1217 LearnersLast updated on June 18th, 2025
Central Limit Theorem
The central limit theorem is a fundamental concept of statistics, which 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.
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 that 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 bigger, it can predict the characteristics of the population.
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What are the Key Components of the Central Limit Theorem
As we have discussed, the central limit theorem. Let us now learn about the key components of the central limit 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 a chance to all the individuals in the population. For example, in political poll if we select the sample based on 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:
The larger the sample, the sampling distribution should be normal. According to the central limit theorem, the perfect sample size should be 30 or more.
How to find the Central Limit Theorem
As we know, the central limit theorem is a fundamental concept in statistics and probability, as it helps to 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. Even if the population distribution is uniform, skewed or binomial.
- The larger the sample size, the narrower the bell curve
- The larger the sample size, the distribution mean is equal to the population mean
Central Limit Theorem Formula
Now let’s see how the formula of limit, if the random variable is X, with an unknown or known 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,
x̄ ∼ N((x̄ - μ) / σ / √n), where σ/ √n, is the standard. The z score of the random variable, x̄ is
Real-world applications of the Central Limit Theorem
The central limit theorem is used in the real world to analyze the population characteristics. Let’s discuss the real-world application 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.
Common Mistakes and How to Avoid Them in Central Limit Theorem
Mistakes are common among students when working on the central limit theorem, as they tend to get confused. So to master the central limit theorem, we can learn a few common mistakes and ways to avoid them.
Mistake 1
Confusing the central limit theorem with the law of large numbers
Students tend to think that the central limit theorem is for all the sample mean will be close to the population mean. Whereas it is wrong as the central limit theorem states that the distribution of the sample mean is always a normal distribution.
Mistake 2
Using central limit theorem for small samples
Using the central limit theorem for a small sample will result in inaccurate conclusion, as it is applicable for the large sample, which is greater than or equal to 30. So before using, try to check the sample size.
Mistake 3
Confusing sample size with population size
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
Thinking that central limit theorem applies to all statistic
Thinking that the central limit theorem is applicable to statistics other than sample mean, but it 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
Not considering the standard error in calculation
Students sometimes forget the role of standard error, which can lead to calculation error. So to avoid them, students should use the standard error formula, σ / n.
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Solved Examples of Central Limit Theorem
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?
The mean of the sample mean is $45000
The standard deviation of the sample mean is $1000
Explanation
The mean of the sample mean is equal to the population mean
So the sample mean is $45000
The standard deviation of the sample is calculated by using the formula, σ / √n
That is, 8000 / √64
= 8000 / 8 = 1000.
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?
The mean of the sample mean is 15 hours
The standard deviation of the sample mean is 0.714 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, σ / √n
That is, 5 / √49 = 5 / 7 = 0.714
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?
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, σ / √n
That is 0.8 / √36
= 0.8 / 6 = 0.133
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
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, σ / √n
That is 10 / √64
= 10 / 8 = 1.25
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?
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 2.5 cups
The standard deviation of the sample is calculated by using the formula, σ / √n
That is 6 / √81
= 6 / 9 = 0.667
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FAQs on Central Limit Theorem
1.What is the central limit theorem?
2.What is the minimum sample size required for the central limit theorem?
3.Is the central limit theorem applicable only to normal distribution?
4.What is the formula for the central limit theorem?
5.What are the real-world applications of the central limit theorem?
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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
Fun Fact
: She compares datasets to puzzle games—the more you play with them, the clearer the picture becomes!
