Central Limit Theorem

Direct Answer

The central limit theorem states that, for a sufficiently large sample size, the sampling distribution of the sample mean will approximate a normal distribution regardless of the population’s original distribution, provided the observations are independent, identically distributed, and have finite variance.

Informal Interpretation

Imagine repeatedly drawing random samples of size n from any population and computing each sample’s mean. When you chart these means, the histogram morphs into the familiar bell curve as n grows, even if the underlying data were skewed or uniform.

Formal Statement

Let X₁, X₂, …, X_n be independent, identically distributed random variables with mean μ and standard deviation σ.

Define the sample mean &overline;X = (1/n)∑_i=1ⁿ X_i.

Then the standardized variable Z = (&overline;X – μ)/(σ/√n) converges in distribution to the standard normal distribution N(0,1) as n → ∞.

Sampling Distribution Parameters

• The mean of the sample mean &overline;X equals the population mean μ.
• The standard deviation of &overline;X, often called the standard error, equals σ/√n.

Conditions and Exceptions

• Observations must be independent and identically distributed.
• The population must have finite variance; distributions with infinite variance (e.g., the Cauchy distribution) do not satisfy the theorem’s premises.
• A common rule of thumb is that n ≥ 30 is sufficient for the sampling distribution to be approximately normal, though larger n improves accuracy.

Examples

1. Sampling n = 50 observations from a skewed or uniform distribution yields sample means that cluster in a bell-shaped pattern.
2. Drawing samples of size n = 100 from a Poisson population produces a near-normal distribution of means, centered at the population’s true mean.

Importance

• Provides the foundation for constructing confidence intervals and hypothesis tests when the population distribution is unknown.
• Justifies using normal-based methods in quality control, survey sampling, and many applied fields.

Next Steps

1. Examine Edgeworth expansions for refined approximations.
2. Study the Lindeberg and Lyapunov conditions that generalize the theorem.
3. Explore applications in bootstrapping and resampling methods.
4. Investigate Bayesian inference approaches to sampling distributions.