What does the central limit theorem state?

The central limit theorem is a foundational concept in statistics that asserts that the sampling distribution of the sample mean will approach a normal distribution as the size of the sample increases, regardless of the shape of the population distribution from which the samples are drawn. This means that even if the underlying population distribution is not normal, the distribution of the sample means will tend to become normal as the sample size grows larger.

This theorem is crucial because it allows statisticians to make inferences about population parameters and to conduct hypothesis testing using the properties of the normal distribution, which is well understood and has many useful statistical tools associated with it.

In contrast, the other options presented do not accurately capture the essence of the central limit theorem. The assertion that the mean of a population will always be normally distributed is incorrect because a population can have various distributions (e.g., skewed, uniform) that are not normal. The statement about the sum of data points being normally distributed does not relate to the central limit theorem, which focuses specifically on the sample mean rather than sums. Finally, the claim that the variance of a sample is independent of sample size is misleading, as larger samples tend to provide more reliable estimates of the population variance, and the sample variance is influenced by sample size