Standardising comparative data

The mean gives the average of a data set. The standard deviation measures its spread relative to that mean: a small standard deviation indicates that values are relatively close to the mean, while a large one indicates greater spread.

Calculate a standardised score using:

$$\text{standardised score}=\frac{x-\mu}{\sigma}$$

Here, $x$ is the raw value, $\mu$ is the mean of the data set it belongs to, and $\sigma$ is that data set's standard deviation.

There are two steps in this calculation. First, subtract the mean to find how far the raw value lies above or below it. Then divide by the standard deviation to express that distance in standard-deviation units. A difference of 10 marks is two standard deviations when the standard deviation is 5 marks, but only one standard deviation when it is 10 marks.

The raw value, mean and standard deviation must use matching units. Those units cancel in the division, so a standardised score has no units.

You may be given the means and standard deviations. If they need to be calculated, find each sample's mean and standard deviation first, then use that sample's summaries to standardise its values. The standard-deviation formulae are provided in the examination formulae sheet; the standardised-score formula is not. Standardising requires a non-zero standard deviation, because division by zero is undefined.

Reading the sign and size

A standardised score of zero means the raw value equals its sample's mean. A positive score means it is above the mean; a negative score means it is below the mean.

For example, a score of $1.5$ means a value is 1.5 standard deviations above its mean. A score of $-2$ means it is two standard deviations below its mean. The further the score is from zero, in either direction, the further the raw value lies from its mean in standard-deviation units.

Comparing performance in two examinations

Michelle's marks and the summaries for all students taking each examination are:

For maths:

$$\frac{80-63.2}{11.9}=1.4117\ldots\approx1.41$$

For English:

$$\frac{72-56.1}{9.3}=1.7096\ldots\approx1.71$$

Both marks are above their respective means. Michelle's maths mark is about 1.41 standard deviations above the maths mean, whereas her English mark is about 1.71 standard deviations above the English mean.

Although her raw maths mark is higher, her standardised English score is higher. Relative to the students taking each examination, her English performance was stronger. This does not mean she earned more marks in English; it answers a different question about her position relative to each examination's results.

Deciding what counts as better

A higher standardised score always indicates a higher position relative to the relevant mean. Whether that is better depends on the measurement.

For examination marks, higher is normally better. For a timed race, lower times are better, so a more negative standardised time can indicate a stronger performance relative to the other competitors. The numerical comparison must therefore be interpreted using the context, rather than automatically treating the larger score as the better result.