14.2 Do Two Distributions Have the Same Means or Variances? 615 that this is wasteful,since it yields much more information than just the median (e.g.,the upper and lower quartile points,the deciles,etc.).In fact,we saw in 88.5 that the element (N+1)/2 can be located in of order N operations.Consult that section for routines. The mode of a probability distribution function p(x)is the value of x where it takes on a maximum value.The mode is useful primarily when there is a single,sharp maximum.in which case it estimates the central value.Occasionally,a distribution will be bimodal.with two relative maxima;then one may wish to know the two modes individually.Note that,in such cases,the mean and median are not very useful,since they will give only a"compromise"value between the two peaks. CITED REFERENCES AND FURTHER READING: Bevington,PR.1969.Data Reduction and Error Analysis for the Physical Sciences (New York: McGraw-Hill),Chapter 2. Stuart,A.,and Ord,J.K.1987,Kendall's Advanced Theory of Statistics,5th ed.(London:Griffin and Co.)[previous eds.published as Kendall,M.,and Stuart,A.,The Advanced Theory of Statistics].vol.1.$10.15 Norusis,M.J.1982.SPSS Introductory Guide:Basic Statistics and Operations:and 1985,SPSS- X Advanced Statistics Guide (New York:McGraw-Hill). 9 Chan,T.F.,Golub,G.H.,and LeVeque,R.J.1983,American Statistician,vol.37,pp.242-247.[1] Cramer,H.1946,Mathematical Methods of Statistics (Princeton:Princeton University Press). 515.10.[2 SCIENTIFIC 14.2 Do Two Distributions Have the Same Means or Variances? Not uncommonly we want to know whether two distributions have the same mean.For example,a first set of measured values may have been gathered before some event,a second set after it.We want to know whether the event,a"treatment" or a "change in a control parameter,"made a difference. Our first thought is to ask"how many standard deviations"one sample mean is Numerical Recipes 10621 43106 from the other.That number may in fact be a useful thing to know.It does relate to the strength or"importance"of a difference of means if that difference is genuine. However,by itself,it says nothing about whether the difference is genuine,that is, (outside 腿 statistically significant.A difference of means can be very small compared to the North standard deviation,and yet very significant,if the number of data points is large. Conversely,a difference may be moderately large but not significant,if the data are sparse.We will be meeting these distinct concepts of strength and significance several times in the next few sections. A quantity that measures the significance of a difference of means is not the number of standard deviations that they are apart,but the number of so-called standard errors that they are apart.The standard error of a set of values measures the accuracy with which the sample mean estimates the population(or"true")mean. Typically the standard error is equal to the sample's standard deviation divided by the square root of the number of points in the sample.14.2 Do Two Distributions Have the Same Means or Variances? 615 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). that this is wasteful, since it yields much more information than just the median (e.g., the upper and lower quartile points, the deciles, etc.). In fact, we saw in §8.5 that the element x(N+1)/2 can be located in of order N operations. Consult that section for routines. The mode of a probability distribution function p(x) is the value of x where it takes on a maximum value. The mode is useful primarily when there is a single, sharp maximum, in which case it estimates the central value. Occasionally, a distribution will be bimodal, with two relative maxima; then one may wish to know the two modes individually. Note that, in such cases, the mean and median are not very useful, since they will give only a “compromise” value between the two peaks. CITED REFERENCES AND FURTHER READING: Bevington, P.R. 1969, Data Reduction and Error Analysis for the Physical Sciences (New York: McGraw-Hill), Chapter 2. Stuart, A., and Ord, J.K. 1987, Kendall’s Advanced Theory of Statistics, 5th ed. (London: Griffin and Co.) [previous eds. published as Kendall, M., and Stuart, A., The Advanced Theory of Statistics], vol. 1, §10.15 Norusis, M.J. 1982, SPSS Introductory Guide: Basic Statistics and Operations; and 1985, SPSS￾X Advanced Statistics Guide (New York: McGraw-Hill). Chan, T.F., Golub, G.H., and LeVeque, R.J. 1983, American Statistician, vol. 37, pp. 242–247. [1] Cram´er, H. 1946, Mathematical Methods of Statistics (Princeton: Princeton University Press), §15.10. [2] 14.2 Do Two Distributions Have the Same Means or Variances? Not uncommonly we want to know whether two distributions have the same mean. For example, a first set of measured values may have been gathered before some event, a second set after it. We want to know whether the event, a “treatment” or a “change in a control parameter,” made a difference. Our first thought is to ask “how many standard deviations” one sample mean is from the other. That number may in fact be a useful thing to know. It does relate to the strength or “importance” of a difference of means if that difference is genuine. However, by itself, it says nothing about whether the difference is genuine, that is, statistically significant. A difference of means can be very small compared to the standard deviation, and yet very significant, if the number of data points is large. Conversely, a difference may be moderately large but not significant, if the data are sparse. We will be meeting these distinct concepts of strength and significance several times in the next few sections. A quantity that measures the significance of a difference of means is not the number of standard deviations that they are apart, but the number of so-called standard errors that they are apart. The standard error of a set of values measures the accuracy with which the sample mean estimates the population (or “true”) mean. Typically the standard error is equal to the sample’s standard deviation divided by the square root of the number of points in the sample