What is the Central Limit Theorem and why is it important?

• A. It asserts that any distribution becomes normal as sample size increases.
• B. It implies that only small samples can be used for inference.
• C. It states that the median of a sample converges to the population mean.
• D. The sum (or average) of independent identically distributed random variables tends toward a normal distribution, enabling inferential statistics with large samples.

Answer: (D)

When you take many independent, identically distributed random quantities with a finite mean and finite variance, the sum or average of those quantities tends toward a normal distribution as the number of terms grows. This happens even if the original distribution isn’t normal, so with enough samples the sampling distribution of the mean (or the sum) becomes bell-shaped.

The practical upshot is powerful: because the mean of these samples behaves approximately normally for large samples, you can use normal-based methods to do inference—construct confidence intervals and perform hypothesis tests—without knowing the exact form of the underlying distribution. The normal approximation has mean equal to the population mean and variance equal to the population variance divided by the sample size, which quantifies the precision you gain as you collect more data.

Other options miss the essential point: not every distribution becomes normal by itself just from taking more data, and the CLT relies on independence and finite mean and variance. It doesn’t say only small samples can be used, and it isn’t about the sample median converging to the mean.