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Last updated on October 6, 2025

Central Limit Theorem

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The central limit theorem is a fundamental concept in statistics. It states that the sampling distribution of the mean will be normal regardless of the shape of the population distribution, as the sample size is large. In this topic, we will learn about it in detail.

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What is Central Limit Theorem?

The central limit theorem states that if we consider numerous random samples from a population, the sample mean distribution will form a normal distribution, which is a bell curve. The distribution of the original population can be skewed, Poisson, or binomial, but the mean distribution of the sample will be normal when the sample size is greater than or equal to 30. So, if the sample size is larger, it allows for better estimation of population characteristics.

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What are the Key Components of the Central Limit Theorem?

So far, we have discussed what the central limit theorem states. Let us now learn about the key components of the theorem. The key concepts are: 

 

  • Sampling is successive: It means that some samples may repeat. For instance, in opinion polls, the population selected can be a part of different studies. 


 

  • Sampling is random: The sample here must be selected randomly, as it gives equal chance to all the individuals in the population. For example, in a political poll, if we select the sample based on a specific group, the result can be biased. 

 

  • Samples should be independent: Each sample should be selected individually; the selection of one should not influence the other. The sample should be independent to follow a true, normal pattern. 

 

  • Large sample size: If the sample is large, the sampling distribution will be normal. According to the central limit theorem, an ideal sample size should be 30 or more. 

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How to Find the Central Limit Theorem?

As we know, the central limit theorem is a fundamental concept in statistics and probability; it helps us understand how the population estimates the behavior under repeated sampling. Now we will discuss how it works and the formulas. 

 

  • When the samples are repeated from the population, the mean will follow a normal distribution, regardless of the type of population distribution (uniform, skewed, or binomial).  

     
  • The larger the sample size, the narrower the bell curve.

     
  • As the sample size increases, the mean of the sample distribution approaches the population mean.
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Central Limit Theorem Formula

Now let’s look at the formula for the central limit theorem. Let X be a random variable with a known or unknown probability distribution. The standard deviation is σ, and the mean of X is μ. According to the central limit theorem, if the large number sample are drawn of size n, then the new random variable is x̄, then,

 

\(\bar{x} \sim N\!\left(\mu, \frac{\sigma}{\sqrt{n}}\right)\), where σ / √n is the standard error. The z score of the random variable x̄ is \( z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}} \)

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Real-World Applications of Central Limit Theorem

The central limit theorem is used in the real world to analyze the population characteristics. Let’s discuss the real-world applications of the central limit theorem. 

 

  • In economics and data science, we use central limit theorem to understand the statistical model and to draw conclusion.


     
  • To understand the public opinion, we use the central limit theorem to analyze and understand the polls and surveys.
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Common Mistakes and How to Avoid Them in Central Limit Theorem

Mistakes are common among students when working on the central limit theorem. So to master the central limit theorem, we can learn about a few common mistakes and ways to avoid them. 

Mistake 1

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Confusing the central limit theorem with the law of large numbers

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Students tend to think that the central limit theorem means that all sample means will be close to the population mean. However, this is incorrect, as the central limit theorem actually states that the distribution of the sample means is always a normal distribution.

Mistake 2

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Using central limit theorem for small samples
 

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Using the central limit theorem for a small sample will result in an inaccurate conclusion; it gives the best possible result when applied for a large sample, which is greater than or equal to 30. So before using, try to check the sample size. 

Mistake 3

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Confusing sample size with population size
 

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When working on central limit theory, students tend to confuse sample size with population size. The central limit theorem focuses on the sample size, not the population size. 

Mistake 4

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Thinking that central limit theorem applies to all stats
 

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Thinking that the central limit theorem is applicable to statistics other than sample mean is wrong. So to avoid this error, students should understand that the central limit theorem is only applicable to sample distribution of the mean. 

Mistake 5

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Not considering the standard error in calculation

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Students sometimes forget the role of standard error, which can lead to mistakes in calculation. So to avoid it, students should use the standard error formula, σ / n.  

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Solved Examples of Central Limit Theorem

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Problem 1

A company reports that the average salary of its employees is $45,000 per year, with a standard deviation of $8,000. If a random sample of 64 employees is taken, what are the mean and standard deviation of the sample mean salaries?

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 The mean of the sample mean is $45,000


The standard deviation of the sample mean is $1,000
 

Explanation

The mean of the sample mean is equal to the population mean


So the sample mean is $45,000


The standard deviation of the sample is calculated by using the formula, \( \sigma \over \sqrt{n}\)


That is, \( {8000 \over \sqrt{64}} \)


\( 8000\over 8\) = 1000. 
 

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Problem 2

The study hours of college students follow a distribution with a mean of 15 hours per week and a standard deviation of 5 hours. If a sample of 49 students is taken, what are the mean and standard deviation of the sample mean study hours?

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The mean of the sample mean is 15 hours.

 

The standard deviation of the sample mean is 0.71 hours.

Explanation

The mean of the sample mean is equal to the population mean


So the sample mean is 15 hours


The standard deviation of the sample is calculated by using the formula, \( \sigma \over \sqrt{n}\)


That is, \( 5\over \sqrt{49}\) = \( 5 \over 7\) = 0.71.

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Problem 3

The daily coffee consumption of people in a city follows a distribution with a mean of 2.5 cups and a standard deviation of 0.8 cups. If a random sample of 36 people is selected, what are the mean and standard deviation of the sample mean daily coffee consumption?

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The mean of the sample mean is 2.5 cups.


The standard deviation of the sample mean is 0.133 cups.
 

Explanation

The mean of the sample mean is equal to the population mean


So the sample mean is 2.5 cups


The standard deviation of the sample is calculated by using the formula, \( \sigma \over \sqrt{n}\)


That is \( 0.8 \over \sqrt{36}\) 


= \( 0.8 \over {6}\) = 0.133.  

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Problem 4

A study on airline flight delays finds that the average delay time is 25 minutes, with a standard deviation of 10 minutes. If a sample of 64 flights is chosen, what are the mean and standard deviation of the sample mean delay times?

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The mean of the sample mean is 25 minutes.


The standard deviation of the sample mean is 1.25 minutes.
 

Explanation

The mean of the sample mean is equal to the population mean


So the sample mean is 25 minutes


The standard deviation of the sample is calculated by using the formula, \( \sigma \over \sqrt{n}\)


That is, \( 10 \over \sqrt{64}\)


= \( 10\over 8\) = 1.25.

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Problem 5

The recorded high temperatures in a city during summer have a mean of 95°F and a standard deviation of 6°F. If a random sample of 81 days is selected, what are the mean and standard deviation of the sample mean temperatures?

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The mean of the sample mean is 95°F.


The standard deviation of the sample mean is 0.667°F.
 

Explanation

The mean of the sample mean is equal to the population mean

 

So the sample mean is 95°F

 

The standard deviation of the sample is calculated by using the formula, \( \sigma \over \sqrt{n}\)


That is, \( 6 \over \sqrt{81}\) 


= \( 6\over 9\) = 0.667

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FAQs on Central Limit Theorem

1.What is the central limit theorem?

The central limit theorem states that the mean of the sampling data will have the normal distribution, regardless of the shapes of the distribution. 

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2.What is the minimum sample size required for the central limit theorem?

The minimum sample size required to apply the central limit theorem is 30.

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3.Is the central limit theorem applicable only to normal distribution?

The central limit theorem is applicable not only to normal distribution, but for all types of distributions. 
   

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4.What is the formula for the central limit theorem?

The formula of the central limit theorem is x̄ ∼ N(μ σ / √n).

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5.What are the real-world applications of the central limit theorem?

The central limit theorem is applicable in different fields to analyze and study the population from a sample. It is used in economics, data science, biology, and so on. 
 

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Jaipreet Kour Wazir

About the Author

Jaipreet Kour Wazir is a data wizard with over 5 years of expertise in simplifying complex data concepts. From crunching numbers to crafting insightful visualizations, she turns raw data into compelling stories. Her journey from analytics to education ref

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Fun Fact

: She compares datasets to puzzle games—the more you play with them, the clearer the picture becomes!

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