A raw value on its own doesn't tell you whether it is big or small — that depends on the distribution it came from. A score of 80 is excellent if most people score 50, but ordinary if most score 78. The z-score fixes this by measuring a value in units of standard deviations from the mean:

It answers one question: how many standard deviations is x away from the mean?

Reading the sign

• z > 0 — the value is above the mean.
• z < 0 — the value is below the mean.
• z = 0 — the value is the mean exactly.

So z = -2 means "two standard deviations below the mean", and z = 1.5 means "one and a half standard deviations above it". The size of |z| tells you how unusual the value is — by the empirical rule, a z beyond \pm 2 is already quite rare.

Comparing across scales

Because a z-score strips away the original units, it lets you compare values from different distributions. If Ann scores z = 1.5 on a maths test and Ben scores z = 2.1 on an English test, Ben's result is the more exceptional one — even though the two tests have different means and spreads. The higher z-score wins.

Standardising every value this way is exactly the step that turns any normal into the standard normal N(0,1), with mean 0 and standard deviation 1.

Slide the z-marker

Drag the slider to move the marker to x = z on the standard normal curve. Read off where it lands: near the centre for small |z|, far out in a thin tail as |z| grows toward 3.