Why Is a Sample Size of 30 Often Considered “Sufficient” for the Central Limit Theorem?

A sample size near 30 is often considered sufficient because, for many population distributions, the sampling distribution of the sample mean is already reasonably close to normal. However, 30 is a practical guideline, not a universal mathematical threshold; for IB Mathematics: Applications and Interpretation HL, the examination convention is specifically that $n>30$ is considered sufficient.

The central limit theorem (CLT) states that, as the sample size $n$ increases, the distribution of the sample mean $\bar{X}$ approaches a normal distribution, even when the original population is not normally distributed. If the population has mean $\mu$ and variance $\sigma^2$, then for sufficiently large $n$,

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

For example, suppose a population has mean $50$, standard deviation $12$, and samples of size $36$ are taken. Then

$$\bar{X}\sim N\left(50,\frac{12^2}{36}\right)=N(50,4),$$

so the standard error is $2$. A probability involving the sample mean is therefore calculated using mean $50$ and standard deviation $2$, provided the observations are independent and the CLT approximation is appropriate.

Therefore, $E(\bar{X})=\mu$, while the standard error is $\frac{\sigma}{\sqrt{n}}$. As increases, the standard error decreases and the sampling distribution becomes more concentrated around .

The value 30 is used because it usually provides an acceptable normal approximation for populations that are not extremely skewed or heavy-tailed. The required sample size still depends on the population's shape.

A common misconception is that every sample of 30 observations is normally distributed. The CLT concerns the distribution of $\bar{X}$ across repeated samples, not the distribution of individual observations.

Exam technique: For AHL 4.15, state that $n>30$ satisfies the IB examination convention, identify $\bar{X}\sim N\left(\mu,\frac{\sigma^2}{n}\right)$, and enter the standard deviation as on the GDC, not .