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242–247. [1]Cramér, H. 1946, Mathematical Methods of Statistics (Princeton: Princeton University Press),§15.10. [2]14.2 Do Two Distributions Have the SameMeans or Variances?Not uncommonly we want to know whether two distributions have the samemean. For example, a first set of measured values may have been gathered beforesome 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 isfrom the other. That number may in fact be a useful thing to know.
It does relate tothe 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 thestandard 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 dataare sparse. We will be meeting these distinct concepts of strength and significanceseveral times in the next few sections.A quantity that measures the significance of a difference of means is not thenumber of standard deviations that they are apart, but the number of so-calledstandard errors that they are apart. The standard error of a set of values measuresthe 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 bythe square root of the number of points in the sample.Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use.
Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (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. Consultthat section for routines.The mode of a probability distribution function p(x) is the value of x where ittakes on a maximum value.
The mode is useful primarily when there is a single, sharpmaximum, in which case it estimates the central value. Occasionally, a distributionwill be bimodal, with two relative maxima; then one may wish to know the twomodes individually. Note that, in such cases, the mean and median are not veryuseful, since they will give only a “compromise” value between the two peaks..