The Central Limit Theorem (CLT) is a fundamental concept in statistics that describes the behavior of the sampling distribution of the sample mean (or other sample statistics) when samples are drawn from a population, regardless of the shape of the population distribution. In simpler terms, it states that as you take larger and larger random samples from any population, the distribution of the sample means will become approximately normally distributed, regardless of the shape of the original population distribution.

Here's a more detailed explanation with an example:

**Central Limit Theorem Example**:

Let's say you have a population with a random variable X, which represents the scores of all students in a school on a particular exam. The scores in this population have a skewed distribution, meaning they are not normally distributed. This population has a mean (μ) and a standard deviation (σ).

The Central Limit Theorem states that if you were to take random samples of, let's say, 30 students each and calculate the mean score for each sample, the distribution of those sample means would be approximately normally distributed, even if the original scores were not.

**Formula of the Central Limit Theorem**:

Mathematically, the CLT can be expressed as follows:

If you have a random sample of n observations (n is sufficiently large, typically n ≥ 30) from any population with mean μ and standard deviation σ, the sampling distribution of the sample mean (x̄) will be approximately normally distributed with mean (μx̄) equal to the population mean μ and standard deviation (σx̄) equal to the population standard deviation σ divided by the square root of n:

μx̄ = μ (same as the population mean)

σx̄ = σ / √n (also called standard error)

In our example, as you take more and more random samples of 30 students each, the distribution of the sample means (average scores) will tend to follow a normal distribution, regardless of whether the original population of scores had a skewed, uniform, or any other type of distribution.

This is a powerful concept in statistics because it allows us to make probabilistic inferences about population parameters (like the population mean) based on sample statistics (like the sample mean), assuming that the sample size is sufficiently large. It's one of the reasons why the normal distribution is frequently encountered in statistical analyses.