Central Limit Theorem

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Central Limit Theorem

Before we talk about Central Limit Theorem, let's understand what is a sampling distribution.

Let's say you manufacture tablets (the medicine and not the electronic device, mind you.)
There is a chemical (let's call it X) in the tablet. Ideally, in each tablet, the chemical X should be 10 mg.
Every day, your plant produces 1 Lakh such tablets. So, in a month (30 days) 30 Lakh tablets have been produced.
Now, if you take a sample of 10 tablets and find the mean of Chemical X in these 10 tablets and note it down. You again repeat this, by taking another set of 10 tablets and note down the mean as follows:

S1 - 9.8
S2 - 9.5
S3 - 10.2
S4 - 10.6
S5 - 9.2
.
.
.
.
S10,000 - 10.4
Where S1 implies the first set of 10 samples you took and S2 is the second set so on and so forth.

Let's say you repeat this 10,000 times. Meaning, 10,000 times you take a sample of 10 tablets find their mean and note it down. And then you make a plot of these mean values.
The resulting distribution that you would get is what is known as Sampling Distribution.

Now, this sampling distribution has some important properties. (As listed below).

1. Sampling distribution’s mean = Population mean,
2. Sampling distribution’s standard deviation (Standard error) = sigma / sqrt(n),
where sigma is the population’s standard deviation and n is the sample size, and
3. For n greater than 30, the sampling distribution becomes a normal distribution.

Irrespective of how the original population (ie. the level of chemical X in the 30 Lakh tablets) was distributed, the sampling distribution would always have the above properties.

These properties put together is what is known as the Central Limit Theorem.

Note: This is just for theoretical demonstration of sampling distribution and it's properties. In reality it is not required to take 10 samples (10,000 times) and plot it's distribution.
That is NOT required.
What we are saying is, IF you did that, these properties of Central Limit Theorem is what you would definitely end up observing.
Thus Central Limit Theorem becomes very important and it is because of these properties that we can calculate Confidence Interval for a given Confidence level.

You can read more about Confidence Level and Confidence Inerval in this post.