The central limit theorem is a fundamental concept in statistics. It states that given a sufficiently large sample size from a population with a finite level of variance, the mean of all samples from the population will be approximately normally distributed. This means that if we take many samples from a population and compute the mean of each sample, the distribution of means will be normal, even if the distribution of the original population is not.

This theorem is important because it allows us to make predictions about a population based on the characteristics of a sample, which can be much easier to measure. For example, we can use the central limit theorem to make predictions about the average weight of all students at a school based on a sample of students whose weights we have measured.

## A Simple Example:

Imagine we have a population of heights of adults in a certain city. The distribution of heights in this population is not necessarily normal - some people might be very tall, some people might be very short, and most people will be somewhere in between. However, if we take a sample of 10 people from this population and calculate the mean height for that sample, the distribution of those sample means will be approximately normal, even if the population is not.

Instead of taking the average of 10 people, if we take the average of 100 people, the distribution of these means will even better resemble a normal distribution.

## Central Limit Theorem Formula (Mean and Standard Deviation)

The Central Limit Theorem formula states that the mean of a sample from a population with finite variance is equal to the population mean, and the standard deviation of the sample is equal to the population standard deviation divided by the square root of the sample size.

### Mean:

The mean of a sampling distribution is equal to the population mean.

Mean:

$$\Large{\mu_{\bar{x}} = \mu}$$

### Standard Deviation:

The standard deviation of a sampling distribution is equal to the population standard deviation divided by the square root of the sample size.

Standard Deviation:

$$\Large{\sigma_{\bar{x}} = \frac{\sigma_x}{\sqrt{n}}}$$

The standard deviation of the sampling distribution of the sample mean, also known as the standard error of the mean, is a measure of the amount of variation we can expect in the mean of a sample if we were to take multiple samples from the same population. This is important because it allows us to determine how confident we can estimate the population mean based on the sample mean.

## Conclusion

The Central Limit Theorem is an important concept in statistics that allows us to make predictions about a population based on the characteristics of a sample.