Standardised scores, the standardised score formula, and using them to compare performance across distributions.

What this dot point is asking

AQA wants you to calculate a standardised score using the mean and standard deviation, interpret its sign and size, and use standardised scores to compare results that come from different distributions. This is the GCSE Statistics version of the z-score, and it links the standard deviation to the normal distribution.

The standardised score formula

The formula does two things at once: subtracting the mean measures how far above or below average the value is, and dividing by the standard deviation expresses that distance in "standard-deviation units" so that tests on different scales become comparable. A standardised score of $0$ is exactly average, $+1$ is one standard deviation above the mean, and $-1$ is one below. Rearranging gives $x = \bar{x} + (\text{standardised score}) \times \sigma$, which lets you recover a raw mark from a standardised score, a common reverse question.

The standardised score is the GCSE Statistics name for the z-score, and it links directly to the normal distribution. When data is normally distributed, a standardised score tells you not just the relative position but the approximate percentage of people above or below it: a score of $+1$ sits at about the $84$th percentile (since $50\% + 34\% = 84\%$ lie below it), a score of $+2$ at about the $97.5$th percentile, and a negative score the mirror image below the mean. This is why the standardised score is so powerful for comparison: it converts a raw mark into a position on a universal scale that applies to any normal distribution, regardless of its mean or spread.

Calculating a standardised score

Comparing across distributions

Comparing raw marks across tests is misleading because the tests may differ in difficulty (different means) and in how spread out the marks are (different standard deviations). Standardising puts every result on the same footing. A mark of $66$ on an easy-marked test (mean $54$) can represent a stronger relative performance than $72$ on a harder one (mean $60$), once the spread is taken into account. The standard deviation matters as well as the mean: the same distance above the mean is more impressive on a test where marks are tightly bunched (small standard deviation) than on one where they are widely spread, because being well above the pack is harder when everyone is close together.