Standard Deviation - Population vs. Sample Probability
Date: Around 1993
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?
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.
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!).
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Update: June 2012