c14-0 (Numerical Recipes in C)

14.0 IntroductionIn this chapter and the next, the concept of data enters the discussion moreprominently than before.Data consist of numbers, of course. But these numbers are fed into the computer,not produced by it. These are numbers to be treated with considerable respect, neitherto be tampered with, nor subjected to a numerical process whose character you donot completely understand. You are well advised to acquire a reverence for data thatis rather different from the “sporty” attitude that is sometimes allowable, or evencommendable, in other numerical tasks.The analysis of data inevitably involves some trafficking with the field ofstatistics, that gray area which is not quite a branch of mathematics — and just assurely not quite a branch of science. In the following sections, you will repeatedlyencounter the following paradigm:• apply some formula to the data to compute “a statistic”• compute where the value of that statistic falls in a probability distributionthat is computed on the basis of some “null hypothesis”• if it falls in a very unlikely spot, way out on a tail of the distribution,conclude that the null hypothesis is false for your data setIf a statistic falls in a reasonable part of the distribution, you must not makethe mistake of concluding that the null hypothesis is “verified” or “proved.” That isthe curse of statistics, that it can never prove things, only disprove them! At best,you can substantiate a hypothesis by ruling out, statistically, a whole long list ofcompeting hypotheses, every one that has ever been proposed.

After a while youradversaries and competitors will give up trying to think of alternative hypotheses,or else they will grow old and die, and then your hypothesis will become accepted.Sounds crazy, we know, but that’s how science works!In this book we make a somewhat arbitrary distinction between data analysisprocedures that are model-independent and those that are model-dependent. In theformer category, we include so-called descriptive statistics that characterize a dataset in general terms: its mean, variance, and so on.

We also include statistical teststhat seek to establish the “sameness” or “differentness” of two or more data sets, orthat seek to establish and measure a degree of correlation between two data sets.These subjects are discussed in this chapter.609Sample 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).Chapter 14.

Statistical Descriptionof Data610Chapter 14.Statistical Description of DataCITED REFERENCES AND FURTHER READING:Bevington, P.R. 1969, Data Reduction and Error Analysis for the Physical Sciences (New York:McGraw-Hill).Stuart, A., and Ord, J.K. 1987, Kendall’s Advanced Theory of Statistics, 5th ed. (London: Griffinand Co.) [previous eds. published as Kendall, M., and Stuart, A., The Advanced Theoryof Statistics].Norusis, M.J.

1982, SPSS Introductory Guide: Basic Statistics and Operations; and 1985, SPSSX Advanced Statistics Guide (New York: McGraw-Hill).Dunn, O.J., and Clark, V.A. 1974, Applied Statistics: Analysis of Variance and Regression (NewYork: Wiley).14.1 Moments of a Distribution: Mean,Variance, Skewness, and So ForthWhen a set of values has a sufficiently strong central tendency, that is, a tendencyto cluster around some particular value, then it may be useful to characterize theset by a few numbers that are related to its moments, the sums of integer powersof the values.Best known is the mean of the values x1 , . . . , xN ,x=N1 XxjN j=1(14.1.1)which estimates the value around which central clustering occurs. Note the use ofan overbar to denote the mean; angle brackets are an equally common notation, e.g.,hxi. You should be aware that the mean is not the only available estimator of thisSample 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).In the other category, model-dependent statistics, we lump the whole subject offitting data to a theory, parameter estimation, least-squares fits, and so on.

Thosesubjects are introduced in Chapter 15.Section 14.1 deals with so-called measures of central tendency, the moments ofa distribution, the median and mode. In §14.2 we learn to test whether different datasets are drawn from distributions with different values of these measures of centraltendency. This leads naturally, in §14.3, to the more general question of whether twodistributions can be shown to be (significantly) different.In §14.4–§14.7, we deal with measures of association for two distributions.We want to determine whether two variables are “correlated” or “dependent” onone another.

If they are, we want to characterize the degree of correlation insome simple ways. The distinction between parametric and nonparametric (rank)methods is emphasized.Section 14.8 introduces the concept of data smoothing, and discusses theparticular case of Savitzky-Golay smoothing filters.This chapter draws mathematically on the material on special functions thatwas presented in Chapter 6, especially §6.1–§6.4. You may wish, at this point,to review those sections..