The Central Limit Theorem states that...

• A. The population distribution becomes normal as sample size increases.
• B. With a large enough sample size, the sampling distribution of the mean is approximately normal regardless of population shape.
• C. The sampling distribution of the variance is always normal.
• D. The sample median converges to the population mean.

Answer: (B)

When you take a large sample, the distribution of the sample mean tends to be normal, even if the underlying population isn’t, as long as the population variance is finite. In other words, if you repeatedly draw samples of size n and compute their means, those means form a roughly bell-shaped distribution as n grows. The center of that distribution is the population mean, and its spread—called the standard error—is the population standard deviation divided by the square root of n. This normal-approximation property is what lets us use normal-based methods for inference about the mean (like confidence intervals and hypothesis tests) even when the original data aren’t normally distributed, provided the sample is large enough and the variance is finite.

The other statements don’t fit this idea. The population distribution itself doesn’t become normal with larger samples—the population is fixed. The central limit theorem is about the sampling distribution of the mean, not the variance, which isn’t guaranteed to be normal. And the median converges to the population median, not the mean, so that line of reasoning isn’t correct for the mean-focused CLT.