Intuitive Explanation of Bessel's Correction?

Question

When calculating a sample variance a factor of (N-1) appears instead of N (see http://en.wikipedia.org/wiki/Sample_variance#Population_variance_and_sample_variance ). Does anybody have an intuitive way of explaining this to students who need to use this fact but maybe haven't taken a statistics course?

Answer 1

Intuitively, when we're calculating the sample variance, it's because we don't know the population variance--or the population mean. When we calculate the sample variance, we're using the sample mean, but that is ITSELF an estimate, because the true population mean is unknown.

So we're estimating the dispersion about a mean that is itself an estimate. This means our estimate is more uncertain (there is more variation) than there otherwise would be. Bessel's correction, intuitively, takes this into account by decrementing the denominator by 1 (representing the loss of 1 degree of freedom), which makes the fraction larger, and correctly reflects the greater uncertainty associated with estimating a parameter based on another parameter that is itself unknown.