The central limit theorem and the standard error of the mean

Why does the sample mean have a distribution?

Draw one sample and you get one sample mean. Draw another and you get a different one. The sample mean is therefore a random variable, and the pattern of values it takes across all possible samples of the same size is its sampling distribution . To judge how close X̄ is likely to be to μ, you need to know the shape, center and spread of that distribution. The central limit theorem supplies all three.

What does the central limit theorem say?

Memorize

The central limit theorem (CLT) says that, for any population with mean μ and finite variance σ², the sampling distribution of the sample mean from random samples of size n is approximately normal with mean μ (the population mean ) and variance σ² ÷ n, once n is large.

Sample mean, X̄large n, any population with finite variance

• Shape Memorize approximately normal (n ≥ 30 as a guide)
• Center Memorize equals the population mean μ
• Spread Memorize variance σ² ÷ n
  • Standard error Memorize σ ÷ √n, or s ÷ √n when σ is unknown

$$\bar{X} \sim N\!\left(\mu,\ \frac{\sigma^2}{n}\right)$$

Press the draw button repeatedly with a small n, then a large one. The population stays skewed, but the sample means pile up in a bell shape that narrows as n grows.

What does the theorem not say?

It does not say the population, or the individual observations, become normal. A skewed return series stays skewed however many observations you collect. What becomes normal is the distribution of the average.

It also does not say the sample mean moves toward a different center. The center is μ for every n; larger n only tightens the spread. And “large” is a guide: n ≥ 30 is the usual rule, but a very non-normal population can need many more observations.

Why it matters: the normal distribution is well tabulated. Because of the CLT we can make probability statements about a population mean from a sample mean, even when returns are skewed or their distribution is unknown. That is the basis of confidence intervals and the hypothesis tests in the next module.

How precise is the sample mean?

Memorize

The variance of X̄ is σ² ÷ n. Its square root, the standard deviation of the sample mean, is the standard error of the sample mean .

$$\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}$$

In practice σ is rarely known, so we replace it with the sample standard deviation s, computed with an n − 1 divisor.

$$s_{\bar{X}} = \frac{s}{\sqrt{n}}$$

Standard error of the sample mean

• Population σ known rare
  • σ ÷ √n Memorize
• Population σ unknown the usual case
  • s ÷ √n Memorize s is the sample standard deviation (n − 1 divisor)

Exam tip The standard error divides by √n. If you divided by n, or used the variance, you have fallen for an option.

Standard error or standard deviation?

They are different concepts and are not interchangeable.

	Standard deviation	Standard error
Measures	spread of the observations around their mean	spread of the sample mean around the population mean
Answers	how variable are the data?	how precise is my estimate?
Effect of larger n	settles near the true σ	shrinks toward zero
Use	describing	inferring

How big must the sample be?

Rearrange the formula: to reach a target standard error, take n = (s ÷ standard error)². Precision is expensive. Halving the target means four times as many observations.

$$n = \left(\frac{s}{s_{\bar{X}}}\right)^2$$

Go deeper A preview of confidence intervals

Because X̄ is approximately normal, about 95% of sample means fall within 1.96 standard errors of μ. Turned around, X̄ ± 1.96 × standard error is a 95% confidence interval for μ. With X̄ = 8% and a standard error of 3%, the interval is 8 ± 1.96 × 3 = 2.12% to 13.88%. When σ is estimated by s, the t-distribution is used in place of the normal, and the interval is a little wider. The next module builds the full testing framework on this idea.

Go deeper Why n ≥ 30 is only a rule of thumb

How fast the sample mean becomes normal depends on how far the population is from normal. Roughly symmetric data converge with a handful of observations. Heavily skewed or fat-tailed returns may need hundreds. Check the shape of the data rather than treating 30 as a guarantee.

Go deeper A common misuse of the theorem

An analyst holds 40 monthly returns from a strategy that sells options: small gains most months, a rare large loss. She argues that, because of the CLT, next month’s return is normally distributed. It is not. The theorem describes the average of the 40 returns, not any single month, and one month’s return keeps the population’s skew. Even for the average, a population this lopsided may need far more than 40 observations before the normal approximation is reliable.