What is Central Limit theorem (CLT)?

Suppose a student being told to find out the height of all the students in his school as a project to be submitted for the Mathematics class. It would be too difficult for him to ask each and every student about their height, so instead what he does is, he randomly selects many students (samples) from each class and then takes an average of each student (sample) height from the class and thus after taking all the means of each sample he plots it on the graph and finds out he gets an approximately a Normal Distribution curve from where he calculate the mean and standard deviation and submit it to the professor and got an A+ grade, this is essentially a Central Limit Theorem (CLT).

Statement of the CLT:

1. The expected value (equivalent to the mean) of a sampling distribution of the mean is equal to the mean of the parent population.
2. The standard error (equivalent to the standard deviation) of a sampling distribution of the mean is equal to the standard deviation of the parent population divided by the square root of the sample size.
3. Irrespective of the underlying distribution of the parent population, the sampling distribution of the mean increasingly reaches the normal distribution, as the sample size increases,. This is one of the most important elements of the Central Limit Theorem and explains why so many natural phenomena can be described with the Normal distribution.

Mathematically,

Notation

I would also like to show y’all a small animation video where I have selected any distribution curve for my parent population and I have initially set my sample size n= 7. Sampling distribution becomes almost Normal distribution as I increase my sample size n=15

Central Limit Theorem

I hope everyone get the idea of what central limit theorem is and where it can be applied. Central Limit theorem is the most important theorem in statistics and probably even the whole of mathematics.

In my next blog, I will bring a real life business challenge which I will solve with Central Limit Theorem so stay tune.

Happy Learning!!