Statistics screen.
Why n minus one
The question
Why does the sample variance divide by n-1 rather than n?
Have a go before you open the solution. If you are stuck after a few minutes, the first paragraph below is usually enough of a nudge.
Show the worked solution
The answer
Because the sample mean is estimated from the same data, costing one degree of freedom.
Deviations are measured from the sample mean, which is itself chosen to minimise the sum of squared deviations, so those deviations are systematically too small.
Dividing by n therefore underestimates the true variance. Dividing by n-1 corrects the bias exactly.
The degrees-of-freedom framing generalises: estimating k parameters costs k, which is why regression residual variance divides by n-k.
Worth adding that the corrected variance is unbiased but its square root is not an unbiased estimate of the standard deviation, which catches most candidates out.
What it is testing
Whether a familiar formula is understood or memorised.
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