Central Limit Theorem

Illustration of Central Limit Theorem for a skewed population of values Panel A shows the population (highly skewed right and truncated at zero); Panels B, C, and D show distributions of the mean for sample sizes of 15, 30, and 60, respectively, as obtained through a computational sampling approach. As indicated by the x axes, the sample means are approximately 3. The y axes indicate the number of computational samples obtained for a given mean value. As would be expected, larger-sized samples give distributions that are closer to normal and have a narrower range of values. – ResearchGate


The Central Limit Theorem (CLT) is a statistical theory that states that given a sufficiently large sample size from a population, the distribution of the mean samples from the population will be normally distributed.

Basically, Central Limit Theorem states that no matter what the distribution of the sample is if you sample batches of data from that distribution(with replacement) and take the mean of each batch. Then the mean values that we got from all those batches will be normally distributed.1

Whatever the form of the population distribution, the sampling distribution tends to a Gaussian, and its dispersion is given by the central limit theorem – Wikipedia


Example 1: Economics
Economists often use the central limit theorem when using sample data to draw conclusions about a population.

Example 2: Biology
Biologists use the central limit theorem whenever they use data from a sample of organisms to draw conclusions about the overall population of organisms.

Example 3: Manufacturing
Manufacturing plants often use the central limit theorem to estimate how many products produced by the plant are defective.

Example 4: Surveys
Human Resources departments often use the central limit theorem when using surveys to draw conclusions about overall employee satisfaction at companies.

Example 5: Agriculture
Agricultural scientists use the central limit theorem whenever they use data from samples to draw conclusions about a larger population. 6


In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usually abbreviated as i.i.d. or iid or IID. IID was first used in statistics. With the development of science, IID has been applied in different fields such as data mining and signal processing.3


See Theoretical Knowledge Vs Practical Application.


In a mathematical point of view, central limit theorem states the importance of the Gaussian distribution as a natural limiting distribution, and for instance, can also be used to prove the existence of the Brownian motion (see Donsker theorem), which is, similarly, a natural limiting process for many random walks.

From a practical point of view, it justifies many assumption related to statistics, for instance the normality of the error terms in linear regression: if we see an error term as the independent sum of many random variables with finite variance (unobservable errors), the we can naturally assume it is normally distributed.

In a concrete way, when you don’t know the distribution of some data, then you can use the CLT to make the assumption about their normality.

However, the downside of the CLT is that it is often used without checking the assumptions, which has been the case in finance for some time, supposing returns were normal, whereas they have a fat tailed distribution, which inherently carries more risks than the normal distribution. (Samy Msly) 5

Visualizing the Central Limit Theorem

We will now try to cement our points via the use of an instructive online tool that you can play around with on your own as well.4

Many of the References and Additional Reading websites and Videos will assist you with understanding the CLT.

As some professors say: “It is intuitively obvious to even the most casual observer.


1 “Central Limit Theorem: A Complete Guide For Beginners”. 2021. Medium. https://ai.plainenglish.io/central-limit-theorem-a-complete-guide-for-beginners-c17a6bd7417.

2 Sampling ”with replacement” means that when a unit selected at random from the population, it is returned to the population (replaced), and then a second element is selected at random.

3 “Independent And Identically Distributed Random Variables – Wikipedia”. 2009. en.wikipedia.org. https://en.wikipedia.org/wiki/Independent_and_identically_distributed_random_variables.

4 Michelakis, Panos. “The Intuition Behind The Central Limit Theorem”. 2022. Medium. https://medium.com/intuition/the-intuition-behind-the-central-limit-theorem-b7f32278ec09.

Probability theory is humankind’s primary weapon when studying the properties of chaos and uncertainty. Despite us having a vast arsenal of mathematical tools, probability theory utilizes elementary mathematics coupled with logic and common sense. It helps us uncover patterns and order amidst the chaos that rules our world. The Central Limit Theorem — or CLT for short — is one of the most profound and useful theorems in probability theory and applied statistics that achieves this goal.

5 “How Can The Central Limit Theorem Be Used?” 2022. Quora. https://www.quora.com/How-can-the-central-limit-theorem-be-used.

6 Zach. “5 Examples Of Using The Central Limit Theorem In Real Life – Statology”. 2021. Statology. https://www.statology.org/central-limit-theorem-real-life-examples/.

Additional Reading

Albright, Jeremy. “The Central Limit Theorem and its Implications for Statistical Inference”. 2019. Methods. https://tutorials.methodsconsultants.com/posts/the-central-limit-theorem-and-its-implications-for-statistical-inference/.

The central limit theorem is perhaps the most fundamental result in all of statistics. It allows us to understand the behavior of estimates across repeated sampling and thereby conclude if a result from a given sample can be declared to be “statistically significant,” that is, different from some null hypothesized value. This brief tutorial explains what the central theorem tells us and why the result is important for statistical inference.

The central limit theorem tells us exactly what the shape of the distribution of means will be when we draw repeated samples from a given population. Specifically, as the sample sizes get larger, the distribution of means calculated from repeated sampling will approach normality. What makes the central limit theorem so remarkable is that this result holds no matter what shape the original population distribution may have been.

Bahl, Aditya. “Understanding The Central Limit Theorem”. 2020. Medium. https://towardsdatascience.com/understanding-the-central-limit-theorem-de6e65385e97.

Central Limit Theorem (CLT for short) is one of the most important concepts in the field of statistics. In this post, I will try to explain this concept in a simple and non-technical manner. Let’s continue from our high school example as discussed in this post here.

“Central Limit Theorem”. 2021. Medium. https://medium.com/intuition/central-limit-theorem-d70571ac26cb.

“Central Limit Theorem”. 2022. sphweb.bumc.bu.edu. https://sphweb.bumc.bu.edu/otlt/MPH-Modules/BS/BS704_Probability/BS704_Probability12.html.

“Central Limit Theorem Explained – Statistics By Jim”. 2022. Sstatisticsbyjim.com. https://statisticsbyjim.com/basics/central-limit-theorem/.

“Central Limit Theorem | Formulas | Proof| Central Limit Theorem Examples”. 2022. BYJUS. https://byjus.com/jee/central-limit-theorem/.

Central limit theorem is a statistical theory which states that when the large sample size has a finite variance, the samples will be normally distributed and the mean of samples will be approximately equal to the mean of the whole population.

In other words, the central limit theorem states that for any population with mean and standard deviation, the distribution of the sample mean for sample size N has mean μ and standard deviation σ/√n.

As the sample size gets bigger and bigger, the mean of the sample will get closer to the actual population mean. If the sample size is small, the actual distribution of the data may or may not be normal, but as the sample size gets bigger, it can be approximated by a normal distribution. This statistical theory is useful in simplifying analysis while dealing with stock indexes and many more.

The CLT can be applied to almost all types of probability distributions. But there are some exceptions. For example, if the population has a finite variance. Also, this theorem applies to independent, identically distributed variables. It can also be used to answer the question of how big a sample you want. Remember that as the sample size grows, the standard deviation of the sample average falls because it is the population standard deviation divided by the square root of the sample size. This theorem is an important topic in statistics. In many real-time applications, a certain random variable of interest is a sum of a large number of independent random variables. In these situations, we can use the CLT to justify using the normal distribution.

“Central Limit Theorem – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/Central_limit_theorem.

The central limit theorem states that if you have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement2, then the distribution of the sample means will be approximately normally distributed. This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large (usually n > 30). If the population is normal, then the theorem holds true even for samples smaller than 30. In fact, this also holds true even if the population is binomial, provided that min(np, n(1-p))> 5, where n is the sample size and p is the probability of success in the population. This means that we can use the normal probability model to quantify uncertainty when making inferences about a population mean based on the sample mean.

Data Note. “Central Limit Theorem: A Complete Guide For Beginners”. 2021. Medium. https://ai.plainenglish.io/central-limit-theorem-a-complete-guide-for-beginners-c17a6bd7417.

The Central Limit Theorem (CLT) is a statistical theory that states that given a sufficiently large sample size from a population, the distribution of the mean samples from the population will be normally distributed. Basically, Central Limit Theorem states that no matter what the distribution of the sample is if you sample batches of data from that distribution(with replacement) and take the mean of each batch. Then the mean values that we got from all those batches will be normally distributed.

Godwin, James Andrew. “On The Importance Of The Central Limit Theorem”. 2020. Medium. https://medium.com/swlh/on-the-importance-of-the-central-limit-theorem-e8cce1f4d253.

While I remembered the number, I never fully appreciated the importance of the CLT and just how fundamental it is to statistics, and for me, its application to machine learning models. The CLT is the phenomenon that allows for what we consider to be statistics: it enables sampling methods. Without it, we could not reliably compute confidence intervals, and most statistical methods and machine learning algorithms rely on the CLT, Hypothesis tests being just one example.

Lim, Russell. “The Central Limit Theorem — Why Is It So?”. 2021. Medium. https://www.cantorsparadise.com/the-central-limit-theorem-why-is-it-so-2ae93edf6e8.

Ramzai, Juhi. “Clearly Explained: The Mighty Central Limit Theorem”. 2020. Medium. https://towardsdatascience.com/clearly-explained-the-mighty-central-limit-theorem-b8152b94258.

Rumsey, Deborah J. “Dummies – Learning Made Easy”. 2022. dummies.com. https://www.dummies.com/article/academics-the-arts/math/statistics/how-the-central-limit-theorem-is-used-in-statistics-169776.

The normal distribution is used to help measure the accuracy of many statistics, including the sample mean, using an important result called the Central Limit Theorem. This theorem gives you the ability to measure how much the means of various samples will vary, without having to take any other sample means to compare it with. By taking this variability into account, you can use your data to answer questions about a population, such as “What’s the mean household income for the whole U.S.?”; or “This report said 75% of all gift cards go unused; is that really true?” (These two particular analyses are made possible by applications of the Central Limit Theorem called confidence intervals and hypothesis tests, respectively.)

The Central Limit Theorem (CLT for short) basically says that for non-normal data, the distribution of the sample means has an approximate normal distribution, no matter what the distribution of the original data looks like, as long as the sample size is large enough (usually at least 30) and all samples have the same size. And it doesn’t just apply to the sample mean; the CLT is also true for other sample statistics, such as the sample proportion. Because statisticians know so much about the normal distribution, these analyses are much easier.

“The Central Limit Theorem”. 2022. stat.ucla.edu. http://www.stat.ucla.edu/~nchristo/introeconometrics/introecon_central_limit_theorem.pdf.

Tomar, Anmol. “Explaining Central Limit Theorem To My Wife”. 2022. Medium. https://anmol3015.medium.com/explaining-central-limit-theorem-to-my-wife-17db37c988.

So, the Central Limit Theorem states that if you take samples from a distribution (we took samples from uniform distribution — each number has an equal likelihood of occurrence) and find the distribution of the sample mean then it will tend to normal as we increase the sampling exercise to a sufficiently large number.

“Using The Central Limit Theorem | Introduction To Statistics”. 2022. courses.lumenlearning.com. https://courses.lumenlearning.com/introstats1/chapter/using-the-central-limit-theorem/.

Yıldırım, Soner. “Central Limit Theorem — Explained With Examples”. 2020. Medium. https://towardsdatascience.com/central-limit-theorem-explained-with-examples-4c10377ee58c.


Central limit theorem | Inferential statistics | Probability and Statistics | Khan Academy


Central Limit Theorem – Sampling Distribution of Sample Means – Stats & Probability


This statistics video tutorial provides a basic introduction into the central limit theorem. It explains that a sampling distribution of sample means will form the shape of a normal distribution regardless of the shape of the population distribution if a large enough sample is taken from the population. This video gives plenty of examples and practice problems.

The Central Limit Theorem, Clearly Explained!!!


The Central Limit Theorem is a big deal, but it’s easy to understand. Here I show you what it is, then I describe why this is useful and fundamental to Statistics!

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