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Conclusion
To get unbiased estimates of the standard error the sum of squares is divided by n-1, rather than n (where n is the sample size). In short it is divided by the number of degrees of freedom rather than the sample size.
The number of degrees of freedom is one property of the sum of squares and is determined by the number of independent linear comparisons that can be made among n observations.
To illustrate what this means: if A+B+C=D+E = 10 then four of these letters can be assigned any number but the fifth is therefore determined by the sum being 10. So the number of degrees of freedom in this case is 4.
In any such linear comparison, the number of degrees of freedom will be n-1.
So deriving a standard error from sample data results in n-1 degrees of freedom because the standard error is the result of the deviations from the mean and given the any one sample mean is fixed then there are n-1 'free' deviations but the last one is determined.
When computing the mean of the population this can be anything and so the number of degrees of freedom is the same as the population size.
