612 Chapter 14.Statistical Description of Data Skewness Kurtosis positive (leptokurtic) negative negative positive (platykurtic） (a) (b) Figure 14.1.1.Distributions whose third and fourth moments are significantly different from a normal (Gaussian)distribution.(a)Skewness or third moment.(b)Kurtosis or fourth moment. That being the case,the skewness or third moment,and the kurtosis or fourth 兰吴州 8 100 moment should be used with caution or,better yet.not at all. The skewness characterizes the degree of asymmetry ofa distribution around its mean.While the mean,standard deviation,and average deviation are dimensional quantities,that is,have the same units as the measured quantities xj,the skewness N兰eo3 (Nort server 令 is conventionally defined in such a way as to make it nondimensional.It is a pure number that characterizes only the shape of the distribution.The usual definition is America Skew(z1...ZN) ,鬥 (14.1.5) 9 =1 where =o(1...N)is the distribution's standard deviation(14.1.3).A positive value of skewness signifies a distribution with an asymmetric tail extending out CIENTIFIC( towards more positive z;a negative value signifies a distribution whose tail extends out towards more negative (see Figure 14.1.1). Of course,any set of N measured values is likely to give a nonzero value for (14.1.5),even if the underlying distribution is in fact symmetrical (has zero skewness). COMPUTING (ISBN 188812920 For (14.1.5)to be meaningful,we need to have some idea of its standard deviation as an estimator of the skewness of the underlying distribution.Unfortunately,that depends on the shape of the underlying distribution,and rather critically on its tails! uurggoglrion Numerical Recipes 10621 For the idealized case of a normal(Gaussian)distribution,the standard deviation of (14.1.5)is approximately 15/N when is the true mean,and 6/N when it is 43106 estimated by the sample mean,(14.1.1).In real life it is good practice to believe in skewnesses only when they are several or many times as large as this. (outside The kurtosis is also a nondimensional quantity.It measures the relative peakedness or flatness of a distribution.Relative to what?A normal distribution, North Software. what else!A distribution with positive kurtosis is termed leptokurtic,the outline of the Matterhorn is an example.A distribution with negative kurtosis is termed platykurtic;the outline of a loaf of bread is an example.(See Figure 14.1.1.)And, as you no doubt expect,an in-between distribution is termed mesokurtic. The conventional definition of the kurtosis is Kurt(z1... - (14.1.6) where the-3 term makes the value zero for a normal distribution612 Chapter 14. Statistical Description of Data Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machine￾readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). (a) (b) Skewness negative positive positive (leptokurtic) negative (platykurtic) Kurtosis Figure 14.1.1. Distributions whose third and fourth moments are significantly different from a normal (Gaussian) distribution. (a) Skewness or third moment. (b) Kurtosis or fourth moment. That being the case, the skewness or third moment, and the kurtosis or fourth moment should be used with caution or, better yet, not at all. The skewness characterizes the degree of asymmetry of a distribution around its mean. While the mean, standard deviation, and average deviation are dimensional quantities, that is, have the same units as the measured quantities xj , the skewness is conventionally defined in such a way as to make it nondimensional. It is a pure number that characterizes only the shape of the distribution. The usual definition is Skew(x1 ...xN ) = 1 N
N j=1 xj − x σ 3 (14.1.5) where σ = σ(x1 ...xN ) is the distribution’s standard deviation (14.1.3). A positive value of skewness signifies a distribution with an asymmetric tail extending out towards more positive x; a negative value signifies a distribution whose tail extends out towards more negative x (see Figure 14.1.1). Of course, any set of N measured values is likely to give a nonzero value for (14.1.5), even if the underlying distribution is in fact symmetrical (has zero skewness). For (14.1.5) to be meaningful, we need to have some idea of its standard deviation as an estimator of the skewness of the underlying distribution. Unfortunately, that depends on the shape of the underlying distribution, and rather critically on its tails! For the idealized case of a normal (Gaussian) distribution, the standard deviation of (14.1.5) is approximately 15/N when x is the true mean, and 6/N when it is estimated by the sample mean, (14.1.1). In real life it is good practice to believe in skewnesses only when they are several or many times as large as this. The kurtosis is also a nondimensional quantity. It measures the relative peakedness or flatness of a distribution. Relative to what? A normal distribution, what else! A distribution with positive kurtosis is termed leptokurtic; the outline of the Matterhorn is an example. A distribution with negative kurtosis is termed platykurtic; the outline of a loaf of bread is an example. (See Figure 14.1.1.) And, as you no doubt expect, an in-between distribution is termed mesokurtic. The conventional definition of the kurtosis is Kurt(x1 ...xN ) =    1 N
N j=1 xj − x σ 4    − 3 (14.1.6) where the −3 term makes the value zero for a normal distribution.