Which statement best describes the Central Limit Theorem?

• A. The distribution of sample means tends toward normal as sample size increases, even if the population is not normal.
• B. The sample means become more skewed as sample size increases.
• C. The population distribution becomes normal as sample size increases.
• D. Sample means have the same distribution regardless of sample size.

Answer: (A)

The Central Limit Theorem describes what happens when you look at the averages of many independent observations. As you increase the sample size, the distribution of those sample means becomes approximately normal, even if the original population distribution is not normal. This happens because averaging tends to smooth out irregularities, and the spread of the sampling distribution shrinks with larger n (the standard error is sigma divided by sqrt(n)). So, the mean of the sampling distribution equals the population mean, and its shape approaches a bell curve as n grows. This is why large samples allow normal-approximation methods for inference.

The other ideas don’t fit because skewness doesn’t increase with larger samples, the population distribution doesn’t have to become normal for the means to be normal, and the distribution of the sample means actually changes with sample size, becoming tighter and more normal as n increases.