Which formula correctly defines the sample variance?

Which formula correctly defines the sample variance?

• A. s^2 = Σ(xi − μ)^2 / N
• B. s^2 = Σ(xi − x̄)^2 / (n − 1)
• C. σ^2 = Σ(xi − μ)^2 / N
• D. σ^2 = Σ(xi − x̄)^2 / (n − 1)

Answer: (B)

The sample variance uses the deviations from the sample mean and divides by the degrees of freedom, n−1, to give an unbiased estimate of the population variance. That’s why the correct form is s^2 = Σ(xi − x̄)^2 / (n − 1). Here xi are the observed values, x̄ is the sample mean, and n is the sample size. If you use the population mean μ or divide by N, you’re describing the true population variance σ^2, not the sample variance. Using the population mean with a sample, or dividing by n instead of n−1, would not give the standard unbiased estimator for a sample.

The sample variance uses the deviations from the sample mean and divides by the degrees of freedom, n−1, to give an unbiased estimate of the population variance. That’s why the correct form is s^2 = Σ(xi − x̄)^2 / (n − 1). Here xi are the observed values, x̄ is the sample mean, and n is the sample size.

If you use the population mean μ or divide by N, you’re describing the true population variance σ^2, not the sample variance. Using the population mean with a sample, or dividing by n instead of n−1, would not give the standard unbiased estimator for a sample.