CLT Examples from Uniform, Exponential, and Mixture Populations

One Theorem, Different Starting Shapes

The Central Limit Theorem (CLT) says that, under appropriate conditions, a standardized sample mean approaches a normal distribution as the sample size grows. But “approaches normal” does not mean every population begins in the same place or settles into the same shape at the same pace. A symmetric uniform population, a strongly right-skewed exponential population, and a population split into separated groups provide three useful contrasts.

Throughout, suppose \(X_1,\ldots,X_n\) are independent observations from the same population, with population mean \(\mu\) and finite, positive population standard deviation \(\sigma\). The sample mean is \(\bar{x}\). Its mean is \(\mu\), and its standard error is \(\sigma/\sqrt{n}\). The CLT states that the standardized sample mean

has a distribution that approaches the standard normal distribution as \(n\) increases. This is an approximation for the distribution of sample means, not a claim that the individual observations become normal. The examples below track both the common scale and the population-specific shape.

Why Skewness and Kurtosis Change With Sample Size

A useful way to compare the examples is to examine standardized skewness and excess kurtosis. Let \(\mu_3=E[(X-\mu)^3]\) and \(\mu_4=E[(X-\mu)^4]\). When these moments exist, population skewness is \(\gamma_1=\mu_3/\sigma^3\), and population excess kurtosis is \(\gamma_2=\mu_4/\sigma^4-3\). Skewness describes asymmetry; excess kurtosis describes a fourth-moment feature of shape relative to a normal distribution. Neither number completely describes a distribution.

For independent observations with finite third moments, the centered sample mean is \(n^{-1}\sum_{i=1}^{n}(X_i-\mu)\). Its variance is \(\sigma^2/n\). In the third moment of the sum, terms involving a factor \(X_i-\mu\) to the first power and an independent centered factor have expectation zero. Only the \(n\) individual third moments remain. Therefore the third central moment of \(\bar{x}\) is \(\mu_3/n^2\). Dividing by its standard deviation cubed, \((\sigma/\sqrt{n})^3\), gives

When fourth moments exist, expanding the fourth power of the centered sum gives \(n\mu_4+3n(n-1)\sigma^4\). After dividing by \(n^4\), this is the fourth central moment of \(\bar{x}\). Standardizing by \((\sigma/\sqrt{n})^4\) and subtracting 3 yields

These results give precise shape clues: skewness shrinks in proportion to \(1/\sqrt{n}\), and excess kurtosis shrinks in proportion to \(1/n\). They do not provide a complete measure of how close a distribution is to normal. In particular, a symmetric distribution can have zero skewness and still have a very nonnormal shape.

A Strongly Right-Skewed Starting Population

An exponential population provides a different starting shape. If \(X\) has an exponential distribution with mean 1, then \(\mu=1\), \(\sigma=1\), population skewness is 2, and excess kurtosis is 6. The individual observations are nonnegative and strongly right-skewed. The CLT still applies to independent observations because the population has a finite variance, but small-sample means can retain visible right-skew.

A Separated Mixture Can Have Clusters

A mixture population is formed by drawing from one of several component distributions according to specified probabilities. When the components are far apart, the population can have distinct groups rather than one central mound. A sample mean averages across both the component assignments and the observations within each component. For a small sample, the number drawn from each group may strongly shape the sample-mean distribution.

Consider the following concrete model. Let \(B\) equal 0 or 1 with probability \(1/2\) each, let \(U\) be Uniform(0, 1), and let them be independent. Define \(X=10B+U\). Thus an observation is drawn uniformly from either the interval [0, 1] or [10, 11], with equal probability. This population is symmetric about 5.5 and has a finite variance, so the CLT applies to independent observations from it.

Comparing the Three Examples

The uniform and mixture examples both have zero skewness, yet the mixture has separated regions when \(n=4\), while the uniform sample mean has a continuous bounded distribution. This contrast is a reminder that a single shape statistic cannot capture all features relevant to normal approximation. The CLT supplies the limiting result under its assumptions; the population’s starting shape helps explain what may happen before that limit is a useful approximation.

Assumptions and a Common Mistake

For the examples in this tutorial, the observations are independent and identically distributed, and the population has finite positive variance. Those conditions give the stated CLT result for the standardized sample mean. The calculations involving skewness and excess kurtosis additionally require the corresponding population moments to exist. The uniform, exponential, and bounded mixture populations used here satisfy these moment conditions.

• Mistake: “The population is symmetric, so its sample means are normal.” Correction: Symmetry implies zero skewness when the moment exists; it does not imply normality. A separated symmetric mixture demonstrates why other aspects of shape matter.
• Mistake: “Because the CLT applies, a normal approximation is accurate for any chosen sample size.” Correction: The theorem describes a limiting approximation as \(n\) grows. The examples show that the starting population can affect how the approach looks, and the theorem alone does not supply a universal cutoff for adequate accuracy.
• Mistake: “A sample of observations becomes normally distributed as its size increases.” Correction: The CLT concerns the distribution of the standardized sample mean across repeated samples. It does not change the population distribution of individual observations.