Central Limit Theorem

The Central Limit theorem defines that the mean of the given sample is the same as the mean of the population (approx). No matter how bigger the population, we can infer the statistics of the population with the help of the sample. 

Central Limit Theorem Definition:

The central limit theorem states that if you have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed.

Central Limit Theorem Example:

Let us take an example to understand the concept of the Central Limit Theorem (CLT):

Suppose you want to calculate the average weight of the country. The first step in this would be to measure the height of all the people individually and then add them. Then, Divide the sum of their weights by the total number of people. This way we will get the average height. But this method will not make sense for long calculations as it would be tiresome and very long.

So, we will use CTL(Central Limit Theorem) to make the calculation easy. In this method, we will randomly pick peoples from different cities and make a sample.  we will make the samples city-wise and each sample will include some peoples. Then, we will follow the following steps to solve it.

Take all these samples and find the mean.

Now, Find the mean of the sample means.

This way we will get the approximate mean height of the people in the country.

We will get a bell curve shape if we will find the histogram of these sample mean heights.

Central Limit Theorem Formula:

The central limit theorem is applicable for a sufficiently large sample size (n≥30). The formula for the central limit theorem can be stated as follows:

Where,

μ = Population mean

σ = Population standard deviation

μx = Sample mean

σx= Sample standard deviation

n = Sample size

Applications of Central Limit Theorem:

Statistical Applications

• If the distribution is not known or not normal, we consider the sample distribution to be normal according to CTL. This method assumes that the population given is normally distributed. This helps in analyzing data in methods like constructing confidence intervals.

• To estimate the population means more accurately, we can increase the samples taken from the population which will ultimately decrease the sample means deviation.

Practical Significance 

• One of the most common applications of CLT is in election polls. To calculate the percentage of persons supporting a candidate which are seen on news as confidence intervals.

• It is also used to measure the mean or average family income of a family in a particular region.