Central Limit Theorem Simulator

What does the central limit theorem state?

The central limit theorem (CLT) is one of the fundamental results of mathematical statistics. It states that, under general conditions, the distribution of the sum (or the mean) of a large number of independent and identically distributed (i.i.d.) random variables tends to the normal distribution, regardless of the original distribution of the variables.

Formally, let \(X_1, X_2, \ldots, X_n\) be i.i.d. random variables with mean \(\mu\) and finite variance \(\sigma^2 < \infty\). Then the sample mean \(\bar{X}_n = \frac{1}{n}\sum_{i=1}^{n} X_i\) satisfies:

Equivalently, \(\bar{X}_n \approx \mathcal{N}\!\left(\mu,\, \frac{\sigma^2}{n}\right)\) for \(n\) sufficiently large. The speed of convergence depends on the skewness of the original distribution: the more symmetric and "well-behaved" the distribution is, the faster the sample mean converges to normal. The Berry-Esseen theorem quantifies this speed with a uniform bound on the approximation error:

where \(\rho = \mathbb{E}[|X-\mu|^3]\) is the third absolute centered moment and \(C \approx 0.4748\). This implies that distributions with greater skewness require larger samples for the normal approximation to be accurate.

What sample size n does the CLT need?

There is no single threshold: the speed of convergence depends on the underlying distribution. As a practical guide:

• Symmetric distributions (Normal, Uniform): with \(n = 2\) to \(5\) the distribution of the mean is already practically normal. The Normal is, trivially, exact for any \(n\).
• Moderately skewed distributions (Exponential, Poisson): the usual rule of thumb of \(n \geq 30\) is typically enough. With the exponential (skewness coefficient \(= 2\)) the approximation is already very good at \(n = 30\).
• Bounded discrete distributions (Bernoulli): with probabilities close to 0 or 1 the skewness is high, but because the variable is bounded, convergence is relatively fast. At \(n = 30\) the approximation is excellent.
• Heavy-tailed distributions (Chi-square with few degrees of freedom): the skewness is more pronounced and \(n \geq 50\) may be needed for a satisfactory approximation.
• Distributions without finite variance (Cauchy): the CLT does not apply in its standard form, since the variance does not exist. In these cases the law of large numbers does not converge either.

The practical rule \(n \geq 30\) found in many textbooks is a conservative heuristic that is valid for the distributions commonly encountered in applied sciences. However, for highly skewed distributions or distributions with very heavy tails, it may be insufficient.

Practical implications of the CLT

The CLT has far-reaching consequences in applied statistics:

• Justifies the use of z and t tests: mean tests (z-test, t-test) assume that the sample mean follows a normal distribution. Thanks to the CLT, this holds even when the individual data are not normal, provided \(n\) is sufficiently large.
• Confidence intervals: the construction of confidence intervals for the population mean relies directly on the CLT, making \(\bar{X} \pm z_{\alpha/2}\,\sigma/\sqrt{n}\) a valid approximation regardless of the original distribution.
• Ubiquity of the normal distribution: many natural and social phenomena result from the accumulation of many small independent effects (measurement errors, genetic variation, economic fluctuations…), which explains why the normal distribution appears so frequently in nature.
• Extension to sums: since \(S_n = n\bar{X}_n\), the CLT applies equally to the sum of i.i.d. variables: \((S_n - n\mu)/(\sigma\sqrt{n}) \xrightarrow{d} \mathcal{N}(0,1)\). This is the basis of the normal approximation to the binomial.
• Experimental design and resampling: the CLT underpins techniques such as the bootstrap and power analysis, by guaranteeing that statistics based on means have asymptotically normal distributions.

Frequently asked questions