Central Limit Theorem (CLT)

Before directly diving into what Central Limit Theorem (CLT) is, first let's understand some basic terms.

What is sample?

Sample is a small part of the population. For example, let's suppose there is a bag of rice, and we pick a handful of rice from the bag then, the bag of rice is population and the rice we picked is a sample.

Fig1. Population and sample

What is sample mean?

Sample mean is the mean of certain property of the sample. For example, we can find the mean of length of each grain of rice of the sample, and we can get the average length of the grain of rice of the sample.

What is normal distribution?

To learn about normal distribution you can visit here. Or in short normal distribution is the distribution of data points which forms a bell shaped symmetric curve.

Now, what is the Central Limit Theorem (CLT)?

Central Limit Theorem states that the distribution of sample means ( ${\textstyle {\bar {X}}}$ ) follows normal distribution or gaussian distribution with mean μ and variance $\sigma ^{2}$ /n. The Average of mean and variance of all the samples equal the population mean and variance.

Sample are randomly selected from the population with replacement.

What is the use of Central Limit Theorem?

When the population is very large so that we cannot gather information from each element of the population because it will be very time-consuming and expensive. So, we take many samples from the population and gather information from them and use CLT to know the properties of the population.

Property of Central Limit Theorem:

$s^{2}$

No matter what is the shape of distribution of the population, as the number of samples increases the sampling distribution becomes almost normal regardless of shape of the population.

Fig2. Population distribution vs CLT distribution

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Central Limit Theorem