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Lesson 1 of 23 · 13 min

The central limit theorem

Whatever the shape of the population, the mean of a large random sample is approximately normally distributed around the true mean, with variance $\sigma^2/n$ .

In short

Central limit theorem (CLT): for any population with mean $\mu$ and finite variance $\sigma^2$ , the sampling distribution of $\bar X$ from random samples of size $n$ is approximately normal with mean $\mu$ and variance $\sigma^2/n$ when $n$ is large.
Three properties: approximately normal shape; mean equal to the population mean; variance equal to the population variance divided by $n$ .
Rule of thumb: $n \ge 30$ is large enough. A very non-normal population may need a sample well above 30.
The population does not have to be normal. The CLT is about the sample mean, not about individual observations.
Why it matters: it lets us make probability statements about $\mu$ , build confidence intervals and test hypotheses even when the population's distribution is unknown or non-normal.

The sample mean has its own distribution

Draw a random sample of daily returns and compute its mean. Draw another sample of the same size and you get a slightly different mean. Repeat many times and plot all the means: that histogram is the sampling distribution of the sample mean.

Now run the experiment with samples of 10, then 50, then 100, then 300 daily returns. Two things happen as the sample size grows. The histogram looks more and more like a bell-shaped normal curve, and it becomes narrower, clustering tightly around the true mean.

In other words, large samples rarely produce a mean far from $\mu$ , and the shape of their sampling distribution becomes predictable. The central limit theorem states this result precisely.

Sampling distributions of

\bar X

for daily returns with

\mu = 0.04\%

and

\sigma = 1.2\%

. All are centred on

\mu

, but the spread

\sigma/\sqrt{n}

shrinks from 0.38% at n = 10 to 0.07% at n = 300, so a mean far from

\mu

becomes rare.

The theorem and its three properties

Central limit theorem: take a population described by any probability distribution with mean $\mu$ and finite variance $\sigma^2$ . The sampling distribution of the sample mean $\bar X$ , computed from random samples of size $n$ , will be approximately normal with mean $\mu$ and variance $\sigma^2/n$ when $n$ is large.

Read the variance part carefully. The population variance $\sigma^2$ is fixed, but dividing it by $n$ makes the spread of $\bar X$ shrink as the sample grows. Averaging cancels out much of the noise in individual observations, so a sample mean far from $\mu$ becomes progressively rarer.

Shape: $\bar X$ is approximately normal for large $n$ , whatever the population's shape.
Centre: the mean of the distribution of $\bar X$ equals $\mu$ , the population mean.
Spread: the variance of $\bar X$ equals $\sigma^2/n$ , so its standard deviation is $\sigma/\sqrt{n}$ .

How large is 'large'?

The usual rule: when $n \ge 30$ , assume the sample mean is approximately normally distributed.

That is a guideline, not a law. If the population is very non-normal, for example strongly skewed or with fat tails, a sample well above 30 may be needed before the normal distribution describes the sample mean well.

Two conditions must hold for the theorem itself: the samples are random, and the population variance is finite. The population's mean and variance need not be known; they only need to exist.

Why it matters, and what it does not say

We rarely know the shape of a population of returns, and it is often not normal. The CLT says we do not need to know it: for a large sample, the sample mean is approximately normal anyway. That lets us make precise probability statements about the population mean, construct confidence intervals (the confidence-interval lesson of this module) and test hypotheses (the hypothesis-testing lessons later in this module).

The CLT says nothing about individual observations. A skewed population stays skewed; only the distribution of the average of a large sample becomes approximately normal.

It also does not imply that diversifying across many investments drives portfolio risk to zero. Security returns are not independent, and portfolio risk depends on how returns move together. The CLT describes the sample mean of random draws, so it is the wrong tool for that claim.

Common misstatements
Claim	Verdict
The population must be normal for $\bar X$ to be normal.	False: any distribution with finite variance works when n is large.
$\bar X$ is approximately normal for any sample size.	False: only when n is large.
The mean of the sampling distribution of $\bar X$ is $\mu$ .	True.
The variance of $\bar X$ is $\sigma^2$ .	False: it is $\sigma^2/n$ .
Individual returns become normal in large samples.	False: only the sample mean does.

Formulas

Mean of the sample mean: $E(\bar X) = \mu$
The sampling distribution of the sample mean is centred on the population mean.
Variance of the sample mean: $\sigma^2_{\bar X} = \dfrac{\sigma^2}{n}$
$\sigma^2$ = population variance; n = sample size. Its square root, $\sigma/\sqrt{n}$ , is the standard error (next lesson).

Worked examples

Example 1 Describing a sampling distribution

Daily returns on an emerging-market index are positively skewed, with $\mu$ = 0.04% and $\sigma$ = 1.2%. You take random samples of 144 days. Describe the sampling distribution of the sample mean.

Example 2 Spot the errors

A colleague writes: 'Because of the CLT, the mean of 12 monthly returns drawn at random from a skewed distribution is normally distributed, with variance equal to the population variance.' What is wrong?

Example 3 Can we rely on the CLT? (exam style)

For which of these can an analyst treat the sample mean as approximately normal? (1) 40 annual returns of a moderately skewed equity index; (2) 15 monthly returns of a hedge fund with extreme tail events; (3) 500 monthly returns of a strongly skewed strategy.

On the exam

Mostly conceptual: which statements about the CLT are true, what the mean and variance of $\bar X$ are, and the $n \ge 30$ rule. A short calculation of the variance or standard deviation of the sample mean is also common.

The population need not be normal; it must have a finite variance.
Variance of $\bar X$ is $\sigma^2/n$ ; its standard deviation is $\sigma/\sqrt{n}$ . Don't divide the standard deviation by n or the variance by $\sqrt{n}$ .
$n \ge 30$ is a guideline; very non-normal populations need more.
The CLT is about the sample mean, not about individual returns or portfolio diversification.

Check yourself

Q1A population has a variance of 400. For random samples of 25, the variance of the sample mean is:
Q2Statement 1: the means of large random samples are approximately normally distributed. Statement 2: this requires a normally distributed population. Which is correct?
Q3According to the central limit theorem, the mean of the sampling distribution of the sample mean is:
Q4The central limit theorem least likely requires:
Q5As a rule of thumb, a sample is large enough for the sample mean to be treated as approximately normal when n is at least:
Q6Annual returns across a large universe of funds have a standard deviation of 15%. For random samples of 100 funds, the standard deviation of the sample mean is closest to:

0/6Check yourself

Answer the 6 questions: +10 XP for each one you get right first time.

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