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# Central Limit Theorem (CLT)?

What is the Central Limit Theorem (CLT)?

Central Limit Theorem is a probability theory which states that the distribution of the sample is approximately a normal distribution also referred to as a bell curve. It means that when independent variables are added their sum tends towards a normal distribution even though the variables are not normally distributed.  It states that as the sample size increases, the sampling distribution of the mean approaches a normal distribution. Given a large sample size with a finite level of variance, the mean of all samples will be approximately equal to the mean of the population. This stands true when the sample size is over 30.

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3 important components in the CLT theorem are:

1. Population Distribution.
2. Increasing sample size.
3. Randomly selected successive samples from data.

How Does the Central Limit Theorem Work?:

With the help of an illustration, we try to understand the working of CLT:

Let’s assume we have population data with mean µ and standard deviation σ. From this population data, we select a random sample data set of size n (x1, x2, x3 …, xn-1, xn). Similarly, we choose many sample sets from the data. Next, we calculate the mean X of every sample. We have to examine the distribution of mean values which is known as the sampling distribution of the mean. The sample distribution of the mean will approach a normal distribution having mean µ and standard deviation σ/n as the size of n goes up. As n increases the normal distribution is reached quickly.

Example:

Let us assume that in a sports tournament that follows a uniform distribution, the minimum age of a player is 18 years and the maximum is 35 years. That is if we pick out a player randomly his age would be anywhere between 18 to 35 years. Calculate the mean.

The mean of this uniform distribution is m = (b + a) / 2

Where,

b = largest value

a = smallest value

In this case it would be (35+18) / 2 = 26.5 years

The variance of uniform distribution is σ2 = (b-a)2 / 12.

= (35-18)2 / 12

= 289 / 12

= 24.08 years

Now taking random samples of 30 we calculate,

The mean of sampling distribution = X̄ = m = 26.5 years

The variance of sampling distribution = s2 = σ2 / n

= 24.08 / 30

= 0.802

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Conclusion:

The shape of the distribution of mean depends on the shape of population distribution. As the sample size increases, the sample distribution approaches a normal distribution. The central limit theorem states that a normal distribution will arise even after the initial distribution. It firmly states that we can treat the sampling distribution as if it were normal. An appropriate sample size and the central limit theorem help us to get around the problem of data from populations that are not normal.

Author: Urvi Surti