The Central Limit Theorem (CLT) is an important idea in statistics. It is based on a few main ideas:

1. Independence:

   • Each observation needs to be independent. This means that one result should not affect another.

2. Sample Size:

   • The sample size (often called $n$) should be big enough. Usually, a size of $n \geq 30$ is good enough for the CLT to work. However, smaller sizes can be okay if the population is nearly normally distributed.

3. Identically Distributed:

   • The samples should come from the same population. They should have a clear average (mean, $\mu$) and a standard way of measuring how spread out the data is (standard deviation, $\sigma$). They should be taken from the same kind of distribution (like normal or uniform).

4. Finite Variance:

   • The main population needs to have a finite variance. If the variance is infinite, the CLT doesn't apply.

5. Sampling Method:

   • It's best to use random sampling. This helps to ensure that the samples truly represent the population.

When these ideas are followed, as the sample size gets bigger, the average of the sample results will get closer to a normal distribution. This is true no matter how the original population looks.

In simpler terms, if we take enough samples, we can expect the distribution of the sample averages to look like a normal distribution. This normal distribution will have a mean of $\mu$ and a standard error of $\frac{\sigma}{\sqrt{n}}$.