How the Central Limit Theorem Turns Any Distribution Into a Normal Curve

1 Central Limit Theorem

In plain terms, the Central Limit Theorem (CLT) states that:

If the sample size is sufficiently large, the sampling distribution of the sample mean approximates a normal distribution, regardless of the original population distribution.

We illustrate the CLT with several examples:

1.1 Uniform Distribution

Given a uniform distribution U(2,5), its probability density function is defined on the interval [2,5].

Randomly drawing 100,000 numbers from this distribution and plotting the histogram yields the theoretical distribution (left) and a sample histogram (right).

Next, we draw k numbers (k = 1, 3, 5, 10) from the distribution, compute their averages, repeat 10,000 times, and plot the histogram of these averages.

As k increases, the shape transforms from the original uniform shape to a normal distribution that becomes increasingly "thin", meaning the standard deviation decreases, while the mean stays around 3.5.

Is this a coincidence? We try another distribution.

1.2 Beta Distribution

The Beta distribution is defined on [0,1] with parameters a and b that control its shape. Its probability density function is given by the Beta function.

Considering a specific case Beta(30,2), we sample 100,000 points and plot the histogram.

Similarly, we draw k numbers (k = 1, 3, 5, 10), compute their averages, repeat 10,000 times, and plot the histogram.

Again, the data shape changes from the original Beta shape to a normal distribution that becomes thinner as k grows, with decreasing standard deviation and a mean around 0.94.

2 Formal Description of the Central Limit Theorem

Central Limit Theorem: Let a sample of size n be drawn from any population with mean μ and variance σ². When n is sufficiently large, the sample mean approximates a normal distribution with mean μ and variance σ²/n.

As the sample size approaches infinity, the sampling distribution approaches a normal distribution, allowing inference about the population from sample observations.

3 Summary

This article uses images and examples to intuitively demonstrate the meaning of the Central Limit Theorem, provides its precise definition, and highlights its importance.

References

https://baijiahao.baidu.com/s?id=1665261046335447411&wfr=spider&for=pc

https://zhuanlan.zhihu.com/p/425388698

https://zhuanlan.zhihu.com/p/46963974