What does the Central Limit Theorem state?

The Central Limit Theorem (CLT) is a fundamental principle in statistics that states that the sum (or average) of a large number of independent random variables, regardless of the original distribution of the variables, will approximate a normal distribution, provided the sample size is sufficiently large. This means that as you collect more samples, the distribution of the sample means will form a bell-shaped curve, following a normal distribution, centered around the population mean.

This theorem is crucial for various statistical procedures, as it allows for the use of normal distribution approximations in many cases, facilitating hypothesis testing and confidence interval estimation even when the underlying population distribution is not normal.

In this context, the assertion that all data sets are normally distributed is incorrect; CLT applies under certain conditions about sample size and independence, and not all data sets will show a normal distribution without adequate sampling. Similarly, the characterizations regarding variances of unrelated variables or a specific sample size requirement for normality do not accurately encapsulate the essence of the Central Limit Theorem. The focus of the theorem is specifically on the behavior of averages or sums of independent random variables in relation to normality as sample sizes increase.