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Name: Name
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Date: Around 1993


Question:
Hi, I am taking an undergraduate level Business Statistics course. The book we are using is Elementary Business Statistics, by Freund, Williams, and Perles. Neither the book, nor the professor, can answer the question: "Why does the sample standard deviation use a denominator on n-1, and the population SD use a denominator of N?" It looks like the sample formula accentuates variance in the data. But, I do not understand what the utility of this is, or if it is desirable for some reason, why not do it for population SD also?



Replies:
I will try to make this short. If denominator N is used for the sample s.d., its role as an estimator of the population s.d. will be biased. That is, if you consider all possible samples of size N and average all of the resulting variances, this average will not be the population variance. When this happens, the related statistic is called "biased". The multiplier N/(N- 1) corrects this bias in theory. Related reading could be found in introductory texts in Probability and/or Statistical Theory where one first must study the mathematics of "expected value" or "expectation" and then "properties of estimators." Many of us have begun or continued to teach basic statistics and applications without studying all of the underlying theory. One such reference is Statistical Theory by B.W. Lindgren, whose second edition was published in 1968. For those wishing to show some details with expectation, the result is E [ sum(Xi - mean squared)/n ] = variance (1-1/n). This is a bit cryptic without notation for summation and Greek letters for mean and variance.

Good luck!!

Tom Elsner


May I recommend another reference which may be easier to find? Statistical Treatment of Experimental Data by Hugh D. Young (McGraw- Hill, New York, 1962). It is a paperback which many libraries have. If your library does not have it, McGraw-Hill just LOVES to send complete catalogs upon request (I get them all the time!).

Topper



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