Which statement best describes the difference between population variance and sample variance?

• A. Population variance uses division by n-1; sample variance uses division by N.
• B. Population variance uses division by N; sample variance uses division by n-1 to be unbiased.
• C. Population variance and sample variance use the same divisor.
• D. Population variance estimates the mean; sample variance estimates the spread.

Answer: (B)

Variance measures how spread out data are. When you know the true population mean, the population variance is defined by dividing the average of squared deviations by the population size, N. But with a sample, you don’t know the true mean, so you estimate it with the sample mean. Using that estimated mean reduces the sum of squared deviations by one degree of freedom, which would bias the estimate downward if you divided by n. To compensate, you divide by n-1, not n, so the expected value of the sample variance equals the true population variance. This makes the sample variance an unbiased estimator of the population variance. So the statement that population variance uses division by N and sample variance uses division by n-1 to be unbiased is the best description. The other options either mix up the divisors or mischaracterize what variance estimates.