Which statement best describes how the variance of the sample mean relates to sample size?

• A. The larger sample size has smaller variance
• B. The larger sample size has larger variance
• C. They have the same variance
• D. It depends on μ

Answer: (A)

Think about how the sample mean behaves across many samples. The spread of those sample means, measured by the variance of X̄, shrinks as you increase how many observations you average. If the population has variance σ^2, then Var(X̄) = σ^2 / n. So doubling the sample size halves the variance, and the standard error is σ / √n. This means the larger the sample, the more precise the estimate of the population mean becomes. Therefore, the statement that a larger sample size has smaller variance is the best description. The other ideas don’t fit because variance does not increase with n, it changes (decreases) with n, and it does not depend on the mean μ.