Understand the distribution of the sample mean, apply the central limit theorem to describe its shape, mean and standard deviation, and use these to compute probabilities for sample means drawn from a population

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 $\mu$.