Central Limit Theorem

Precise definition

For independent identically distributed variables $X_1,\ldots,X_n$ with mean $\mu$ and finite positive variance $\sigma^2$, the central limit theorem states $\sqrt n(\bar X-\mu)/\sigma$ converges in distribution to $N(0,1)$ as $n\to\infty$. Thus $\bar X$ is approximately normal with mean $\mu$ and standard error $\sigma/\sqrt n$ for sufficiently large $n$.

Notation and mathematical language

The theorem concerns the sampling distribution of a standardized sum or mean, not the distribution of individual observations. Convergence is asymptotic; there is no universal sample-size cutoff. Skew, heavy tails, dependence, and rare-event probabilities can require much larger $n$.

Conceptual picture

A sum combines many independent contributions. After centring and scaling, no single finite-variance contribution dominates in the iid setting, and the aggregate shape approaches the normal curve. The mean's spread shrinks at rate $1/\sqrt n$.

Conditions and key results

The basic theorem assumes iid sampling and finite variance. Sampling without replacement from a finite population introduces dependence and may require a finite-population correction or small sampling fraction. The CLT does not make biased data representative and does not prove observations are normal.

A reliable strategy

1. Identify the random variable, sampling design, target mean, variance, and sample size.
2. Check independence or justified weak dependence, identical distribution if using the basic form, and finite variance.
3. Use mean $\mu$ and standard error $\sigma/\sqrt n$, then standardize the requested sample-mean probability.
4. Label the result approximate, assess skew or tails, and use simulation or an exact distribution when conditions are doubtful.

Fully worked example

Interpretation and application

CLT reasoning supports confidence intervals, quality control, polling, and aggregated risk. It quantifies sampling variability under a model; it does not correct selection bias, measurement bias, confounding, or causal ambiguity.

Common mistakes

Verification and reasonableness

• Confirm the standardized bounds and that the probability lies in $[0,1]$.
• Simulate sample means under a plausible parent distribution and compare shapes.
• Check how the approximation changes with $n$ and with an exact model when available.

Practice

1. If $\sigma=20,n=100$, find the standard error.
2. What distribution becomes approximately normal?
3. Does CLT remove sampling bias?

Further deduction

For Bernoulli data, the sample proportion $\hat p$ is a mean of 0–1 variables, so the CLT gives approximate standard error $\sqrt{p(1-p)/n}$. Rules such as $np$ and $n(1-p)$ both at least 10 are heuristics, not theorem hypotheses. When counts are small or probabilities near 0 or 1, exact binomial or improved interval methods are preferable.

Berry–Esseen bounds quantify approximation rate under a finite third absolute moment: the maximum CDF difference is bounded by a constant times $E|X-\mu|^3/(\sigma^3\sqrt n)$. The theorem need not be used computationally in an introductory course, but it explains why stronger skew or tail weight slows normal approximation and why no single $n=30$ rule is universal.

Related topics

• Normal Distribution
• Confidence Intervals
• Sampling