Central Limit Theorem Formula

Central Limit Theorem Formula (Table of Contents)

• Central Limit Theorem Formula
• Examples of Central Limit Theorem Formula (With Excel Template)
• Central Limit Theorem Formula Calculator

Central Limit Theorem Formula

The central limit theorem formula is being widely used in the probability distribution and sampling techniques. The central limit theorem states that as the sample size gets larger and larger the sample approaches a normal distribution. No matter what the shape of the population distribution is, the fact essentially holds true as the sample size is over 30 data points. The central limit theorem essentially have the following characteristics:-

• Mean of Sample is the same as the mean of the population.
• The standard deviation which is calculated is the same as the standard deviation of the population divided by the square root of the sample size.

A formula for Central Limit Theorem is given by:

Where,

• σ = Population Standard Deviation
• σ_x¯ = Sample Standard Deviation
• n = Sample size

Examples of Central Limit Theorem Formula (With Excel Template)

Let’s take an example to understand the calculation of Central Limit Theorem formula in a better manner.

Example #1

In a country located in the middle east region, the recorded weights of the male population are following a normal distribution. The mean and the standard deviations are 70 kg and 15 kg respectively. If a person is eager to find the record of 50 males in the population then what would mean and the standard deviation of the chosen sample?

Solution:

Mean of Sample is the same as the mean of the population.

The mean of the population is 70 since the sample size > 30.

Sample Standard Deviation is calculated using the formula given below

σ_x= σ/√n

• Sample Standard Deviation = 15 / √50
• Sample Standard Deviation = 2.12

Example #2

A certain group of people gives their annual pension benefit of Rs. 110 per week with a standard deviation of Rs. 20 per week. If a random sample of 50 people are taken what will be the mean and the standard deviation of the pension benefits received?

Solution:

Mean of Sample is the same as the mean of the population.

The mean of the population is 110 since the sample size > 30.

Sample Standard Deviation is calculated using the formula given below

σ_x= σ/√n

• Sample Standard Deviation = 20 / √50
• Sample Standard Deviation = 2.83

Example #3

A certain group of people gives their annual hardship allowance benefit of Rs. 150 per month with a standard deviation of Rs. 40 per month. If a random sample of 45 people are taken what will be the mean and the standard deviation of the pension benefits received?

Solution:

Mean of Sample is the same as the mean of the population.

The mean of the population is 150 since the sample size > 30.

Sample Standard Deviation is calculated using the formula given below

σ_x= σ/√n

• Sample Standard Deviation = 40 / √45
• Sample Standard Deviation = 5.96

Explanation

The formula of the central limit theorem states that the with an infinite number of successive random samples which are taken in the population the sampling distribution of the selected random variables will become approximately normally distributed in nature as the sample size get larger and larger in size

Relevance and Uses of Central Limit Theorem

• The central limit theorem is widely used in sampling and probability distribution and statistical analysis where a large sample of data is considered and needs to be analyzed in detail.
• The central limit theorem is also used in finance to analyze stocks and index which simplifies many procedures of analysis as generally and most of the times you will have a sample size which is greater than 50.
• Investors of all types rely on the CLT to analyze stock returns, construct portfolios and manage risk.
• A central limit theorem is also used in binomial probability which places an active role in the analysis of statistical data in detail.

Central Limit Theorem Formula Calculator

You can use the following Central Limit Theorem Calculator

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