is an exponential distribution. The conventional example of a Poisson processes is concerned with occurence of radioactive emissions in a substantial sample of radioactive with a half-life very much longer than the total observation period. • polarity of a variable the positivity or negativity of a variable; its direction polygon a closed planar geometric figure consisting of line segments (tlle sides), each of which intersects exactly two others at endpoints forming the polygon'S angles. Each point of intersection is a vertex of the polygon.

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polygonal region the union of a polygon and its lntenor • polyhedron a solid whose surface consists of polygons and tlleir interiors, each of which is a face. A line

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P01,;,i1W I population stand4rd deviation

segment where two faces intersect is an edge. A point of intersection of three or more edges is a vertex.

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of a box of numb.ered tickets is the mean of the list comprised of all the numbers on I all the tickets. The population I mean is a parameter. C.f. : sample mean. I

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- polyomino a planar figure consisting 0 f I congruent squares joined edge- : - population percentage to-edge. the percentage of units in a I population that possess a specified property. For example, the percentage of a given collection I of registered voters who are registered as Republicans. If each unit that possesses the I property is labelled with "1," and each lmit that does not possess the property is labelled - population I with "0," the population pera definable set of individual I centage is the same as the mean units to which the findings of that list of zeros and ones; from statistical examination of I that is, the population percenta sample subset are intended I age is the population mean for to be applied. The population a population of zeros and ones. will generally much outnum- The population percentage is a ber the sample. In re- I parameter. c.f. sample percentrandomisation statistics the age. process of applying inferences based upon the sample to the - population standard deviation population is essentially inforI the standard deviation of the mal. values of a variable for a popu- population mean lation. This is a parameter, not the mean of the numbers in a I a statistic. C.f. sample standard numerical' population. For ex- deviation. ample, the population mean

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*================= • population a collection of units being studied. Units can be people, places, objects, epochs, drugs, procedures, or many other things. Much of statistics is concerned with estimating numerical properties (parameters) of an entire population from a random sample of units from the population. • post hoc ergo propter hoc after this, therefore because of this. A fallacy of logic known since classical times: inferring a causal relation from correlation. Don't do this at home! • postulates premises in a deductive system accepted without proof. • power this is the probability that a statistical test will detect a defined pattern in data and declare the extent of the pattern as showing statistical significance. power is related to type-2 error by the simple formula: power = (I-beta) ; the motive for this re-definition is so that an increase in value for power shall represent im-

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provement of performance of a statistical test. practical number a practical number is a positive integer m such that every natural munber n not exceeding m is a sum of distinct divisors of

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m. • precision the closeness of repeated measurements to the same value.

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• preform space the space corresponding to centered objects, i. e., differences in location have been removed. It is of k(P-I) dimensions.

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• preimage the original object that is reflected premises (in a deductive sy~tem) Statements (including undefined terms, definitions, properties of algebra and equality, postulates, and theorems) used to prove further conclusions.

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preshape space the space corresponding to figures that have been centered and scaled but not rotated to alignment. It is of k(P-l)-1 dimensions.

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""12""6==========,,,,p"=·nu;aCie I p1-incipal components atlalysis

• prima facie latin for "at tirst glance." "On the face of it." Prima facie evidence for something is information that at first glance supports the conclusion. On closer examination, that might not be true; there could be another explanation for the evidence. • prime a prime number is an integer larger than 1 whose only positive divisors are 1 and itself.

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• prime factorisation the unique set of factors of a number, all of which are prime numbers. • primitive pythagorean triangle a right triangle whose sides are relatively prime integers.

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its effect on a circle or sphere. An affine transformation takes circles into ellipses. The principal axes of the shape change are the directions of the diameters of the circle that are mapped into the major and minor axes of the ellipse. The principal strains of the change are the ratios of the lengths of the axes of the ellipse to the diameter of the circle. In the case of the tetrahedron, there are three principal axes, the axes of the ellipsoid into which a sphen: is deformed. One has the greatest principal strain (ratio of axis length to diameter of sphere), one the least, and there is a third perpendicular to both, having an intermediate principal strain.

principal components analysis • primitive root of unity the complex number z is a the eigenanalysis of the primitive nth root of unity if ~ sample covariance matrix. Zll = 1 but Zk is not equal to 1 for ; Principal components (PC's) any positive integer k less than can be defined as the set of I vectors that are orthogonal n. both with respect to the iden• principal axes and strains tity matrix and the sample a change of one triangle into I covariance matrix. They can another, or of one tetrahedron also be defined sequentially: into another, can be modelled the first is the linear combias an affine transformation I nation with the largest variwhich can be parameterised by I •

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II principal wa1'"Ps I prism..

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ance of all those with coeffi- I set of p 2D landm~irks (p-4 for cients summing in square to I 3D data) they form a finite seI; the second has the largest ries . Together with the uniform variance (when normalised I terms, the partial warps, which that way) of all that are , are projections (shadows) of the uncorrelated with the first principal warps, supply an orone; etc. One way to compute thonormal basis for a space that principal components is to use I is tangent to Kendall's shape a singular value decomposi- space in the vicinity of a mean tion. Relative warps are prin- form. cipal components of partial I • prism warp scores. There is a lot to I a polyhedron with two congrube said about PC's; see any of ent polygons in parallel planes the colored books. as bases. Line segments (lateral • principal warps I edges) connect the correspondprincipal warps are ing bases to form lateral faces, eigenfunctions of the bending- which are parallelograms. An energy matrix interpreted as I altitude is a line segment beactual warped surfaces (thin- I tween, and perpendicular to, the plate splines) over the picture planes of the bases. The height of the original landmark con- I is the length of an altitude. If figuration. Principal warps are I the lateral edges are perpenlike the harmonics in a Fourier dicular to the bases, the prism analysis (for circular shape) or is a right prism; otherwise it is Legendre polynomials (for lin- I oblique. ear shape) in that together they decompose the relation of any sample shape to the sample av- , erage shape as a unique swumation of multiples of eigenfunctions of bending en- , ergy. They differ from these more familiar analogues in that there are only p-3 of them for a I

MRthemilties=================== II

""12,,,,8========== .probability I probabilitydistrilmtilm "

• probability I have probability density functhe probability of an event is a I nons. number between zero and I • probability distribution 100%. The meaning (interpre- the probability distribution of tation) of probability is the sub- I a random variable specifies ject of theories of probability, I the chance that the variable which differ in their interpretatakes a value in any subset of tions. However, any rule for I the real numbers . (The subassigning probabilities to events sets have to satisfy some techhas to satisfy the axioms of ~ nical conditions that are not probability. important for this course.) I The probability distribution of • probability density function a random variable is comthe chance that a continuous pletely characterised by the random variable is in any I cumulative probability distrirange of values can be calcu- bution function; the terms lated as the area under a curve sometimes are used synonyover that range of values. The I mously. The probability districurve is the probability den- I bution of a discrete random sity function of the random variable can be characterised variable. That is, if X is a con- I by the chance that the random tinuous random variable, I variable takes each of its posthere is a function f(x) such sible values. For example, the that for every pair of numbers probability distribution of the a<=b, P(a<= X <=b) = I total number of spots S show(area under f between a and ing on the roll of two fair dice b); f is the probability density can be written as a table: P(S=s) function of X. For example, I s the probability density func- 2 1/36 tion of a random variable with 3 2/36 3/36 a standard normal distribu- I 4 tion is the normal curve. Only 5 4/36 continuous random variables 6 5/36 I 7 6/36

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II probability wmts I probability} th~0""if==========1""2,,,,9 8 5/36 9 4/36 10 3/36 11 2/36 12 1/36 The probability distribution of a continuous random variable can be characterised by its probability density ftillction. • probability events the set of all possible outcomes of an experiment is the sample space; any subset of the sample space IS an event. • probability histogram a probability histogram for a ran d om varia bl e is ana Iogous to a histogram of data, but instead of plotting the area of the bins proportional to the relative frequency of observations in the class interval, one plots the area of the bins proportional to the probability that the random variable is in the class interval.

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~ • probability, theories of I a theory of probability is a way : of assigning meaning to prob~ ability statements such as "the I chance that a thumbtack lands ~ point-up is 2/3." That is, a theory of probability connects I the ma-thematics of probabilI ity, which is the set of conseq uences of the axioms of ~ probability, with the real ; world of observation and ex~ periment. There are several . common theories of probabilI ity. According to the fre; quency theory of probability; the probability of an event is I the limit of the percentage of times that the event occurs in repeated, independent trials ~ under essentially the same cir; cumstances. According to the : subjective theory of probabil~­ ~ ity; a probability is a number ; that measures how strongly : we believe an event will occur. ~ The number is on a scale of

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;ocrusteidistance Iprocrustes methods 11

0% to 100%, with 0% indicat- I most subsequent morphometing that we are completely I ric analyses . sure it won't occur, and 100% I • procrustes methods indicating that we are coma term for least-squares methpletely sure that it will occur. I ods for estimating nuisance According to the theory of : parameters of the Euclidean equally likely outcomes, if an ~ similarity transformations. experiment has n possible out- I The adjective "Procrustes" comes, ana (for example, by I refers to the Greek giant who symmetry) there is no reason would stretch or shorten victhat any of the n possible out- I tims to fit a bed and was first comes should occur preferenI used in the context of supertially to any of the others, then imposition methods by Hudey the chance of each outcome is I and Cattell, 1962, The 100%jn. Each of these theoI Procrustes program: producries has its limitations, its proing a direct rotation to test an ponents, and its detractors. hypothesised factor structure, • procrustes distance I Behav. Sci. 7:258-262. Modapproximately ,the square root I ern workers have often cited · of the sum of squared differ- : Mosier (1939), a psychomeences benveen the positions of ~ trician, as the earliest known the landmarks in two optimally; developer of these methods. (by least-squares) superim- However, Cole (1996) reports posed configurations at centroid that Franz Boas in 1905 sugsize. This is the distance that I gested the "method of least (ordinary defines the metric for Kendall's differences" shape space. Procrustes analysis) as a I means of comparing homolo• procrustes mean the shape that has the least I gous points to address obvious problems with the stansummed squared Procrustes distance to all the configurations I dard point-line registrations of a sample; the best choice of ~ (Boas, 1905). Cole further points out that one of Boas' consensus configuration for ; students extended the method

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II procrustes residttals I proportional

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*================= the construction of mean configurations from the superimposition of multiple specimens using either the standard registrations of Boas' method (Phelps, 1932). The latter being essentially a Generalized Procrustes Analysis. to

• procrustes residuals the set of vectors connecting the landmarks of a specimen to corresponding landmarks in the consensus configuration after a Procrustes fit. The sum of squared lengths of these vectors is approximately the squared Procrustes distance between the specimen and the consensus in Kendall's shape space. The partial warp scores are an orthogonal rotation of the full set of these residuals.

procrustes superimposition the construction of a two-form superimpOSitIOn by least squares using orthogonal or affine transformations.

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• pronic number a number of the form n(n+ 1) . • proof a sequence of justified conclusions used to prove the validity of an if-then statement • proper divisor the integer d is a proper divisor of the integer n if O
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• properties of equality reflexive property (a = a), symmetric property (if a = b, then b = a), and transitive property (if a = band b = c, then a = c).

• procrustes scatter I a collection of forms all superimposed by ordinary orthogonal Procrustes fit over one I single consensus configuration that is their Procrustes • proportion mean; a scatter of all the I a statement of equality between Procrustes residuals each cen- two ratios. tered at the corresponding • proportional landmark of the Procrustes lone of four numbers that form mean shape. I a true proportion

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proportWnality Ip-palue

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• proportionality a relationship described by a constant ratio. • proposition, logical proposition a logical proposition is a statement that can be either true or false. For example, "the sun is shining in Berkeley right now" is a proposition. • protractor a tool used to measure the size of an angle if'. degrees.

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• prove use logical arguments, definitions, theorems, and properties to show that a relationship is true for all numbers or specific set of figures. • pseudo-random a source of data which is effectively unpredictable although generated by a determinate

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process. Successive pseudo-random data are produced by a fLxed calculation process acting upon preceding data from the pseudo-random sequence. To start the sequence it is necessary to decide arbitrarily upon a first datum, which is termed the seed value .. • p-value suppose we have a family of hypothesis tests of a null hypothesis that let us test the hypothesis at any significance level p between 0 and 100% we choose. The P value of the null hypothesis given the data is the smallest significance level p for which any of the tests would have rejected the null hypothesis. For example, let X be a test statistic, and for p between 0 and 100%, let xp be the smallest number such that, under the null hypothesis, P( X < = x ) > = p. Then for any p between o and 100%, the rule reject the null hypothesis if X < xp tests the null hypothesis at significance level p. If we observed X = x, the P-value of the null hypothesis given the data would be the smallest p such that x < xp.

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II pyramid I qef

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• pyramid a polyhedron with a polygon base and line segments connecting the vertices of the base with a single point (the vertex of the pyramid) that is not coplanar with the base. The altitude is the line segment from the vertex ending at and perpendicular to the plane of the base. The height is the length of the altitude. If the line segment connecting the vertex to the center of the base is perpendicular to the base, then the pyramid is right; otherwise it is oblique.

• pythagorean fractal similar figures figures that have the same shape but not necessarily the same size. Their corresponding sides are proportional.

I squares of the lengths of the ; legs equals the square of the : length of the hypotenuse.

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: • pythagorean triangle ~ a right triangle whose sides are I integers. ; • pythagorean triple : three positive integers with I : the property that the sum of I the squares of two of the in: tegers equals the square of the third. If the three integers I have no common integer facI tors, then the triple is primi: tive. If the three integers have I a common factor, then the ; triple is a multiple. ; • qed : ((quod erat demonstrandum)) ~ (Latin) This stems from medi; eval translators' habitual ten: dency of translating the Greek ~ for "this was to be demon; strated" to the Latin phrase : above. This appeared originally ~ at the end of many of Euclid's I propositions, signifying that he ~ had proved what he set out to : prove. I

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• pythagorean theorem I ((quod erat faciendum)) is the the theorem that says that, in a ~ latin for "which was to be right triangle, the sum of the : done" It appears in Latin MAthmuJti&s=======

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translations of Euclid's works signifying that he had demonstrated what he had set out to demonstrated. • quadrangle a closed broken line in the plane consisting of 4 line segments. • quadrangular prism a prism whose base is a quadrilateral. • quadrangular pyramid a pyranlid whose base is a quadrilateral.

.. qw:ulrangle I quantitative l1ariable 1/

quadrature the quadrature of a geometric figure is the determination of its area.

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quadric curve the graph of a second degree equation in two variables.

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• quadric surface the graph of a second degree equation in three variables. quadrilateral a four-sided polygon ..

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• quadrinomial • quadrant : an algebraic expression consistanyone of the four portions of I ing of 4 terms. the plane into wruch the plane :I • qualitative variable is divided by the coordinate a qualitative variable is one axes. whose values are adjectives, • quadratfrie I such as colours, genders, nasquare free tionalities, etc. C.f. quantitative variable .and categorical vari• quadratic equation I able. an equation of the form f(x) =0 where f(x) is a second degree I • quantitative variable polynomial. That IS, I a variable that takes numerical ax2 +bx+c=0. values for which arithmetic I makes sense, for example, • quadratic term of an I counts, temperatures, weights, equation amounts of money, etc. For the term AX2 in a quadratic I some variables that take nuequation I merical values, arithmetic with those values does not make sense; such variables are not

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IllJUllrtic polynomial I radical axis

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*================ quantitative. For example, adding and subtracting social security numbers does not make sense. Quantitative variables typically have units of measurement, such as inches, people, or pounds. - quartic polynomial a polynomial of degree 4. - quartiles there are three quartiles. The first or lower quartile (LQ) of . IS . a number (not necesa hst sarily a number in the list) such that at least 1/4 of the numbers in the list are no larger than it, and at least 3/4 of the numbers in the list are no smaller than it. The second quartile is the median. The third or upper quartile (UQ) is a number such that at least 3/4 of the entries in the list are no larger than it, and at least 1/4 of the numbers in the list are no smaller than it. To find the quartiles, first sort the list into increasing order. Find the smallest integer that is at least as big as the number of entries in the list divided by four. Call that integer k. The kth element of the sorted list is the lower quartile. Find the small-

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est integer that is at least as ; big as the number of entries : in the list divided by two. Call ~ that integer 1. The lth element ; of the sorted list is the me: dian. Find the smallest integer ~ that is at least as large as the 1 number of entries in the list ~ times 3/4. Call that integer m. : The mth element of the sorted 1 list is the upper quartile. ~ Quota Sampling. : .. I 'al - qUlnnc po ynoml 1 . ~ a polynoffilal of degree 5. : - quotient 1 the result of a division. ;_R · I

rotation

;- r : radius ~ _ radian 1 a unit of angular measurement such that there are 2 pi radians in a complete circle. One radian ~ = ISOjpi degrees. One radian ; is approximately 57.30. ; - radical axis : the locus of points of equal 1 • h : power Wit respect to two 1 circle.

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each time a measurement is made, and behaves like a number drawn with replacement from a box of numbered tickets whose average is zero.

random experiment an experiment or trial whose , outcome is not perfectly pre\. I dictable, but for which the long• radical center run relative frequency of outthe radical center of three circles I comes of different types in reis the common point of; peated trials is predictable. Note interesection of the radical axes that "random" is different from of each pair of circles. "haphazard," which does not I necessarily imply long-term • radii regularity. plural form of radius \

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• radix point the generalisation of decimal point to bases of numeration other than base 10. • random error all measurements are subject to error, which can often be broken down into two components: a bias or systematic error, which affects all measurements the same way; and a random error, which is in general different

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• random sample a random sample is a sample whose members are chosen at random from a given population in such a way that the chance of obtaining any particular sample can be computed. The number of units in the sample is called the sample size, often denoted n. The number of units in the population often is denoted N. Random samples can be drawn with or without replacing objects between draws; that is, drawing all n objects in the sample at once (a random sample without replace-

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ment), or drawing the objects ~ 1 to the outcome {T, T, H}, the one at a time, replacing them ; number to to the outcome {T: in the population between : H, H}, and the number 3 to the draws (a random sample with ~ outcome {H, H, H}. replacement). In a random ~ _ randomisation sample with replacement, any ; the process of arranging for given member of the popula- : data-collection, in accordance tion can occur in the sample ~ with the experimental design, more than once. In a random I such that there should be no sample without replacement, : foreseeable possibilty of any any given member of the ~ systematic relationship bepopulation can be in the ~ tween the data and any sample at most once. A ran; measureable characteristic of dom sample without replace- : the procedure by which the isamp d . ment in which. every . hsubset of I d ata was e . t h"IS IS usuf h e N umts m t e popu- : 11 not d bOO Y asslgnmg ex1anon IS equa11y 11ke1y IS a1so I a Y. arrange I' 11 d O l d : penmenta umts to groups, ca e 1 a slm p e random ~ and repeated measures to exsamp e. Th e term ran om· l' I 1 h i d I pen menta umts, on a stnct y samp e WIt rep acement e-: random basis. notes a random sample drawn I in such a way that every n- : - randomisation distributuple of units in the popula- ~ tion tion is equally likely. ; a collection of values of the test : statistic obtained by undertak- random variable ~ ing a number of rea random variable is an assignI randomisations of the actual ment of numbers to possible ~ data within the randomisation outcomes of a random experi: set. ment. For example, consider " olDlsabon set tossing three coins. The num- :I - rand ber of heads showing when the ~ the collection of possible recoins land is a random variable: ; randomisations of data within it assigns the number 0 to the : the constraints of the experioutcome {T, T, T}, the number ~ mental design 0

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MIIth_tics================== /I

",1""3,,,,8===========;ndmnisation test I ratio ofsimilitude

• randomisation test the rationale of a randomisation test involves exploring rerandomisations of the actual data to form the randomisation distribution of values of the test statistic. the outcome value value of the test statistic is judged in terms of its relative position within the rerandomisation distribution. if the outcome value is near to one extreme of the rerandomisation distribution then it may be judged that it is in the extreme tail of the distribution, with reference to a nominal alpha criterion value, and thus judged to show statistical significance. • randomised controlled experiment an experiment in which chance is deliberately introduced in assigning subjects to the treatment and control groups. For example, we could write an identifying number for each subject on a slip of paper, stir up the slips of paper, and draw slips without replacement until we have drawn half of them. The subjects identified on the slips drawn could then be as-

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signed to treatment, and the rest to control. Randomising the assignment tends to decrease confoimding of the treatment effect with other factors, by making the treatment and control groups roughly comparable in all respects but the treatment. • range the range of a set of num bers is the largest value in the set minus the smallest value in the set. Note that as a statistical term, the range is a single number, not a range of numbers.

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• ranked data this refers to the practice of taking a set of N data, to be regarded as ordinal-scale, and replacing each datum by its rank (1 .. N) within the set. • rate a ratio where the quantities are . of different kinds; example: 60 miles per hour • ratio the quotient of two numbers. ratio of similitude the simplest form ratio of the measures of corresponding parts of similar figures.

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ratio scale I recursiveform

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*================ - ratio scale this is a type of measurement scale for which it is meaningful to reason in terms of differences in scores and also in terms of ratios of scores. Such a scale will have a zero point which is meaningful in the sense that it indicates complete absence of the property which the scale measures. The ratio scale may be either unipolar (negative values not meaningful) or bipolar (both positive and negative values meaningful), and either continuous or discrete.

~ - real part ; the real number x is called ther : eal part of the complex number ~ x+iy where x and y are real and ; i=sqrt( -1).

; - real variable : a variable whose value ranges ~ over the real numbers. I • I : - reclproca I the reciprocal of the number x : is the number l/x. I

; - recompose : put addends or factors of a I given number back together in a way different from the original arrangement or decompoI sition.

- rational number a rational number is a number ~ - rectangle that is the ratio of two integers. ; a quadrilateral with 4 right. All other real numbers are said : angles. to be irrational. I : - rectangular solid - ray I the union of a box and its inteall points on a line that lie on I nor one side of a specified point, the ray's endpoint. A ray is referred ; - recursion to by giving the names of two : the process of generating a sepoints, first the endpoint and ~ quence (or pattern) by speci; fying a first term and then apthen any point on the ray. : plying a rule to obtain any suc- real axis ~ ceeding term from the previthe x-axis of an Argand diagram. I ous term. - real numbers ; - recursive form rational and irrational numbers. a formula for the next term of

""l40==========~eforencellngle I rejlectionalsymmetry II a sequence given the term before it. • reference angle the angle of less than 360 degrees that corresponds to an angle of over 360 degrees; In order to get the reference angle, you must subtract 360 degrees from the given angle until there is less than 360 degrees left.

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• reference configuration I in the context of superimposition methods, this is the con- I figuration to which data are fit. It may be another specimen in the sample but usually I it will be the average (consensus) configuration for a sample. The construction of ~ two-point shape coordinates; does not involve a reference specimen, though the intelli- I gent choice of baseline for the I construction usually does. The I reference configuration corresponds to the point of tan- I gency of the linear tangent space used to approximate Kendall's shape space. The I mean configuration is usually I used as the reference in order to mInImise distortions I caused by this approximation. I When splines and warps are

part of the analysis, the bending energy that goes with them is computed using the geometry of the grand mean shape, and the orthogonality that characterises the partial warps is with respect to this particular formula for bending energy. There has been some controversery regarding the choice of reference . refine to change a conjecture slightly so that it is true

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• reflection an isometry under which every point and its image are on opposite sides of a fL\:ed line (the line of reflection, or mirror line) and are the same distance from the line. • reflection image of a figure the set of all of the reflection . fi images of pomts in the Igure reflection notation rm(ABC), \vhich stands for the reflection over line m of figure ABC •

reflectional symmetry the property of a figure that it coincides with its image under at least one reflection. Also

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II rejlectWn-symmetricM'tre I regres~oward the mmn, regression effict

141

called line symmetry or mirror ~ ; symmetry. : • reflection-symmetric I . figure

• regression fallacy the regression fallacy is to attribute the regression effect to an external cause.

a figure that shows reflection symmetry

• regression toward the mean, regression effect suppose one measures two variabIes for each member of a group of individuals, and that the correlation coefficient of the variables is positive (negative). If the value of the first variable for that individual is above average, the value of the second variable for that individual is likely to be above (below) average, but by fewer standard deviations than the first variable is. That is, the second observation is likely to be closer to the mean in standard units. For example, suppose one measures the heights of fathers and sons. Each individual is a (father, son) pair; the two variables measured are the height of the father and the height of the son. These two variables will tend to have a positive correlation coefficient: fathers who are taller than average tend to have sons who are taller than average. Consider a (father, son) pair chosen at random from this

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I • reflex angle : an angle between 1800 and

3600.

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• reflex polygon : a polygon for which 2 or more I of its sides intersect each other • reflexive property of congruence I the property of every geomet- I ric object that it is congruent to I itself. I

• region the tullOn of a figure and its interior • regression a model for predicting one variable from another. Due to Francis Galton, the word comes from the faG: that when measurements of offspring, whether peas or people, were plotted against the same measurements of their parents, the offspring measurements "went back" or regressed towards the mean.

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regres;, linearngnssWn I regularpyramid

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group . Suppose the father's I zontal axis, the regression line height is 3SD above the aver- I passes through the point of avage of all the fathers' heights. erages, and has slope equal to (The SD is the standard devia- I the correlation coefficient times tion of the fathers' heights.) I the SD ofY divided by the SD Then the son's height is also of X. ~kely to be above the average • regular hexagon of the sons' heights, but by I a six-sided figure whose sides fewer than 3SD (here the SD is I are of equal length and whose the standard deviation of the angles are of equal measure. sons' heights). In an hypothesis I test using a test statistic, the I • regular polygon rejection region is the set of : a convex polygon whose angles values of the test statistic for I and sides are all congruent which we reject the null hypoth·· I • regular polyhedron eSlS. a polyhedron whose faces are • regression, linear regresenclosed by congruent, regular sion I polygons that meet at all vertilinear regression fits a line to a I ces in exactly the same way. scatterplot in such a way as to minimise the sum of the I squares of the residuals. The I resulting regression line, together with the standard deviations of the two variables or I their correlation coefficient, can be a reasonable summary of a scatterplot if the scatterplot is I roughly football-shaped. In other cases, it is a poor sum- • regular pyramid mary. If we are regressing the I a pyramid whose base is a regu" variable Y on the variable X, and lar polygon and whose vertex if Y is plotted on the vertical forms a segment with the cenaxis and X is plotted on the hori - I

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II reguJartessellatWn I repeated-meaJ1t~ ==========14~3 ter of the polygon perpendicu- ~ differences. Relative warps can ; be computed from Procrustes lar to its plane : residuals or from partial warps. _ regular tessellation I

an edge-to-edge tessellation in : - remote interior angles which tiles are congruent regu- ~ (of the exterior angle of a tri; angle), the two interior angles lar polygons. : that do not share a vertex with - relative power ~ the e:A'terior angle. a comparison of two or more statistical tests, for the same ~ - rep digit experimental design, sample I an integer all of whose digits size, and nominal alpha crite- : are the same. I rion value, in terms of the re- : _ repeated median spective values of power. ~ a median of medians. Repeated _ relative warps I medians are used to estimate relative warps are principal com- ~ some superimposition paramponents of a distribution of : eters in the resistant-fit methshapes in a space tangent to ~ ods. For example, the resistantKendall's shape space. Theyare ; fit rotation estimate is the methe axes of the "ellipsoid" oc- : dian of the estimates obtained cupied by the sample of shapes ~ for each landmark, which is, in in a geometry in which spheres ; tmn, the median of angular difare defined by Procrustes dis- : ferences between the reference rance. Each relative warp, as a ~ configuration and the configudirection of shape change about ; ration being fit of the line segthe mean form, can be inter- : ments defined using that land.: preted as specifying multiples of ~ mark and the other n-l landone single transformation, a I marks. Repeated medians are transformation that can often : insensitive to larger subsets of be usefully drawn out as a thin- ~ extremely deviant values than plate spline. In a relative warps I simple medians. analysis, the parameter can be .I _ repeated-measures used to weight shape variation ~ this is a feature of an experiBy the geometric scale of shape : mental design whereby sevI

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replications I resitlfuU

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eral observations measured ~ • repunit on a common scale refer to ; an integer consisting onlyofl's. the same sampling unit. Identification of the relation of the I • re-randomisation individual observations to the I the process of generating al. ternative arrangements of expenmental design is crucial I given data which would be to this definition. Examples : the measurement of water I consistent "\vith the experimental design. Ievel at a particular site on several systematically-defined : • re-randomisation statistics occasions; measurement of I also known as permutation or reaction-time of an individual : randomisation statistics. These using right hand and left hand are the specific area of concern I of this present glossary. separatel. I • resampling stats • replications this is a feature of an experi- this is the name of an educamental design whereby obser- tional initiative involving the vations on an experimental I use of a programming lanunit are repeated under the I guage, in the form of an intersame conditions. Identifica- preter, allowing the user to tion of the position of a .par- I specify monte-carloresampling ticular observation within the I of a set of data and accumulasequence of replications is ir- tion of the randomisation disI tribution of a defmed test starelevant. I tistic . • representative patterns in a sample of units I • residual may reasonably be attributed the deviations of an observed to the population from which : value or vector of values from the sample is drawn, only if I some expectation, e.g., the difthe sample is representative. ~ ferences between a shape and in practical terms, to ensure its prediction by an allometric that a sample is representative I regression expressed in any set almost always means ensuring of shape coordinates. that it is a random sample.

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• residual plot ~ mean is not resistant; the mea residual plot for a regression ; dian is. is a plot of the residuals from i • resistant-fit superimposithe regression against the extion planatory variable. ~ superimposition methods that • residual ; use medianand repeated-methe difference between a datum : dian-based estimates of fitting and the value predicted for it by ~ parameters rather than leasta model. In linear regression of I squares estimates. Resistant-fit a variable plotted on the verti- : procedures are less sensitive to cal axis onto a variable plotted ~ subsets of extreme values than on the horizontal axis, a re- I those of comparable leastsidual is the "vertical" distance ~ squares methods. As such, their from a datum to the line. Re- : results may provide a simple siduals can be positive (if the ~ description of differences in datum is above the line) or ; shape that are due to changes negative (if the datum is below : in the positions of just a few the line). Plots of residuals can ~ landmarks. However, resistantreveal computational errors in i fit methods lack the well-devellinear regression, as well as con- : oped distributional theory assoditions under which linear re- ~ ciated with the least-squares fitgression is inappropriate, such I ting methods. as non-linearity and; • resolution heteroscedasticity. If linear re- : the smallest scale distinguishgression is performed properly, ~ able by a digitising, imaging, or the sum of the residuals from ~ display device. the regression line must be zero; otherwise, there is a com- I • resultant vector ~ the result of combining two vecputational error somewhere. : tors. To fmd the resultant vec• resistant I tor, slide the original vectors so a statistic is said to be resistant ~ that their tails intersect. The if corrupting a datum cannot : resultant vector's tail is the comchange the statistic much. The ~ mon tail. Its head is the image

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of the head of one of the vec- ~ tors after you translate it along; the other vector. Also known as I a vector sum.

• right cylinder a cylinder whose direction of sliding is perpendicular to the plane of the base right prism a prism whose direction of sliding is perpendicular to the plane of the base

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• review mandala a circular design arranged in concentric arcs.

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• rhombus a parallelogram with four equilateral sides

• right triangle I a triangle with a right angle. • ridge curve : ridge curves are curves on a sur- The side opposite the right face along which the curvature ~ angle is the hypotenuse. The perpendicular to the curve is a I other two sides are the legs. local maximum. For instance on I • rigid motion a skull, the line of the jaw or a motion that preserves shape the rim of an orbit. and size. • right angle an angle whose measure is 90 degrees • right cone a cone whose axis is perpendicular to the plane containing its base

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• rigid rotation an orthogonal transformation of a real vector space with respect to the Euclidean distance metric. Such transformations leave distances between points and angles between vectors unchanged. A principal compo-

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nents analysis represents a rigid ~ of the elements in the list. It is rotation to new orthogonal ; a measure of the average "size" axes. A canonical variates : of the elements of the list. To analysis does not. ~ compute the rms of a list, you ; square all the entries, average Y2 : the numbers you get, and take I : the square-root of that average. I

- rms error of regression the rms error of regression is the rms of the vertical residuals from the regression line. For regressing Y on X, the nns error of regression is equal to (I r2)!f2xSDY, where r is the correlation coefficient between X and Y and SDY is the standard deviation of the values of Y.

: - Root-Mean-Square Error I (RMSE) : the RMSE of an an estimator ~ of a parameter is the square~ root of the mean squared error ; (MSE) of the estimator. In sym: boIs, if X is an estimator of the ~ parameter t, then RMSE(X) = ; (E( (X-t)2 ) ) !f2. The RMSE of : an estimator is a measure of the ~ expected error of the estimator. ; The units of RMSE are the : same as the units of the estima~ tor.

~ - rotation ; in effect, a rotation is a turning : of the plane about a point (the ~ center of rotation) by an angle I (the angle of rotation). For: mally, a rotation is an isometry ~ that is the composition of re- root of unity I flections through two lines that a solution of the equation xn = I, ~ intersect at the center of the where n is a positive integer. : rotation. The angle of rotation - root-mean-square (rms) ~ has twice the measure of the the rms of a list is the square- ; smaller angle formed by the root of the mean of the squares : lines. - RNG acronym for Random Number Generator. This is a process which uses a arithmetic algorithm to generate seyuences of pseudo-random numbers.

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=l48==~==~====;tUnudsymmetry I samplepercentage II • rotational symmetry a figure has rotational symmetry if it can be rotated (turned) less than 360 degrees about a point so that it appears the same as the original figure. • ruled surface a surface formed by mqving a straight line (called the generator).

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a pair of compasses that are fixed open in a given position. • sa surface area • sample a set of individual units, drawn from some definable population of units, and generally a small proportion of the population, to be used for a statistical examination of which the findings are intended to be applied to the population. it is essential for such inference that the sample should be representative. in rerandomisation statistics the process of applying inferences based upon the sample to the population is essentially inforI rna.

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• sample mean the arithmetic mean of a random sample from a population. It is a statistic commonly used to estimate the population mean. Suppose there are n data, {xl, x2, ... ,xn}. The sample mean is (xl + x2 + .. . + xn)jn. The expected value of the sample mean is the population mean. For sampling with replacement, the SE of the sample mean is the population standard deviation, divided by the squareroot of the sample size. For sampling without replacement, the SE of the sample mean is the finite-population correction ((N-n)j(N-l»V2 times the SE of the sample mean for sampling with replacement, with N the size of the population and !1 the size of the sample.

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sample percentage the percentage of a random sample with a certain properry, such as the percentage of voters registered as Democrats in a simple random sample of voters. The sample mean is a statistic commonly used to estimate the popula-

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tion percentage. The expected value of the sample percentage from a simple random sample or a random sample with replacement is the population percentage. The SE of the sample percentage for sampling with replacement is (p(l-p)/n )V2, where p is the population percentage and n is the sample size. The SE of the sample percentage for sampling without replacement is the finite-population correction ((N-n)/(N-I»V2 times the SE of the sample percentage for saQlpling with replacement, with N the size of the population and n the size of the sample. The SE of the sample percentage is often estimated by the bootstrap. • sample size the number of experimental units on which observations are considered. this may be less than the number of observations in a data-set, due to the possible multiplying effects of multiple variables and/or repeated measures within the experimental design.

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• sample standard deviation the sample standard deviation S is an estimator of the standard deviation of a population based on a random sample from the population. The sample standard deviation is a statistic that measures how "spread out" the sample is around the sample mean. It is quite similar to the standard deviation of the sample, but instead of averaging the squared deviations (to get the rms of the deviations of the data from the sample mean) it divides the sum of the squared deviations by (number of data 1) before taking the square-root. Suppose there are n data, {xl, x2, ... , xn}, with mean M = (xl + x2 + ... + xn)/n. Then s = ( ((xl M)2 + (x2 M)2 + ... + (xn M)2)/(n-l) )lh The square of the sample standard deviation, S2 (the sample variance) is an unbiased estimator of the square of the SD of the population (the variance of the population).

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• sample sum ; the sum of a random sample : from a population. The ex~ pected value of the sample

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sum is the sample size times the population mean. For sampIing with replacement, the SE of the sample sum is the population standard deviation, times the square-root of the sample size. For sampling without replacement, the SE of the sample sum is the finite-population correction «N-n)j(N-l))V2 times the SE of the sample sum for sampIing with replacement, with N h . f th I· t e Slze .0 e popu atl~n and n the Slze of the samp e. - sample survey a survey based on the responses of a sample of individuals, rather than the entire population.

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- sample variance I the sample variance is the square of the sample standard I deviation S. It is an unbiased I estimator of the square of the : population standard deviation, ~ which is also called the variance I of the population. - sample a sample is a collection of units from a population. - sampling distribution

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the sampling distribution of an estimator is the probability distribution of the estimator when it is applied to random samples. The tool on this page allows you to explore empirically the sampling distribution of the sample mean and the sample percentage of random draws with or without replacement draws from a box of numbered tickets. _ lin samp g error in estimating from a random sample, the difference between the estimator and the parameter can be written as the sum of two components: bias and sampling error. The bias is the average error of the estimator over all pos,sible' samples. The bias is not random. Sampling error is the component of error that varies from sample to sample. The sampling error is random: it comes from "the luck of the draw" in which units happen to be in the sample. It is the chance variation of the estimator. The average of the sampling error over all possible samples (the expected value of the sampling error)

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sampling unit I s c o r e . .

is zero. The standard error of the estimator is a measure of the typical size of the sampling error. • sampling unit a sample from a population can be drawn one unit at a time, or more than one unit at a time (one can sample clusters of units). The fundamental unit of the sample is called the sampling unit. It need not be a unit of the population. • scale factor the ratio of corresponding lengths in similar figures . • scalene triangle a triangle with three sides of different lengths.

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• scatterplot a scatterplot is a way to visualise bivariate data. A scatterplot is a plot of pairs of measurements on a collection of "individuals" (which need not be people). For example, suppose we record the heights and weights of a group of 100 people. The scatterplot of those data would be 100 points. Each point represents one person's height and weight. In a scatterplot of

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weight against height, the xcoordinate of each point would be height of one person, the y-coordinate of that point would be the weight of the same person . In a scatterplot of height against weight, the x-coordinates would be the weights and the y-coordinates would be the heights . • scientific notation a notation for expressing very large and very small numbers as a product of a decimal number greater than or equal to one and less than ten and a power of ten. • score a linear combination of an observed set of measured variables. The coefficients for the linear combination are usually determined by some matrix computation. Multivariate statistical findings in the form of coefficient vectors can usually be more easily interpreted if scores are also shown case by case, their scatters, their loadings (correlations with the original variables), etc.

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sd line I selection bias

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=========* • sd line for a scatterplot, a line that goes through the point of averages, with slope equal to the ratio of the standard deviations of the two plotted variables. If the variable plotted on the horizontal axis is called X and the variable plotted on the vertical axis is called Y, the slope of the SD line is the SD of Y, divided by the SD of X. • se(sample mean) = n -V2xSD(box,where SD(box) is the standard deviation of the list of the numbers on all the tickets in the box (including repeated values). • se(sample sum) = nl/2 x SD(box),and the standard error of the sample mean of n random draws with replacement from a box of tickets is

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• section (of a solid) An intersection with a plane. • sector of a circle the region between a central angle and the arc it intercepts.

• segment aka line segment; the set of points consisting of two distinct points and all inbetween them. • segment of a circle the region between a chord and the included arc.

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• secant a line that intersects a circle in two points.

secular trend a linear association (trend) with time.

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• selection bias a systematic tendency for a sampling procedure to include and/or exclude units of a certain type. For example, in a quota sample, unconscious prejudices or predilections on the part of the interviewer can result in selection bias. Selection bias is a potential problem whenever a human has latitude in selecting individual

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units for the sample; it tends to be eliminated by probability sampling schemes in which the interviewer is told exactly whom to contact (with no room for individual choice).

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*================ ~ - semicircle i an arc of a circle whose end: points are the endpoints of a ~ diamete~ -

~ - semi-magic square i a square array of n numbers - self-selection : such that sum of the n numbers self-selection occurs when in- ~ in any row or column is a condividuals decide for them- I stant (known as the magic selves whether they are in the ~ sum). control group or the treat'gu1ar t esse11a0'on ".. _ semtre . ment group" Self-selectIon IS I t " t'lllg 0 f " " " " a esse 11 a t"lon conS1S qUlte common III studles of i l l all f h human behaviour. For ex- : reg~ ar PI? ygons hOw" ose f h ffi f I vertIces le on ot er vertIces, " amp1e, stud les 0 tee ect "0 : an d"tn W h"lC h every vertex IS " ki h h alth smo ng on ~an "e " . tn- I surrounded by the same arvolve self-selectIon: tndlVldu- : t f' I ( t' J: h 1 I range men 0 po ygons 0 a Is choose lor t emse ves : one or more k'In d s )"III t h e whether or not to smoke. ~ same order. Also called a 1Self-selection precludes an ex- i uniform tiling. periment; it results in an observational study. When there i- septagon is self-selection, one must be : a seven-sided polygon I wary of possible confounding : - sequence from factors that influence ~ a collection of numbers in a preindividuals' decisions to be- ; scribed order: a ~, ~, a , ••• p 4 long to the treatment group. i-series - self-similarity : the sum of a fmite or infinite the property of a figure that it :I sequence is similar to, or approximately ~ - set similar to, a part of itself. i a set is a collection of things, :I without regard to their order. r

set ofdam ptlints I shape lIariable

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• set of data points data collected and placed into ordered pairs for the purpose of graphing. • seven bridges of konisberg network a collection of designated points connected by paths.

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shape coordinates in the past, any system of distance-ratios and perpendicular projections permitting the exact reconstrUction of a system of landmarks by a rigid trusswork. Now, more generally, coordinates with respect to any basis for the tangent space to Kendall's shape space in the vicinity of a mean form.

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• shape the geometric properties of a I configuration of points that . h . mvanant to c anges m . are . . translation, rotation, and I scale. In morphometries, we represent the shape of an ob- I . ject by a point in a space of ~ shape variables, which are' ~ measurements of a geometric ; object that are unchanged under similarity transforma- ; tions. For data that are con- : figurations of landmarks, I there is also a representation I of shapes per se, without any : nuisance parameters (posi- ~ tion, rotation, scale), as single I points in a space, Kendall's shape space, with a geometry given by Procrustes distance. I Other sorts of shapes (e.g., I those of outlines, surfaces, or : functions) correspond to quite I different statistical spaces.

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shape space a space in which the shape of a figure is represented by a single point. It is of 2p-4 dimens ions for 2-dimensional coordinate data and 3p-7 dimens ions for 3-dimensional coordinate data.

• shape variable any measure of the geometry of a biological form, or the image of a form, that does not change under similarity transformations: translations, rotations, and changes of geometric scale (enlargements or reductions). Useful shape variables include angles, ratios of distances, and any of the sets of shape coordinates that arise in geometric morphometrics.

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*================ • shear in two-dimensional problems, shape aspects of any affine transformation can be diagrammed as a pure shear, a map taking a square to a parallelogram of unchanged base segment and height. This is a transformation that leaves one Cartesian coordinate, y, invariant and alters the other by a translation that is a multiple ofy: for instance, what happens when you slide the top of a square sideways without altering its vertical position or the length of the horizontal edges. The score for such a translation, together with a separate score for change in the horizontal/vertical ratio, supplies one orthonormal basis for the subspace of uniform shape changes of twodimensional data. Without the adjective "pure," geometric morphometricians usually use the word "shear" as an informal synonym for "affme transformation," since any 2D uniform transformation can be drawn as one if you wish. In multivariate morphometries, a somewhat different use of pure shear is in a transformation of the "shape principal compo-

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nents" of an allometric analysis of distances to be uncorrelated with within-group size . • side (of a polygon) A line segment connecting consecutive vertices of a polygon.

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: • side of a polygon ~ a single segment from the union I that forms a polygon I • sides : (of an angle) The two rays, hav~ ing a common endpoint, that ~ form an angle. I • Sierpinski triangle ~ a type of fractal. : .• significance ~ also known as , significance I level, statistiCal significance. : The significance level of an hy~ pothesis test is the chance that ~ the test erroneously rejects the ; null hypothesis when the null : hypothesis is true.

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: • similar figures ~ two geometric figures are simi; lar if their sides are in propor~ tion and all their angles are the . same. ~ • similar polygons ; polygons whose corresponding : angles are congruent and whose I

MJr.themtlties======~ II

!!!!:1!!!!!56~~~~~~~~~~:laritytransjimJmtiun lsimulatiun

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simple random sample a simple random sample of n units from a population is a ran6 I dom sample drawn by a proceL I dure that is equally likely to give every collection of n units 4 from the population; that is, the I probability that the sample will r consist of any given subset of n of the N units in the population I is Iren. Simple random sampling is sampling at random without replacement (without • similarity transformation I replacing the units between a change of Cartesian coordi- draws). A simple random nate system that leaves all ra- sample of size n from a poputios of distances unchanged. I lation of N > = n units can be The term proper or special simi- I constructed by assigning a ranlarity group of similarities is dom number between zero and sometimes used when the trans- one to each unit in the populaformations do not involve re- I tion, then taking those units flection. Similarities are arbi- that were assigned the n largtrary combinations of transla- : est random numbers to be the tions, rotations, and changes of I sample. scale. : I • Simpson's paradox what is true for the parts is not • simple events a single activity in a probability necessarily true for the whole. experiment such as flipping a :I • Slm • ul . ation com. ~ ·an experiment that has the corresponding sides are proportional.

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; same number of outcomes as a • simple polygons convex, closed shapes bounded given situation but is easier or by line segments joined end to I more practical to carry out than the given situation. end.

Ii sine I slide rule - sine (ofcin acute angle) The ratio of the. length of the opposite side to the length of the hypotenuse in any right triangle containing the angle. - singular value decomposition any mxn matrix X may be decomposed into three matrices U, D, V (with dimensions mxm, mxn, and nxn, respectively) in the form: X= UDVt, where the columns of U are . orthogonal, D is a diagonal matrix of singular values, and the columns of V are orthogonal. The singular value decomposition of a variance-covariance matrix S is written as S=ELEt, where L is the diagonal matrix of eigenvalues and E the matrix of eigenvectors. - size change factor size change magnitude

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measures of dimension one, i areas are size measures of di: mension two, etc. •

: - skeleton division ~ a long division in which most i or all of the digits have been : replaced by asterisks to form a :• cryptan. . ·thm ·• _skew i two lines are skew if they do not ~ intersect and are noncoplanar. : - skew lines ~ non-coplanar lines that don't • intersect

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~ - skewed distribution ; a distribution that is not sym: metrical. •

.: - slant height - size measure ~ the height of each triangular in general, some measure of a ~ lateral face of a pyramid. form (i. e., an invariant under the group of isometries) that • - slide rule scales as a positive power of the : a calculating device consisting geometric scale of the form. ~ of two sliding logarithmic Interlandmark lengths are size • scales.

Mathematics=================== II

158 • slope the ratio of the increase in the y-values to the increase in the x-values between any two ordered pairs.

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slope-intercept form the form of a linear equation y = fiX + b where m represents the slope and b represents the y-intercept.

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• slope of a line I • small circle in a coordinate plane, the the circle formed by the interamount of vertical change : sectiCJn of a sphere and a plane (change in y) for each unit of I that doesn't contain the center horizontal change (change inx). ; • solid The slope of a vertical line is the union of the surface and the undefined. You can calculate the I region of space enclosed by a 3slope m of a line (or line seg- I D figure; examples: conic solid, ment) through points with co- cylindric solid, rectangular solid ordinates (xl,yl) and (x2,y2) I • solid geometry using the formula m = the study of figures in three-di(Y2 - Yl) I mensional space ( ~ - Xl)

solid of revolution a solid formed by rotating a two-dimensional figure about a line.

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• slope triangle a right triangle used to help fmd the slope of a line or line I segment through two points, I • solidus which are used as the end- the slanted line in a fraction such points of the hypotenuse. The as a/b dividing the numerator length of the triangle's verti- I from the denominator. cal leg is the "rise." The length of the horizontal leg is • space the "run." Signs are attached I in statistics, a collection of obto each quantity depending on : jects or measurements of obthe direction of travel along ~ jects, treated as if they were the legs between the points. ; points in a plane, a volume, on the surface of a sphere, or on any higher-dimensional

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II sphere IsttuuJ,srd units generalisation of these intuitive strUctur~s. Examples are: Euclidean spaces, sample spaces, shape spaces, linear vector spaces, etc.

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of deviations between each ele; ment of the set and the mean : of the set. I

: • standard error ~ the Standard Error of a random ; variable is a measure of how far • sphere the locus of pointsin three-space : it is likely to be from its exthat are a fIXed distance froma ~ pected value; that is, its scatter given point (called the center). I in repeated experiments. The :I SE of a random variable X is .sphericru oigonometty the branch of mathematics deal- : defmed to be SE(X) = [E( (X I E(X))2 )] V2. That is, the staning with measurements on the ~ dard error is the square-root of sphere. : the expected squared difference I between the random variable • square a quadrilateral with 4 equal ; and its expected value. The SE sides and 4 right angles. : of a random variable is analo~ gous to the SD of a list. • square free an integer is said to be square ~ • standard form free if it is not divisible by a ; the form of a number expressed : as a sum of products involving perfect square, n2, for n> 1. • square number ~ powers of ten. a number of the form n2 • ~ • standard units • square-root law ; a variable (a set of data) is said the Square-Root Law says that: to be in standard units if its the standard error (SE) of the ~ me~n .is z.ero and its standard sample sum of n random draws ~ dCV1anon IS ~ne. You transfor.m with replacement from a box of ; a set of dat~ mto standard uruts tickets with numbers on them : by subtracnng the mean from IS I each element of the list, and di~ viding the results by the stan• standard deviation : dard deviation. A random varithe standard deviation of a set ~ able is said to be in standard of numbers is the rms of the set Mathcmatics=======

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units if its expected value is zero and its standard error is one. You transform a random variable to standard units by subtracting its expected value then dividing by its standard error. • standardise to transform into standard units.

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scale versus interval scale versus ratio scale. straight angle an angle whose measure is 180 degrees, forming a line with its sides

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straightedge a tool used to construct straight lines.

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• statistic : • straightedge, unmarked a number that can be computed I just how it sounds, an unfrom data, involving no un- : marked tool used to draw known parameters. As a func- I straight lines tion of a random sample, a statistic is a random variable. Sta- • stratified tistics are used to estimate pa- I this is a feature of an experirameters, and to test hypoth- mental design whereby a scheme of observations is reeses. I peated entirely using further • stem and leaf plots sets (strata) of experimental a method of displaying data I units, with each such further where the leading digit (s) are I set distinguished by a level of the stem and the ending single a categorical variable which is dio1ts are arranged in ascend- I 0distinct from any categorical ing order to the side represent- I variables used to define the ing the leaves. I experimnad design within a single set (stratum). The data • Stevens' typology this is widely-observed scheme : from the various strata are reof distinctions between types of I garded as distinct. This situameasurement scales according : tion occurs when attempting to the meaningfulness of arith- to make inferences based metic which may be performed I upon the results of several upon data values. The types are similar independent experi: nominal scale versus ordinal ments.

II stmriftetlsmnple I stulimt'stcurPe • stratified. sample in a stratified sample, subsets of sampling units are selected separately from different strata, rather than from the frame as a whole.

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~ vary enormously with location. ; We might divide the country : into states, then divide each ~ state into urban,. suburban, and ; rural areas; then draw random : samples separately from each ~ such division.

• stratified. sampling the act of drawing a stratified ~ • studentised score ; the observed value a statissample. :" tic, minus the expected value of • stratum ~ the statistic, divided by the esin random sampling, someI timated standard error of the times the sample is drawn separately from different disjoint ; statistic.

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subsets of the population. Each such subset is called a stratum. (The plural of stratum is strata.) Samples drawn in such a way are called stratified samples. Estimators based on stratified random samples can have smaller sampling errors than estimators computed from simple random samples of the same size, if the average variability of the variable of interest within strata is smaller than it is across the entire population; that is, if stratum membership is associated with the variable. For example, to determine average home prices in the India., it would be advantageous to stratify on geography, because average home prices

~ • student's t curve : student's t curve is a family of ~ curves indexed by a parameter ; called the degrees of freedom, : which can take the values 1, ~ 2, ... Student's t curve is used ; to approximate some prob: ability histograms. Consider a ~ population of numbers that ; are nearly normally distrib: uted and have population ~ mean is J.L. Consider drawing I a random sample of size n with : replacement from the popula~ tion, and computing the ~ sample mean M and the ; sample standard deviation S. : Define the random variable T ~ = (M J.L)/(S/nV2). If the ; sample size n is large, the : probability histogram ofT can I

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be approximated accurately ~ ment of the subset must beby the normal curve. How- ; long to the original set, but ever, for small and intermedi- not every element of the origiate values of n, Student's t 1 nal set need' be in a subset curve with n I degrees of free- I (otherwise, a subset would dom gives a better approxima- always be identical to the set tion. That is, P(a < T < b) is it came from). approximately the area under I . •• Student's T curve with n I : - successlve apprOXimatiOn 1 a sequence of approximations, h cl th degrees of freedom, from a to· b. Student's t curve can be ; eac one oser to e desired : value, used to test hypotheses about the population mean and con- ~ - sufficient condition struct confidence intervals for 1 a version of a conditional that the population mean, when tells you when you can use the the population distribution is term defined, where the term known to be nearly normally 1 is in the consequent; a condition distributed. This page con- 1 that implies a preset conclusion. tains a tool that shows _ superimposition Student's t curve and lets you ~ the transformation of one or find the area under parts of the 1 more figures to achieve some curve. I geometric relationship to an_ subject, experimental other figure. The transformaI tions are usually affine transsubject a member of the control group I formations or similarities. or the treattnent group. : They can be computed by ~ matching two or three land- subroutine 1 marks, by least-squares optia previously known algorithm: mization of squared residuals used in another algorithm ~ at all landmarks, or in other _ subset ~ ways. Sometimes informally a subset of a given set is a col- ; referred to as a "fit" or "fitlection of things that belong: ring," e.g., a resistant fit. to the original set. Every ele- I

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~ gram will be symmetrical about • supplementary two angels are supplementary ; a vertical line drawn at x=a. of they add up to 18()<>. ; • symmetric property of congruence • supplementary angles 2 angles whose measures, when ~ the property of congruent figadded together, equal 180 de- ; ures that, if one geometric fig: ure is congruent to a second figgrees ~ ure, then the second figure is • surface ~ congruent to the first. the boundary of a 3-D figure I • symmetry diagonal • surface area : the diagonal that perpendicu(of a solid), the sum of the ar~ larly bisects, the other and is a eas of all the surfaces. ~ symmetry line for the kite • symmedian reflection of a median of a triangle about the corresponding I angle bisector.

• symmetric distribution the probability distribution of a random variable X is symmetric if there is a number a such that the chance that X> =a + b is the same as the chance that X < =a-b for every value ofb. A list of numbers has a symmetric distribution if there is a number a such that the fraction of numbers in the list that are greater than or equal to a + b is the same as the fraction of numbers in the list that are less than or equal to a-b, for every value of b. In either case, the histogram or the probability his to-

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symmetry line ~ the line of reflection in a reflec: tion-symmetric figure I •

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: • system of equations ~ a set of two or more equations. ~

• systematic error ; an error that affects all the : measurements similarly. For ~ example, if a ruler is too short,

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systematic random sample I tl-tm

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=================* everything measured with it will appear to be longer than it really is (ignoring random error) . If your watch runs fast, every time interval you measure with it will appear to be longer than it really is (again, ignoring random error). Systematic errors do not tend to average out.

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they were random, if the order in which the units appears in the list is haphazard. Systematic samples are a special case of cluster samples. t test an hypothesis test based on approximating the probability histogram of the test statistic by Student's t curve. t tests usually are used to test hypothes.es about the mean of a population when the sample size is intermediate and the distribution of the population is known to be nearly normal. • t2 statistic

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• systematic random sample a systematic sample starting I at a random point in the listing of units in the of frame, instead of starting at the first I unit. Svstematic random sampIing is better than systematic sampling, but typically not as I good as simple random sam- I piing. I • systematic sample a systematic sample from a I frame of units is one drawn by I listing the units and selecting I every kth element of the list. For example, if there are N I units ifi the frame, and we I want a sample of size N/I0, we would take every tenth I unit: the first unit, the eJev- I enth unit, the 21st unit, eLc. Systematic samples are not I random samples, but they of- I ten behave essentially as if :

a multivariate generalisation of the univariate t 2 statistic. It is . 0 f th e t he square 0 f t h e ratIo group mean difference to the standard error of that difference. Used in the 'P-test. t 2 -test a test due to Hotelling for comparing an observed mean vector to a parametric mean; or comparing the difference between two mean vectors to a parametric difference (usually the zero vector). If the observations are independently multivariate normal, then the 'P-

•

II = = = = = = = M R . t h e m R . t i c s

II :able ojJlalues I tangent space

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*================= test may be used to test null hypotheses using the F-distribution. T2 is also closely related to Mahalanobis D2. • table of values a table of two colLUnns, the first representing values of the independent variable, the second representing the values of the dependent variable.

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actual outcome as part of the tail.

tangent (of an acute angle) The ratio of I the length of the opposite side I to the length of the adjacent side in any right triangle containing the angle. I • 1 : . tangent clr~ es I •

I circles that are tangent to the : same line at the san1e point. • tail an area at the extreme of a ~ They can be internally tangent randomisation distribution, I or externally tangent. where the degree of extremity is sufficient to be notable judged against some nominal I alpha criterion value.

• tail definition policy this is a defined method for dividing a discrete distribution into a tail area and a body area. the scope for differing policies arises due to the noninfinitesmal amount of probability measure which may be associated with the actual outome value. The conventional policy, based upon considerations of simplicity and of conservatism in terms of alpha, is to include the whole of the weight of outcomes equal to the

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: • tangent line ~ a line that lies in the plane of a ; circle and that intersects the : circle at exactly one point (the ~ point of tangency). ~ • tangent segment ; a line segment that lies on a tan: gent line to a circle, with one ~ endpoint at the point of tanI gency. I • tangent space : informally, if S is a curving space ~ and P a point in it, the tangent

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tangmtitU veIoeity I tensor

II

========* space to S at P is a linear space I it left the circle along a tangent T having points with the same I line. "names" as the points in S and I • tautology in which the metric on S "in the a sentence that is true because vicinity of P" is very nearly the I of its logical structure. ordinary Euclidean metric on T. One can visualise T as the pro- I • tensor jection of S onto a "tangent an example of a tensor in plane" "touching" at P just like morphometrics is the represena map is a projection of the sur- tation of a uniform component face of the earth onto flat pa- I of shape change as a transforIn geometric I mation matrix. The transformaper. morphometrics, the most rel- tion matrix assigns to each vecevant tangent space is a linear I tor in a starting (or average) vector space that is tangent to form a vector in a second form. Kendall's shape space at a point A rigorous, general definition corresponding to the shape of ~ of a tensor would be beyond a reference configuration (usu- . the scope of this glossary, but a intUltIve ally taken as the mean of a reasonably sample of shapes). If variation I characterisation comes from in shape is small then Euclid- I Misner, Thorne, and Wheeler, ean distances in the tangent Gravitation (Freeman, 1973): a space can be used to approxi- I tensor is a "geometric machine" mate Procrustes distances in I that is fed one or more vectors Kendall's shape space. Since the in an arbitrary Cartesian coortangent space is linear, it is pos- dinate system and that produces sible to apply conventional sta- I scalar values (ordinary decimal tistical methods to study varia- numbers) that are independent tion in shape. of that coordinate system. In I morphometrics, these "num"• tangential velocity bers" will be ordinary geomet(0f an object moving in a circle) I ric entities like lengths, areas, The speed of the moving object I in the direction it would take if : or angles: anything that doesn't I change when the coordinate system changes. For the represen-

II=======MII~

II terminal side I tetrahedron

167

*=================

tation of a uniform component ~. - tessellation as a transformation matrix, the I an arrangement of shapes "scalars" of the Misner.:rhorne- : (called tiles) that completely Wheeler metaphor arc the ~ LOvers a plane without overlaps lengths of the resulting vectors I or gaps. and the angles among them. A different tensor representing the same uniform transforma- I tion is the relative metric tensor, which you probably know as the ellipse of principal axes and I principal strains. This tensor produces the necessary numerical invariants (distances in the ~ second form as a function of ; . . . - test statistic coordmates on the first form) : . . d h th directly. Other tensors include ~ a sta~ch use hto . test ypo ~ the metric tensor of a curving ; eses. dYPOb t des1s'dt~st can di . constructe y eCl 109 to re, sun.ace Which expresses stance·. th ull h h' h th on the surface as a function of ~ Ject e n ypot eSls.w. e~ . e . wh'1Ch sunc.ace I. value of the test statistic. IS 10 the parameters 10 . d d h some range or collection of pomts are expresse an t e T . h cUrPature tensor of the same sur- I ran~es. J.~ ~et a test WIt a c. Which expresses teh way·10 I speCified slgmficance level, the . lace · h the surlace C. "C._II chance when the null hvpothes1s lauS awa y" . -. . w h1C fi' I I IS true that the test statistic falls r~m itS tangent p ane at any ~ in the range where the hvpothpomt. . wo uld be rqecte . dmust ' be : eS1S - terminal side ~ at most the specified signifithe side that the measurement ; cance level. The Z statistic is a of an angle ends at : common test statistic. I

- tesselate : the ability of a regIOn to ~ tessalate ; : I

- tetrahedron a polyhedron with four faces. The regular tetrahedron is one of the Platonic solids.

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• tetromino a four-square polyomino.

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• the five platonic solids the five regular polyhedrons: regular tetrahedron, regular icosahedron, regular octahedron, regular hexahedron, and regular dodecahedron. • theorem important mathematical statements which can be proven by postulates, definitions, and/or previously proved theorems. • thin-plate spline in continuum mechanics, a thin-plate spline models the form taken by a metal plate that is constrained at some combination of points and lines and otherwise free to adopt the form that minimises bending energy. (The extent of bending is taken as so small

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that elastic energy stretches and shrinks in the plane of the original plate can be neglected.) One particular version of this problem an infinite, uniform plate constrained only by displacements at a set of discrete points-can be solved algebraically by a simple matrix inversion. In that form, the technique is a convenient general approach to the problem of surface interpolation for computer graphics and computer-aided design. In morphometrics, the same interpolation (applied once for each Cartesian coordinate) provides a unique solution to the construction of D'Arcy Thompson-type deformation grids for data in the form of two landmark configurations. three types of proofs direct proof a proof in which you state premises, then use valid forms of reasoning to arrive directly at a conclusion.

I •

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• three-dimensional having length, width, and thickness (i.e., space)

II =======MRthmulriu

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hometrics • =========1=6=9

tied rilniu I trlUlitiolUJl mM1

- tied ranks in a non parametric test involving ranked data, if two data have tied values then they will deserve to receive the same rank value. it is generally agreed that this should be the average of the ranks which would have been assigned if the values had been discernably unequal. Thus, the ranks assigned to a set of 6 data, with ties present might emerge as sets such as : 1,3,3,3,5,6 or 1,2,3.5,3.5,5,6. The possibility of tied ranks leads to elaborations in the otherwise-standard tasks of computing or tabulating randomisation distributions where data are replaced by ranks. _ tied values where data are represented by ranks, tied values lead to tied ranks. whether or not data are rep [resnted by ranks, for any test statistic the occurrence of tied values will increase the extent to which a randomisation distribution will be a discrete distribution rather than a continuous distribution.

~

what happens when you kick ; the pinball machine too hard. ; - Toeplitz matrix a matrix in which all the ele~ ments are the same along any ; diagonal that slopes from north: west to southeast. I

all a l2 an !lIZa]1 a]Z I

a l3 a lZ all

al3 I

al2 aJ3 a l2 all

; - torus a 3-D figure formed by rolling I a rectangle into a cylirtder and I bending the cylinder until its bases meet; a "doughnut". I

: - traee ~ the trace of a matrix is the sum ; of the terms along the princi: pal diagonal. I

: - traditional morphometries ~ application of multivariate I statistical methods to arbi: trary collections of size or ~ shape variables such as · disI tances and angles. "Tradi~ tional morphometries" differs - tilt : from the geometric the measure of an angle as ~ morphometries discussed here compared to a horizontal line;

MRtm-#Cs==================== II

=17=O=========~talnumber I transformation II in that even though the dis- I formations are used to put tances or measurements are I variables in standard units . In defined to record biologically that case, you subtract the meaningful aspects of the or- I mean and divide the results by ganism, but the geometrical I the SD. This is equivalent to relationships between these multiplying by the reciprocal measurements are not taken of the SD and adding the into account. Traditional I negative of the mean, divided morphometrics makes no ref- by the SD, so it is an affine erence to Procrustes distance transformation. Affine transor any other aspect of ~ formations with positive mulKendall's shape space. . tiplicative constants have a • transcendental number simple effect on the mean, I median, mode, quartiles, and a number that is not algebraic. other percentiles: the new • transformation value of any of these is the old transformations turn lists into lone, transformed using exother lists, or variables into I acdy the same formula. When other variables. For example, the multiplicative constant is to transform a list of tem- I negative, the mean, median, peratures in degrees Celsius I mode, are still transformed by into the corresponding list of : the same rule, but quartiles temperatures in degrees Fahr- ~ and percentiles are reversed: enheit, you multiply each ele- I the qth quantile of the transment by 9/5, and add 32 to formed distribution is the each product. This is an ex- transformed value of the 1ample of an affine transforma- I qth quantile of the original tion: multiply by something distribution (ignoring the efand add something (y = ax + fect of data spacing). The efb is the general affine trans- I feet of an affine transformaformation of X; it's the famil-tion on the SD, range, and iar equation of a straight line). IQR, is to make the new value In a linear transformation, I the old value times the absoyou only multiply by some- I lute value of the number you thing (y = ax). Affine trans- multiplied the first list by:

II

tmnsformationnotation

I treatment~==========17=1 ~

of the translation), and have the ; same length (the distance of the : translation) . I

: - translation vector ~ see translation ~ - transversal ; a line that intersects 2 others

; - transversible a network in which all arcs can -2 ~ what you added does not affect be traced without going over lone more than once them. - transformation notation t(P), which stands for the transformation of P; also Sk where the transformation S that maps (x, y) onto (kx, ky) and k is the magnitude of that transformation

I - trapezium : a quadrilateral in which no sides I : are parallel.

:I - trapeZOl'd I a quadrilateral with exactly one ~ pair of parallel sides. The par: allel sides are called bases. A ~ pair of angles that have a base - transitive property of ; as a common side are called a congruence : the property that, if one geo- I pair of base angles. metric object is congruent to a second object, which in turn is , congruent to a third, then the first and third objects are congruent to each other.

,

- translation : - treatment effect an isometry under which the I the effect of the treatment on vectors (any of which is a trans- ~ the variable of interest. Establation vector) between each : lishing whether the treatment point and its image are all parallel (determining the direction

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172

matmmtgroup I trinomud

II

=================* has an effect is the point of an experiment. • treatment group the individuals who receive the treatment, as opposed to those in the control group, who do not. • treatment the substance or procedure studied in an experiment or observational study. At issue is whether the treatment has an effect on the outcome or variable of interest. • tree a tree is a graph with the property that there is a unique path from any vertex to any other vertex traveling along the edges. • tree diagram a concept map in the form of tl1e branches of a tree. You can use it to show the relationships among members of a family of concepts. • triangle a polygon with three sides. • triangle inequality the property that states that the length of any side of a triangle

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is less than the sum of the lengths of the other two sides. triangular numbers numbers of dots that can be put into triangular arrangements; equivalently, sums of consecutive positive integers beginning with 1.

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• triangulate to divide a polygon into triangles

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• tridecagon a 13-sided polygon

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• trigon a three-sided polygon. trigonometric ratios and the unit circle unit circle a circle on the coordinate plane with center (0, 0) and radius 1 unit. periodic curve A curve that repeats in a regular pattern. period The horizontal distance between corresponding points on adjacent cycles of a periodic curve.

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. ; : ~

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• trigonometry the study of the relationships between the measures of sides and angles of triangles. trinomial an algebraic expression consisting of 3 terms.

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173

tromino I two-sided hypothesis test

*================ - tromino a three-square polyomino.

~

'. truncated pyramid part of a pyramid remaining I after truncating the vertex with a plane parallel to the ~ base.

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: - twin primes I two prime numbers that differ : by 2. For example, 11 and 13 I are twin primes. ~ : _ two-column proof a form of proof in which each statement in the argument is written in the left column, and the reason for each statement is written directly across from it in the right column. _ two-column proof a form of proof in which each statement in the argument is written in the left column, and the reason for each statement is written directly across from it in the right column. - two-dimensional having both width and length, but no thickness - two-point perspective a method of perspective drawing that uses two vanishing points.

~

; :

_ two-point shape coordinates a conveniel1t system of shape coordinates, originally Francis Galton's, rediscovered by Bookstein, consisting (for twodimensional data) of the coordinates of landmarks 3, 4, ... after forms are rescaled and .. repositloned so that landmark 1 is fixed at (0,0) and landmark 2 is fixed at (1, 0) in a Cartesian coordinate system. Also referred to as Bookstein coordinates or Bookstein's shape coordinates.

I

: - two-sided hypothesis test ~ c.f. one-sided test. An hypoth; esis test of the null hypothesis : that the value of a parameter, ~ /-L, is equal to a null value, /-LO, I designed to have power against : the alternative hypothesis that ~ either /-L < /-LO or /-L > /-LO (the I alternative hypothesis contains ~ values on both sides of the null : value). For example, a signifII cance level 5%, two-sided z test ~ of the null hypothesis that the : mean of a population equals ~ zero against the alternative that ; it is greater than zero would : reject the null hypothesis for ~ values of

1\

174

=================* (sample mean) I Izl=----->

1.96.SE(sample mean)

~

true. A 1YPe 2 error occurs if ; the null hypothesis is not re: jected when it is in fact false. I

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• type 2 landmark a mathematical point whose claimed homology from case to case is supported only by geometric, not histological, evidence: for instance, the sharpest curvature of a tooth.

I • two-way table a representation of suitable data in a table organised as rows and columns, such that the rows rep- I resent one scheme of alternatives covering the whole of the • type 3 landmark the data represented, the col- I a landmark having at least one umns represent a further I deficient coordinate, for inscheme of alternatives covering stance, either end of a longest the whole of the data repre- I diameter, or the bottom of a sented, and the entries in the I concavity. Type 3 landmarks two-way table are the counts of : characterise more than one renumbers of observations con- I gion of the form. The multiforming to the respective cells I variate machinery of geometric of the two-way classification. I morphometries permits them to be treated as landmark • type 1 landmark a mathematical point whose I points in some analyses, but the claimed homology from case to I deficiency they embody must case is supported by the stron- be kept in mind in the course gest evidence, such as a local of any geometric or biological pattern of juxtaposition of tis- I interpretation. sue types or a small patch of : estimator some unusual histology. ~ •anunbiased estimator, , that has as its expected value the parametric • type land type 2 errors these refer to hypothesis test- I value, q, it is intended to estiing. A Type 1 error occurs when mate: .. the null hypothesis is rejected erroneously when it is in fact I • unbiased not biased; having zero bias.

II uncontrolled experiment I uniformS;""cmn='-P""onetJ=""t======",,1~7~5 _ uncontrolled experiment an experiment in which there is no control group; i.e., in which the method of comparison is not used: the experimenter decides who gets the treatment, but the outcome of the treated group is not compared with the outcome of a control group that does not receive treatment.

~ - undefined term ; in a deductive system, terms : that are assumed, and assigned ~ no properties, "and whose mean; ing is derived only from the : postulates or axioms that use ~ them. In our (Euclidean) sysI tern, the undefined terms are ~ point, line, plane, and space.

: - uniform shape component ~ that part of the difference in I shape between a set of con~ figurations that can be mod: eled by an affine transforma~ tion. Once a metric is supplied ; for shape space one can ascer: tain which such transforma~ tion takes a reference form ; closest to a particular target : form. For the Procrustes met~ ric (the geometry of Kendall's ; shape space), that uniform _ uncountable : a set is uncountable if it is not transformation is computed ~ by a formula based in countable. I Procrustes residuals or by an_undecagon ~ other based in two-point an eleven-sided polygon. : shape coordinates. Together ~ with the partial warps, the uni; form component defined in : this way supplies an orthonor~ mal basis for all of shape space ; in the vicinity of a mean form. : In this setting, the uniform ~ shape component may also be

_ uncorrelated a set of bivariate data is uncorrelated if its correlation coefficient is zero. 1\:vo random variables are uncorrelated if the expected value of their product equals the product of their expected values. If two random variables are independent, they are uncorrelated. (The converse is not true in general.)

MIIthmuJtics==========

II

""1,,,,76===========* unilateralsuiftue I unitarydi'Pisor \I

interpreted as the projection ~. - union of a shape difference (be- I the union of two or more sets tween two group means, or is the set of objects contained between a mean and a particu- I by at least one of the sets. The lar specimen) into the plane I union of the events A and B is (or hyperplane for data of di- denoted "A+B", "A or B", mension greater than two) and '1\.UB". C.f. intersection. through that mean form and I _ unit analysis all nearby forms related to it I the process of using conversion by affine transformations. For factors to change from one descriptive purposes, the uni- I measure or rate to another. form component is parameterised not by a vector, - unit circle like the partial warps, but by I a unit circle is a circle with raa representation as a tensor, I dius 1. in terms of sets of shears and - unit cube dilations with respect to a I a cube with edge length 1. fixed, orthogonal set of Car- I - unit fraction tesian axes. I with one as a numerator and a - unilateral surface natural number as a denominaa surface with only one S1"de, I tor. such as a Moebius strip. _ unit of attribute _ unimodal I the unit chosen depends upon a finite sequence is unimodal if : the attribute being measured it first increases and then de- ~ (e.g., for the attribute "length", creases. I a tmit might be "meters")" _ unimodular - unit square a square matrix is unimodular I a unit square is a square of side if its determinant is 1. length 1. _ union of two sets a and b _ unitary divisor the set of elements in A, B, or I a divisor d of c is called unitary both; written AUB I if gcd(d,c/d) = 1.

II ===================MJJth_tics

177 *==~~~~~~~

- unity one - univariate having or having to do with a single variable. Some univariate techniques and statistics include the histogram, IQR, mean, median, percentiles, quantiles, and SD. C.f. bivariate. _ universal statement a conditional that uses the words 'all' or 'everything' _ universe in a Venn diagram, everything that is outside the sets

_ variable i a numerical value or a charac: teristic that can differ from in~ dividual to individual. Variance, ; population variance The vari: ance of a list is the square of the ~ standard deviation of the list, ; that is, the average of the : squares of the deviations of the ~ numbers in the list from their I mean. The variance of a random ~ variable X, Var(X), is the ex: pected value of the squared difI ference between the variable ; and its expected value: Var(X) : = E( (X E(X) )2). The variance ~ of a random variable is the ; square of the standard error : (SE) of the variable. ~

- upper bound any number above which a function value may approach but not I : pass. ~ - valid reasoning ; an argument that reaches its : conclusion through accepted ~ forms of reasoning. I _ vanishing line : the horiwn; in a drawing it is ~ at the height of viewer's eye ~ _ vanishing point a point toward which lines in a perspective drawing converge if they represent parallel lines that recede from the foreground to the background.

- vector a quantity that has both magnitude and direction, usually represented by an arrow with a head and tail. The vector's direction is indicated from the tail to the head. The vector's magnitude is the length of the line segment.

I - velocity ~ the rate of change of position. : _ Venn diagram ~ a pictorial way of showing the I relations among sets or :

178

events. The universal set or I • vertex arrangement outcome space is usually I a notation that uses positive drawn as a rectangle; sets are integers and other symbols to regions within the rectangle. I describe the arrangement of The overlap of the regions I regular polygons about verticorresponds to the intersec- ces of a semiregular tessellation of the sets. If the regions tion. Also called the numerido not overlap, the sets are I cal name. disjoint. The part of the rect- I • vertex of a conic solid angle included in one or more the point that marks the thinof the regions corresponds to I nest part of a conic solid the union of the sets. This page contains a tool that illus- • vertex of a polygon trates Venn diagrams; the tool ~ an endpoint of a segment in a represents the probability of ; polygon an event by the area of the • vertex of an angle event. the common endpoint of the I two rays vertical angles non-adjacent angles formed by the intersection of two lines .

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• vertex (of a polygon) The point of intersection of three or more edges.

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• vertex angles (of a kite) The angles between the pairs of congruent sides.

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• vertical line a line that goes straight up and down, and whose slope is defined as infinite or undefined . vertices plural form of vertex; the point of intersection of the rays of an angle, "corner" point of any geometric figure bounded by lines, planes, or lines and planes.

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11l'iew I WU~ test

179

• view ~ • whole number a drawing of a side of an object ; a natural number. • vigesimal related to intervals of 20. • vinculum the horizontal bar in a fraction separating the numerator from the denominator. • volume the amount of space a 3-D object can hold. • vulgar fraction a common fraction. • weak inequality an inequality that permits the equality case. For example, a is less than or equal to b. • weight matrix w matrix :fhe matrix of partial warp scores, together with the uniform component, for a sample of shapes. The weight ~atrix is com~uted as a ~otanon of the Procrustes-resIdual shape coordinates; like them, they are a set of shape coordinates for which the sum of squared differences is the squared Procrustes distance between any two specimens. wff • a well-formed formula.

; • Wilcoxon test : named after the statistician F, ~ Wilcoxon . This test applies to I an experimental design involv~ ing ~o repeated measure ob: servanons on a common set of I experimental units, which need ~ be only ordinal-scale. the pur: pose is to measure shift in scale I location between the two levels ~ of the repeated measure distinc: tion. the test statistic is derived ~ from the set of differences be; tween the two levels of the re: peated measure distinction one ~ difference score for each obser; vational unit. the procedure is : somewhat variable between ~ authors, although the variants I each correspond to valid well: sized exact tests. Wilcoxon's ~ original procedure commences I by discarding entirely the obser~ vations from any experimental : units for which the data values ~ are equal at each level of the ; repeated measure comparison. : thus or otherwise, the next step ~ is ranking the differences, pro; viding a rank for each retained : experimental unit; the ranks are ~. according to the absolute values

Mtldlmufriu======;;;;;;;;;;;

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wintling number I x-pentoltlitllJ

180

II

of the differences. The ranks are summed separately into two or three categories: negative differences; zero differences (if any); positive differences. the test statistic is the smaller of the outer categories, plus an adjustment for the middle (zero-difference) category.

I proceeds to fmd additional fac; tors to fit to the residuals, and : so on until the data is ad~ equately fit. ;• x

• winding number the number of times a closed curve in the plane passes around a given point in the counterclockwise direction.

• x-intercept the point at which a line crosses the x-axis. • XOR, exclusive disjunction XOR is an operation on two logical propositions. If p and q are two propositions, (p XOR q) is a proposition that is true if either p is true or if q is true, but not both. (pXOR q) is logically equivalent to «p OR q) AND NOT (pANDq».

• witch of agnesi a curve whose equation xl y=4a 2 (2a-y).

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x-axis the horiwntal axis in the plane.

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IS I

• work the result of force applied over some distance, given by the formula w = fd, where w = work, f = force, and d = distance.

roman numeral for 10.

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I • x-pentomino • Wright factor analysis a pentomino in the shape of the a version of factor analysis, due I letter X. to Sewall Wright, in which a I path model is used to describe the relation between the mea-, ~ sured variables and the factors I of interest. It is usually exploratory, in that one fits a simple one factor model iteratively to I maximally explain the correlations among variables, and then

181 *==~~========~ • yard ~ nal)jSE(original). a measure of length equal to 3 ~ • zero

feet.

;0

• y-axis the vertical axis in the plane.

; • zero angle : an angle whose measure is O. • year ~ In a zero angle, both the initial a measure of time equal to the ; and terminal sides are the same. period of one revolution of the ; • zero divisors earth about the sun. Approxi- : nonzero elements of a ring mately equal to 365 days. ~ whose product is O. • y-intercept ~ • zero element the point at which a line crosses ; the element 0 is a zero element the y-axis. : ofagroupifa+O=aandO+a=a I : for all elements a. I . I .• zero-di menSlona ~ having no dimension; a point z-intercept ~ the point at which a line crosses : the z-axis. I •

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• z statistic a Z statistic is a test statistic whose distribution under the null hypothesis has expected value zero and can be approximated well by the normal curve. Usually, Z statistics are constructed by standardising some other statistic. The Z statistic is related to the original statistic by Z = (original expected value of origi-

: • zone ~ the portion of a sphere between ; two parallel planes. ; • z-score : the observed value of the Z staI risric. z-test an hypothesis test based on approximating the probability histogram of the Z statistic under the null hypothesis by the normal curve.

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; : ~

; : I