How Does the Distribution of the Sample Mean Relate to the Population Distribution?

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This is the key relationship in AHL 4.15: Central limit theorem. Imagine repeatedly taking independent random samples of the same size and calculating each sample mean. The distribution formed by all those values of $\bar{X}$ is the sampling distribution.

If the population itself is normal, then the result is exact for every sample size:

If the population is not normal, the central limit theorem states that the distribution of $\bar{X}$ approaches a normal distribution as $n$ increases. In IB examinations, $n>30$ is considered sufficient for applying the theorem.

For example, if $\mu=50$, $\sigma=12$, and $n=36$, then

Thus, individual values vary with standard deviation $12$, while sample means vary with standard deviation only $2$. This reduced spread means larger samples produce more precise estimates of $\mu$: increasing $n$ does not change the centre, but decreases the standard error.

Using the same example, the probability that a sample mean exceeds $54$ is found from $\bar{X}\sim N(50,4)$, where $4$ is the variance. A GDC gives $P(\bar{X}>54)\approx 0.0228$. Enter standard deviation $2$, not variance $4$, in the normal-distribution routine.

The population distribution describes individual observations; the sampling distribution describes means from repeated samples. They are different random variables, although both are centred at $\mu$.

A common misconception is that the sample mean has standard deviation $\sigma$. Its standard deviation is the standard error, $\frac{\sigma}{\sqrt{n}}$. In an IB Math AI HL response, state the distribution of $\bar{X}$, show this parameter adjustment, use the GDC for the probability, and interpret the result in context.