Central Limit Theorem Formula: Proof, Statistics, Examples

The Central Limit Theorem states that the sampling distribution of the sampling means approaches a normal distribution as the sample size increases, regardless of the shape of the data distribution.

• In general, the central limit theorem requires a sample size of at least 30.
• The fact of this theorem is that the average of the sample mean and standard deviation approximates the population mean and standard deviation.
• The average sample means will be the population mean, which is a crucial component of the Central Limit Theorem.

Key Terms: Central limit theorem, Population mean, Mean, Standard deviation, Data, Average sample, Normal distribution, Variable

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What is Central Limit Theorem?

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According to the central limit theorem, if large enough samples are obtained from the population, the sampling distribution of the mean for that random variable will approach a normal distribution.

The central limit theorem states that if large samples of a population are collected and their sums calculated, the sums form their own normal distribution.

• Furthermore, by applying the law of large numbers, this sum approaches the population mean.
• The central limit theorem is sometimes abbreviated as CLT.
• This is true for samples that are more than or equal to 30.
• In other words, as sample sizes increase, the sample mean graph begins to resemble a normal distribution.

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Central Limit Theorem Formula

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The formula for the central limit theorem can be given by

\(\LARGE \mu _{\overline{x}}=\mu\)

\(and\)

\(\LARGE \sigma _{\overline{x}}=\frac{\sigma }{\sqrt{n}}\)

Where

• σ is the population standard deviation
• μ is the population mean
• \(\LARGE \sigma _{\overline{x}}\) is the sample standard deviation
• \(\LARGE \mu _{\overline{x}}\) is the sample mean
• n is the sample size.

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Central Limit Theorem Application

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The central limit theorem is commonly used in cases where population characteristics must be found but complete population analysis is difficult. Other applications of the central limit theorem are mentioned below:

• In data science, the central limit theorem is used to determine accurate population assumptions to create a reliable statistical model.
• In applied machine learning, the CLT helps in making predictions regarding model performance.
• In statistical hypothesis testing, the central limit theorem is used to determine if a particular sample belongs to a specific population.

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Solved Examples

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Ques. The record of weights of the female population follows a normal distribution. Its mean and standard deviations are 65 kg and 14 kg, respectively. If a researcher considers the records of 50 females, then find out the standard deviation of the chosen sample.

Ans.  Given

• Mean of the population μ = 65 kg
• The standard deviation of the population = 14 kg
• Sample size n = 50

The standard deviation is given by

\(\LARGE \sigma _{\overline{x}}=\frac{\sigma }{\sqrt{n}}\)

On substituting the values, we get

\(\LARGE \sigma _{\overline{x}}\) = 14/√50

⇒ \(\LARGE \sigma _{\overline{x}}\) = 14/7.071 

⇒ \(\LARGE \sigma _{\overline{x}}\) = 1.97

Ques. A set of samples have been collected from a larger sample and the sample mean values are 13.8, 11.9, 12.4, 14.3, 13.5, 15.6, 16, 8, 12.7. Find the population mean.

Ans. The given sample mean values are 13.8, 11.9, 12.4, 14.3, 13.5, 15.6, 16, 8, 12.7.

The population mean values are an average of the above sample mean values

μ = (13.8 + 11.9 + 12.4 + 14.3 + 13.5 + 15.6 + 16 + 8 +12.7) / 9

⇒ μ = 118.2 / 9

⇒ μ = 13.13

Hence, the mean population is 13.13

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Things to Remember

• The central limit theorem formula establishes a link between the sample and population means.
• It is assumed that the samples were drawn from a larger population with replacement. 
• The sample means are normally distributed for a sufficiently large set of samples.
• The central limit theorem formula aids in concluding the sample mean and population mean values. 
• The average sample means will be the population mean, which is a crucial component of the Central Limit Theorem.
• The sample mean is the same as the population mean. The sample standard deviation equals the population standard deviation divided by the square root of the sample size.

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