Normal and Correlation Tests

If individual values follow $N\left(\mu ,{\sigma }^2\right)$, then the mean of a sample of $n$ of them follows $N\left(\mu ,\frac{{\sigma }^2}{n}\right)$. The centre is unchanged, but the variance is divided by $n$.

The reason is that high and low values partially cancel when averaged. A single observation can be extreme; for the mean of 25 observations to be equally extreme, most of them would have to be extreme in the same direction, which is far less likely.

Taking the square root, the standard deviation of $\bar{X}$ is $\frac{\sigma }{\sqrt{n}}$ — the $\text{standard error}$. Quadrupling the sample size halves it, which is the practical statement of diminishing returns in sampling.

The structure is unchanged from the binomial test. State hypotheses about $\mu$, assume $H_0$, work out how extreme the observed sample mean is, and compare with the significance level.

The test statistic standardises the sample mean using the standard error rather than $\sigma$ itself — that is the only substantive difference from a routine normal calculation.

This test assumes the population variance is known and the population is normally distributed, or that $n$ is large enough for the sample mean to be approximately normal regardless.

The correlation test asks whether an observed sample correlation $r$ is large enough to indicate a genuine relationship in the population, or whether it could plausibly have arisen from chance in a sample of that size.

The hypotheses concern the $\text{population}$ correlation coefficient $\rho$: $H_0:\rho =0$ means no linear relationship. The alternative is $\rho \ne 0$, $\rho >0$ or $\rho <0$ depending on the wording.

The method is a table lookup. AQA provides critical values of $r$ indexed by sample size and significance level; compare the magnitude of your $r$ with the critical value and reject $H_0$ if it is larger.

Sample size matters enormously. With $n=5$ an $r$ of $0.8$ is not significant, because small samples produce large correlations by chance routinely; with $n=30$ a much smaller $r$ is convincing.

Both tests are about the $\text{population}$, using the sample as evidence. A significant result says the observed data would be unlikely if the null were true — nothing stronger.

The mean test assumes the population variance is known, which is a substantial assumption; in practice it is usually estimated, which leads to methods beyond this specification.

The correlation test detects only $\text{linear}$ relationships, so a non-significant $r$ does not rule out a strong curved relationship. And with a large sample, a correlation too small to be practically interesting can still be statistically significant — significance and importance are different questions.