614 Chapter 14.Statistical Description of Data Semi-Invariants The mean and variance of independent random variables are additive:If and y are drawn independently from two,possibly different,probability distributions,then (x+y)=+Var(x+y)=Var(x)+Var(z) (14.1.9) Higher moments are not,in general,additive.However,certain combinations of them, called semi-imariants,are in fact additive.If the centered moments of a distribution are denoted M, M三(e:-到》 (14.1.10) so that,e.g.,M2 Var(),then the first few semi-invariants,denoted I&are given by I2=M2I3=M3I4=M4-3M (14.1.11) I5=M5-10M2M3I6=M6-15M2M4-10M+30M Notice that the skewness and kurtosis,equations(14.1.5)and(14.1.6)are simple powers / 3 of the semi-invariants, Skew()=Is/I2 Kurt()=I/I (14.1.12) 9 A Gaussian distribution has all its semi-invariants higher than I2 equal to zero.A Poisson distribution has all of its semi-invariants equal to its mean.For more details,see [2]. 9 Median and Mode 9a The median of a probability distribution function p(x)is the value xmed for which larger and smaller values of are equally probable: med p(a)dr=1 p(r)dx (14.1.13) The median of a distribution is estimated from a sample of values 1,..., N by finding that value i which has equal numbers of values above it and below 10.621 it.Of course,this is not possible when N is even.In that case it is conventional Numerica to estimate the median as the mean of the unique two central values.If the values 431 jj=1,...,N are sorted into ascending (or,for that matter,descending)order, Recipes then the formula for the median is [E(N+1)/2: N odd Imed North 1(红NW2+xN2+1), (14.1.14) N even If a distribution has a strong central tendency,so that most of its area is under a single peak,then the median is an estimator of the central value.It is a more robust estimator than the mean is:The median fails as an estimator only if the area in the tails is large,while the mean fails if the first moment of the tails is large; it is easy to construct examples where the first moment of the tails is large even though their area is negligible. To find the median of a set of values,one can proceed by sorting the set and then applying(14.1.14).This is a process of order N log N.You might rightly think614 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). Semi-Invariants The mean and variance of independent random variables are additive: If x and y are drawn independently from two, possibly different, probability distributions, then (x + y) = x + y Var(x + y) = Var(x) + Var(x) (14.1.9) Higher moments are not, in general, additive. However, certain combinations of them, called semi-invariants, are in fact additive. If the centered moments of a distribution are denoted Mk, Mk ≡  (xi − x) k  (14.1.10) so that, e.g., M2 = Var(x), then the first few semi-invariants, denoted Ik are given by I2 = M2 I3 = M3 I4 = M4 − 3M2 2 I5 = M5 − 10M2M3 I6 = M6 − 15M2M4 − 10M2 3 + 30M3 2 (14.1.11) Notice that the skewness and kurtosis, equations (14.1.5) and (14.1.6) are simple powers of the semi-invariants, Skew(x) = I3/I3/2 2 Kurt(x) = I4/I2 2 (14.1.12) A Gaussian distribution has all its semi-invariants higher than I2 equal to zero. A Poisson distribution has all of its semi-invariants equal to its mean. For more details, see [2]. Median and Mode The median of a probability distribution function p(x) is the value x med for which larger and smaller values of x are equally probable:  xmed −∞ p(x) dx = 1 2 =  ∞ xmed p(x) dx (14.1.13) The median of a distribution is estimated from a sample of values x1,..., xN by finding that value xi which has equal numbers of values above it and below it. Of course, this is not possible when N is even. In that case it is conventional to estimate the median as the mean of the unique two central values. If the values xj j = 1,...,N are sorted into ascending (or, for that matter, descending) order, then the formula for the median is xmed =  x(N+1)/2, N odd 1 2 (xN/2 + x(N/2)+1), N even (14.1.14) If a distribution has a strong central tendency, so that most of its area is under a single peak, then the median is an estimator of the central value. It is a more robust estimator than the mean is: The median fails as an estimator only if the area in the tails is large, while the mean fails if the first moment of the tails is large; it is easy to construct examples where the first moment of the tails is large even though their area is negligible. To find the median of a set of values, one can proceed by sorting the set and then applying (14.1.14). This is a process of order N log N. You might rightly think