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What is Central Limit Theorem? Properties, Best Practices, Examples & Everything to Know

central limit theorem

Statistics is a fascinating field, and if you’re familiar with Lean Six Sigma, you may have encountered the phrase “central limit theorem.” This article tackles the subject of the central limit theorem, what it is, why it’s important, its properties, and other valuable information.

So, for the uninitiated, let’s define the central limit theorem.

What’s the Central Limit Theorem?

The Central Limit Theorem (CLT for short) is a statistical concept that says the distribution of the sample mean can be approximated by a near-normal distribution if the sample size is large enough, even if the original population is non-normal. The theorem says sampling distribution as the sample size grows, despite the original sample’s distribution.

So, to put it a different way, if you have a population with a mean (mu) and standard deviation (sigma), then take a sufficiently large random, independent sample from the population with replacement, and the sample means distribution will approach a normal distribution.

This formula is the central limit theorem calculator:

The formula tells us that as the denominator (sample size) increases, the Mean’s SE (Standard for Error) decreases.

We will get a clearer idea of how this works when we check out the central limit theorem example presented later.

So, a sample is a group of observations derived from a broader population encompassing all possible observations that somebody could make if somebody tested them. It contains the following characteristics:

1) Observation. The results were gathered from one trial of an experiment.

2) Sample. A group of results was collected from separate independent trials.

3) Population. The space of all possible observations that could be observed from the trial.

The graph’s mean moves towards a normal distribution as an analyst increases the sample’s numbers and size. A critical part of the theorem shows us that the sample’s mean will end up as the mean for the entire population. Thus, if you calculate a given population’s mean multiple samples, add them together, and discover their average, your result will be a summation of the population’s mean.

As a side note, if you don’t want to work with the formula, plenty of online central limit theorem calculators are available.

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The Central Limit Theorem’s Properties

Normal distributions have two parameters: mean and standard deviations. As the sample size grows, the sample distribution’s amplitude comes together on a normal distribution where the means equals the population mean, and the standard deviation equals σ/√n.


“σ” represents the population standard deviation, and “n” stands for the sample size. As the sample size (or “n”) expands, the sampling distribution’s standard deviation shrinks.

Additionally, the CLT works on three assumptions:

1) First, the data must be randomly sampled.

2) The samples must not be related to one another; consequently, no sample should impact the others.

3) Finally, the sample size should be at most 10 percent of the population if the samples are taken without replacement.

The History of the Central Limit Theorem

Now that we’ve settled the central limit theorem definition, let’s look at its origins. The first appearance of the Central Limit Theorem was in an article published in 1733 by Abraham DeMoivre, a French mathematician. In his article, DeMoivre used the normal distribution to determine the number of times heads came up from multiple coin flips.

In 1812, Pierre-Simon Laplace, another French mathematician, revived DeMoivre’s concept. Laplace re-introduced the idea of normal distribution in his work, “Théorie Analytique des Probabilités.” Laplace attempted to approximate a binomial distribution with the normal distribution in his published work.

Then, almost one hundred years later, in 1901, Aleksandr Lyapunov, a Russian mathematician, took things a step further, defining the concept in general terms so he could prove how the idea worked mathematically. Our modern probability theory adopted the characteristic functions he used to support the theorem.

Thus, the early work in Central Limit Theorem statistics laid the groundwork for how statisticians work today, impacting everything from commerce to politics.

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Why Is It So Essential to Understand the Central Limit Theorem?

If you’re working with statistics and probability, it’s essential to understand the central limit theorem definition and why the theorem is vital. This importance boils down to three reasons.

1) The assumption of normality. The CLT is essential for statistics because it lets statisticians safely assume that the mean’s sampling distribution will eventually approach normality. This assumption allows statisticians to take advantage of statistical processes and functions that accept a normal distribution.

2) It imparts flexibility with different distributions. The Central Limit Theorem can be employed with continuous and discrete data, so statisticians have flexibility and choice regarding its use in various applications.

3) It’s independent of the underlying distribution. The sampling distribution’s normality will happen regardless of the underlying distribution’s shape. Thus, you can safely sample from any kind of distribution shape and be assured that the sampling distribution will resemble a normal distribution.

Explaining the Concept of Distribution of the Variable in the Population

The CLT’s definition includes the phrase “regardless of the variable’s distribution in the population.” Regarding population, the variable’s values can follow a variety of probability distributions. For example, the distributions can be normal, right-skewed, left-skewed, and uniform. There are other distributions as well.

Although the Central Limit Theorem applies to all kinds of probability distributions, it comes with some implicit expectations. For instance, the population must include finite variance.

Additionally, the population distribution is also the variable’s probability distribution when we select a random case from the population.

Central Limit Theorem’s Best Practices

Here are three critical tips you need to apply the Central Limit Theorem properly.

1) Choose an appropriate number of samples and sample size. The ideal sample size is about 30. The Central Limit Theorem’s outcome should improve as the number of samples you collect increases.

2) Perform a Measurement System Analysis (MSA). Since you depend on sample data to make your decisions, conduct an MSA to validate your measurement system.

3) Check your normality. Because the desired result of the Central Limit Theorem is to establish a normal means distribution, you must test whether that result has occurred with either a statistical method, such as the Anderson-Darling test, or a Normal Probability Plot.

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Presenting an Example of the Central Limit Theorem

Let’s check out a central limit theorem example. Manufacturing facilities often employ the CLT to estimate how many of their products are defective.

For example, the factory manager could randomly choose 100 products produced by the plant on a given day, then count how many of the products have defects. The manager can then use the proportion of defective products in their sample to estimate the proportion of faulty products that the whole facility produces.

So, suppose the manager discovers that 3.5 percent of the products sampled contain defects. In that case, their best estimate for the proportion of defective products versus successfully created products manufactured by the whole plant is also 3.5 percent.

Now, if that same manager decides to sample more products, say, another 25, the mean will shift towards a normal distribution. This way, the manager will have a figure closer to reality. The more products the manager adds to the sampling, the closer the mean will get to the normal distribution, and consequently, there is less risk of error.

Note:Six Sigma offers a range of powerful tools, such as DMAIC, Control Charts, and Root Cause Analysis, that can help organizations identify process inefficiencies and make data-driven decisions for continuous improvement.

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