Which statement correctly distinguishes sample variance from population variance?

Which statement correctly distinguishes sample variance from population variance?

• A. Sample variance uses n-1 in the denominator, while population variance uses n
• B. Population variance uses n-1, while sample variance uses n
• C. They both use n-1
• D. They both use n

Answer: (A)

The main idea is why we use a different denominator when estimating variance from a sample. For the population variance, you divide by the total number of observations, n, because you’re measuring squared deviations from the true mean mu. When you estimate from a sample, you replace mu with the sample mean, x̄, but the deviations are constrained to sum to zero, so you effectively lose one degree of freedom. Dividing by n−1 compensates for this and makes the estimator unbiased for the population variance. That’s why the sample variance uses n−1 while the population variance uses n. The other options misstate this relationship or treat both cases the same, which doesn’t reflect the bias correction needed when estimating from a sample.

The main idea is why we use a different denominator when estimating variance from a sample. For the population variance, you divide by the total number of observations, n, because you’re measuring squared deviations from the true mean mu. When you estimate from a sample, you replace mu with the sample mean, x̄, but the deviations are constrained to sum to zero, so you effectively lose one degree of freedom. Dividing by n−1 compensates for this and makes the estimator unbiased for the population variance. That’s why the sample variance uses n−1 while the population variance uses n. The other options misstate this relationship or treat both cases the same, which doesn’t reflect the bias correction needed when estimating from a sample.