Re: Population SD when sample size > 20?


Kevin E. Thorpe wrote:
Data Matter wrote:

I'd advise consulting a real statistics textbook. the information on
this web page is rather dubious.

I agree.

The meaning of SD is given as "measurement of the spread in individual
data points to reflect the uncertainty of a single measurement". One
of the worst definitions I have ever seen. What is this "single
measurement"?

The reason why you would need to take a sample is because it is
impractical or unrealistic or impossible or too costly to measure all
units of a population. As such, the population mean or population SD
should be treated as unknowable whenever your data contain a sample of
that population. It is then irrelevant what the formula is.

In the case where you can measure every item in a population (i.e. the
population is finite, and realistically small), then you can use the
population mean/SD formulas with N as the denominator.

The estimator using n-1 is commonly known as the unbiased
estimate of the population sd (or variance if you drop the
square root). On the other hand, a divisor of n gives you
the maximum likelihood estimate of the population sd.

Just a little side-note:

The sample variance calculated with n-1 in the denominator is an
unbiased estimate of the population variance, but when you take the
square root of the sample variance, then you do not get an unbiased
estimator of the population standard deviation. But the bias is quite
small and is usually ignored.

When you calculate the sample variance with n in the denominator, then
you do get the MLE of the population variance (assuming normality, as
pointed out). Due to the properties of maximum likelihood estimators,
taking the square root of the sample variance then yields the MLE of
the population standard deviation. Both of these MLEs are biased.

m00es

.

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