z-scores (standardised values)

A z-score says how many standard deviations a value lies above or below the mean, which allows comparison across different scales.

z = \frac{x - \xbar}{s} \qquad \text{or} \qquad z = \frac{x - \mu}{\sigma}

Sample-based and population-based standardisation.

$x$: The value being standardised.
$\xbar,\ s$: Sample mean and sample SD ( $n - 1$ ).
$\mu,\ \sigma$: Population (or model) mean and SD.

When to use

Comparing relative standing across variables measured on different scales (a test score vs a race time).
Flagging unusual values (|z| > 2 or 3 in roughly bell-shaped data).
Converting to the standard normal to compute probabilities (see normal distribution).

How to interpret

$z = 1.5$ : 1.5 SDs above the mean; $z = -0.5$ : half an SD below. Standardised data have mean 0 and SD 1, but the shape does not change — standardising skewed data does not make it normal.

Common mistakes

Reading normal-table percentiles from z-scores of clearly non-normal data.
Mixing population and sample SDs within one comparison.
Forgetting the sign: for times (lower is better) a negative z can be the better performance.

Example

A score of 82 in a class with mean 70 and SD 8: $z = (82 - 70)/8 = 1.5$ . For the data 2, 4, 4, 4, 5, 5, 7, 9 ( $\xbar = 5$ , $s = 2.138$ ), the value 9 has $z = 4/2.138 = 1.87$ .