Z-score (standard score): formula, calculator and worked example

What the symbols mean

$z$: the z-score: positive above the mean, negative below
$x$: the value
$μ$: the mean of the distribution
$σ$: the standard deviation (greater than 0)

When to use it

Use it to compare results measured on different scales (a mark of 72 in one exam against 65 in a harder one), to spot unusual values (|z| > 2 is unusual, |z| > 3 very rare for normal data), and as the first step of every normal-distribution probability. Solving for x turns a z-score back into a raw value, as in "what score is 1.5 standard deviations above the mean?"

Worked example

IQ scores have mean 100 and standard deviation 15. What is the z-score of an IQ of 130? (x = 130, μ = 100, σ = 15)

Start from the formula.
$z = \frac{x - μ}{σ}$
Substitute $x$ = $130$ , $μ$ = $100$ , $σ$ = $15$ and work it out.
$z = \frac{130 - 100}{15}$
$= \frac{30}{15}$
$= 2$
$x$ is 2 standard deviations above the mean. If the data are normally distributed, about 97.7% of values lie below it (Φ(z) = 0.9772).

Answer z = 2

Common mistakes

Subtracting the wrong way round: $x − μ$ , so values below the mean get negative z-scores.
Dividing by the variance instead of the standard deviation.
Reading z as a probability. A z-score is a distance; the normal distribution turns it into a probability.

Z-score (standard score)

Calculator

See it move

What the symbols mean

When to use it

Worked example

Common mistakes