n vs. n-1

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n vs. n-1
Message posted by Rich on April 27, 2000 at 12:00 AM (ET)

In calculating standard deviations, and in particular RSD's of data sets, when is it proper to use n rather than n-1. I seem to remember being taught that "n" was used for small populations, I believe the number given was 20 sample points. At the time, this was related as 'degrees of freedom', which I now believe may have been explained to me incorrectly. For references' sake, the RSD being measured is of a known and finite set of measurement (area counts from chromatographic injections). Any help here or via private email will be appreciated. Thanks, Rich

READERS RESPOND:
(In chronological order. Most recent at the bottom.)

Re: n vs. n-1
Message posted by Phil Rosenkrantz on April 30, 2000 at 12:00 AM (ET)

In addition to the previous responder, the reason behind the n vs. n-1 choice related to the idea of an "unbiased estimator". This is of importance to mathematical statisticians and probably only a few others. [An "unbiased estimator" is one where the "Expected Value" of the estimator yields the parameter of interest.]

You will notice that as "n" increases, the difference between using n or n-1 becomes almost negilible. Many people teach that after about n=25 or n=30, you can use n instead of n-1 without a problem.

For the t-statistic, n-1 is called the "degrees-of-freedom."

Unless you a statistics major, don't worry about why.

Re: n vs. n-1
Message posted by Jack Tomsky on May 1, 2000 at 12:00 AM (ET)

Theoretically, dividing the sum of squares about the sample mean by n-1 gives you an unbiased estimate of the population variance (sigma^2). Dividing by n gives you the maximum-likelihood estimate. Dividing by n+1 gives you the minimum mean-squared-error (minimax, for a quadratic loss function).

If the mean happens to be known, then dividing the sum of squares about this known mean by n is an unbiased estimate of sigma^2. Thus, you have several choices, each one having some desirable property.

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