Central Limit Theorem Formula
The central limit theorem formula provides a relationship between the sample means and the population mean. For this it is assumed that the samples have been collected from a larger population with replacement. For a sufficient larger set of samples, the sample means are normally distributed. The central limit theorem formula helps to make inferences about the sample mean and the population mean values. Further, central limit theorem formula helps us to identify if the sample belongs to the referred population.
What is Central Limit Theorem Formula?
The formula is based on the central limit theorem, which states that the average of the sample means taken from a large population is equal to the population mean. Here \(\bar x_1 \), \( \bar x_2 \), \( \bar x_3\), ......\(\bar x_n\) are the individual means of the samples picked from larger population, and \(\mu \) is the population mean. Then, the central limit theorem formula is given as:
\[ \mu = \dfrac{\bar x_1 + \bar x_2 + \bar x_3 + ......\bar x_n}{n}\]
Let us try out a few examples to more clearly understand the central limit theorem formula.
Solved Examples on Central Limit Theorem Formula

Example 1: A set of samples have been collected from a larger sample and the sample mean values are 12.8, 10.9, 11.4, 14.2, 12.5, 13.6, 15, 9, 12.6. Find the population mean.
Solution:
The given sample mean values are 12.8, 10.9, 11.4, 14.2, 12.5, 13.6, 15, 9, 12.6.
The population mean values are an average of the above sample mean values.
\(\begin{align}\mu &= \frac{12.8 +10.9 +11.4 + 14.2 +12.5 +13.6 +15 + 9 +12.6}{9} \\&=\frac{112}{9} \\ \mu &= 12.4 \end{align}\)
Answer: Hence the population mean is 12.4 
Example 2: The sample means of the data picked from a larger population is 14, 12.8, X, 13.2, 15, and the population mean is 13.6. Find the value of the sample mean X.
Solution:
The sample means are 14, 12.8, X, 13.2, 15.
Population mean \(\mu \) = 13.6.
\( \begin{align} \mu &= \frac{\bar x_1 + \bar x_2 + \bar x_3 + \bar x_4 + \bar x_5}{n} \\ 13.6 &= \frac{14 +12.8 + X +13.2 +15}{5} \\13.6 \times 5 &=55 + X \\68 &=55 +X \\X &= 68  55 \\X &=13 \end{align} \)
Answer: Hence the mean of the sample is 13.
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