Center limit theorem in statistics

What is central limit theorem?

If you do random sampling from any type of distribution and calculate the mean of the samples and plot it in a graph then the distribution of the mean will be normal distribution and the curve shape will be bell shape curve. This sampling technique must be with replacement sampling technique. With replacement sampling means that when you select a data from the population at random, it is returned to the population again and for this reason the data which is selected has a chance to again get selected while creating second sample. For this reason, in with replacement sampling, samples can contain unique data and also duplicate data.

Suppose you have a population X. From X population you are doing random sampling. After sampling, you have a total of 100 samples. Now calculate the mean of each sample. After finding the mean plot a histogram of mean values and the histogram will be a normal distribution where mean(µ) will be equal to population(X) mean and variance will be σ²/n.

Why we use central limit theorem(CLT)?

This theorem is based on hypothesis testing. This theorem gives knowledge about the practical application of inferential statistics of data. You will use it in confidence intervals, DOE, regression analysis, etc.

Example:

In the image, population data is normally distributed. The first time take sample size is two and plot the graph and you can see that the graph is normally distributed. Then second time takes sample size=8, then third time takes sample size=16, and the fourth time takes 32. If you plot the graph and each time the distribution is the normal distribution.