What does the Law of Large Numbers state in terms of sample means?

• A. The sample mean becomes more variable as n grows
• B. The sample mean always equals μ for any n
• C. As n increases, the sample mean converges to the population mean μ.
• D. The sample mean equals the population median

Answer: (C)

The main idea here is that averaging many independent observations reduces randomness and the estimated average gets closer to the true population mean as you collect more data. This is the essence of the Law of Large Numbers: as the sample size grows, the sample mean tends to approach the population mean μ.

Intuitively, the spread of the sample mean shrinks with more data—the standard error is σ/√n—so with large n the average you compute is very likely to be near μ. That’s why the correct statement says the sample mean converges to μ as n increases.

The other ideas don’t fit: the sample mean does not become more variable with more observations; it varies less as n grows, not more. It also doesn’t guarantee equality to μ for any finite n, only in the limiting sense as n becomes large. And the mean need not equal the population median unless the distribution is symmetric or has a special relationship between mean and median.