What Is the Central Limit Theorem?

The central limit theorem (CLT) states that, for a sufficiently large random sample, the distribution of the sample mean is approximately normal, even when the original population is not normally distributed. In IB Mathematics: Applications and Interpretation HL, $n>30$ is considered sufficiently large.

It describes what would happen if many samples of the same size were taken: each sample produces one mean, and those means form a new distribution. It does not claim that the original population becomes normal.

This is HL-only content from AHL 4.15 Central limit theorem. It can be tested in any GDC-active HL paper style.

The mechanism

Suppose a population has mean $\mu$ and finite standard deviation $\sigma$. For independent observations in a random sample of size $n$, the sampling distribution of the sample mean $\bar{X}$ is approximately

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

Therefore,

$$E(\bar{X})=\mu \quad \text{and} \quad \operatorname{SD}(\bar{X})=\frac{\sigma}{\sqrt{n}}.$$

The quantity $\frac{\sigma}{\sqrt{n}}$ is the standard error. As $$ increases, the sample means become less spread out around .

This result depends on random sampling, independence, and a finite population mean and variance. The approximation generally improves as $n$ increases; $n>30$ is the IB examination convention, not a universal guarantee for every extremely skewed population.

Worked example

Delivery times have mean $12$ minutes and standard deviation $4$ minutes, but their population distribution is not normal. For random samples of $64$ deliveries,

$$\bar{X}\sim N\left(12,\frac{4^2}{64}\right)=N(12,0.25).$$

Thus the standard deviation of $\bar{X}$ is $0.5$ minutes. Using the GDC normal distribution function,

$$P(\bar{X}<13)\approx 0.977.$$

Exam technique

Write the distribution and its parameters before using the GDC. A common misconception is using $\sigma$ as the standard deviation of $\bar{X}$; the correct value is $\frac{\sigma}{\sqrt{n}}$. Also distinguish a single observation from the sample mean , and interpret the final probability in context. Always state what the probability represents.