What happens to the sample mean as the sample size increases?

• A. It becomes more volatile.
• B. It diverges from the population mean.
• C. It becomes biased.
• D. It approaches a normal distribution.

Answer: (D)

As you increase the sample size, the distribution of the sample mean across many samples becomes approximately normal. This comes from the Central Limit Theorem, which says that with enough observations, the means of samples will form a bell-shaped distribution centered at the population mean, with spread that shrinks as n grows (variance equals the population variance divided by n). So the best way to describe what happens is that the sampling distribution of the sample mean tends toward a normal distribution as sample size increases. The other ideas don’t fit: the sample mean becomes less volatile with larger samples, not more; it is an unbiased estimator and does not systematically diverge from the population mean; and the point being that a single sample mean isn’t said to become normal, but the distribution of sample means across repeated samples does.