Central Limit Theorem
The theorem that averages of many independent random variables become approximately normally distributed, regardless of the original distribution.
The central limit theorem (CLT) states that the sum — equivalently the average — of a large number of independent, identically distributed random variables with finite mean and variance is approximately normally distributed, regardless of the shape of the original distribution. If X₁ … Xₙ are independent draws with mean μ and standard deviation σ, then for large n the standardized average (X̄ − μ)/(σ/√n) converges in distribution to the standard normal N(0, 1).
What makes the theorem remarkable is that the parent distribution does not matter: whether the individual values are binary, uniform, skewed, or heavy-tailed (within finite-variance limits), the average becomes bell-shaped as the sample grows. A common rule of thumb is that roughly 30 independent observations suffice for approximately symmetric distributions, while strongly skewed distributions may need far more.
Two practical consequences follow directly. The standard error of the mean is σ/√n, which explains why larger samples give more precise estimates and why averaging cancels noise. And the theorem justifies the normal approximation to the binomial distribution, along with confidence intervals and hypothesis tests throughout applied statistics.
The theorem requires independence and finite variance. When variables are dependent, or when the distribution has infinite variance, the CLT fails and other limit laws apply — for example, stable laws with heavier tails. These boundary cases are not curiosities: they arise in finance, network traffic, and other domains with extreme values.
Tags
distributions mathematics probability sampling statistics
Related articles
- Standard Deviation
- Bayes' Theorem
- Fundamental Theorem of Arithmetic
- Non-Euclidean Geometry
- Pythagorean Theorem
Click here for easy-to-read helpful e-books for anyone, anywhere, and about anything