Distribution of the sample mean and the central limit theorem (QCE Specialist Mathematics Unit 4)

What this dot point is asking

QCAA wants you to understand that the sample mean $\bar{X}$ is itself a random variable with its own distribution, to state and apply the central limit theorem, and to use the resulting normal model to compute probabilities about sample means. This is the foundation of statistical inference, assessed in IA3 and the external assessment, and it precedes confidence-interval work.

The answer

The sample mean is random

If you take a random sample of size $n$ from a population and compute its mean $\bar{X}$, a different sample gives a different mean. So $\bar{X}$ is a random variable. Its distribution is called the sampling distribution of the mean, and it has its own mean and standard deviation.

Mean and standard error

If the population has mean $\mu$ and standard deviation $\sigma$, then for a sample of size $n$:

The standard deviation of the sample mean, $\dfrac{\sigma}{\sqrt{n}}$, is called the standard error. It is smaller than the population standard deviation, and it shrinks as $n$ grows: larger samples give more reliable estimates of $\mu$. Crucially the divisor is $\sqrt{n}$, not $n$.

The central limit theorem

The central limit theorem states that for a sufficiently large sample size $n$, the distribution of the sample mean $\bar{X}$ is approximately normal,

regardless of the shape of the original population. If the population is already normal, $\bar{X}$ is exactly normal for any $n$. A common rule of thumb takes $n \geq 30$ as large enough for the approximation when the population is not too skewed.

Standardising to find probabilities

To compute a probability for $\bar{X}$, standardise using the standard error:

This $Z$ has the standard normal distribution $N(0,1)$, so probabilities follow from the normal model. The only change from a single-observation calculation is dividing by $\dfrac{\sigma}{\sqrt{n}}$ rather than $\sigma$.

Effect of sample size

Because the standard error is $\dfrac{\sigma}{\sqrt{n}}$, quadrupling the sample size halves the spread of $\bar{X}$. This is why larger samples produce tighter estimates and narrower confidence intervals: the sampling distribution concentrates around $\mu$.

Working backwards to find a sample size

A favourite extended-response task gives a probability statement about $\bar{X}$ and asks for the sample size. The method reverses standardising: convert the probability to a critical $z$-value, then solve $\dfrac{\text{(value} - \mu)}{\sigma/\sqrt{n}} = z$ for $n$. Because $n$ must be a whole number of observations, round the result, and state explicitly that a sample size is an integer. Reading the correct $z$ from the stated tail probability (lower tail, upper tail, or central region) is where care is needed, and a sketch of the normal curve with the area shaded prevents sign errors.

Normal population versus the central limit theorem

It is worth distinguishing two reasons $\bar{X}$ might be normal. If the population itself is normal, then $\bar{X}$ is exactly normal for every sample size, however small, because a sum of normal variables is normal. If the population is not normal, $\bar{X}$ is only approximately normal, and only for a large enough $n$, by the central limit theorem. An exam question that asks you to justify normality is testing exactly this distinction: cite the normal population if one is given, and cite the central limit theorem only when the population shape is unknown or non-normal and $n$ is large.

Single observation versus the mean of a sample

The most consequential modelling decision is whether a question concerns one randomly chosen value or the average of a sample. A single observation $X$ has standard deviation $\sigma$; the sample mean $\bar{X}$ has the much smaller standard error $\dfrac{\sigma}{\sqrt{n}}$. Using the wrong one is the difference between a correct and an incorrect probability. Read the wording carefully: phrases like "the mean of the sample" or "the average" signal $\bar{X}$, while "a randomly selected" individual signals $X$.