Mathematics.

inferential statistics

Central Limit Theorem

Statistics30 minDifficulty4 out of 10

You should know: sampling distributions, normal distribution

Overview

The central limit theorem (CLT) states that if you draw sufficiently large independent random samples from any population with finite mean μ and finite variance σ², the distribution of the sample means approaches a normal distribution, regardless of the shape of the original population's distribution. This holds even if the underlying population is skewed, bimodal, or otherwise far from normal. As a practical rule of thumb, a sample size of n ≥ 30 is often considered large enough for the approximation to be good, though more skewed populations may require larger n. The CLT is the reason normal-distribution-based methods (confidence intervals, hypothesis tests) work so broadly across statistics, even when raw data isn't normally distributed.

Intuition

Even if a single die roll is uniformly distributed (flat, not bell-shaped), the sum or average of many die rolls tends to pile up near the middle and taper off at the extremes — this is the CLT in action. Averaging washes out the individual quirks of the original distribution because extreme values in one direction tend to be balanced by extreme values in the other across many independent draws, leaving behind the bell-shaped pattern predicted by the CLT.

Formal Definition

Definition

Let X₁, X₂, ..., Xₙ be independent, identically distributed random variables with mean μ and variance σ² < ∞. As n → ∞, the standardized sample mean converges in distribution to the standard normal:

Zn=Xˉn−μσ/n→dN(0,1)Z_n = \frac{\bar{X}_n - \mu}{\sigma/\sqrt{n}} \xrightarrow{d} N(0,1)
CLT convergence statement
Xˉn≈N ⁣(μ,σ2n)for large n\bar{X}_n \approx N\!\left(\mu, \frac{\sigma^2}{n}\right) \quad \text{for large } n
Practical normal approximation

Worked Examples

  1. By the CLT, since n = 36 is reasonably large, the sample mean is approximately normal with mean μ and standard error σ/√n.

    σXˉ=1236=126=2\sigma_{\bar{X}} = \frac{12}{\sqrt{36}} = \frac{12}{6} = 2

  2. State the approximating normal distribution.

    Xˉ≈N(40,22)\bar{X} \approx N(40, 2^2)

Answer: The sample mean is approximately N(40, 4), i.e., mean 40 and standard deviation 2.

Practice Problems

Difficulty 4/10

A population has μ = 100, σ = 15. For n = 25, find the standard error of the sample mean and the approximating normal distribution.

Difficulty 5/10

Using the distribution from the previous problem (X̄ ≈ N(100, 9), standard error 3), find P(X̄ < 97).

Difficulty 5/10

A factory's individual part weights are heavily right-skewed (not normal), with mean 50 g and standard deviation 8 g. Explain why the average weight of a random sample of 64 parts can still be treated as approximately normal.

Quiz

The central limit theorem states that the sampling distribution of the sample mean approaches a normal distribution as sample size grows, provided:
As sample size n increases, the standard error of the sample mean (σ/√n):
Why is the central limit theorem so important in practice?

Summary

  • The CLT says the sampling distribution of the sample mean approaches normal as n grows, regardless of the population's original shape.
  • The approximating distribution is N(μ, σ²/n); a common rule of thumb is n ≥ 30 for a reasonable approximation.
  • The CLT underlies why normal-based confidence intervals and hypothesis tests work broadly, even for non-normal populations.

References

  1. WebsiteWikipedia — Central limit theorem