Z-score

Z-score (or standard score) measures how many standard deviations a value sits above or below the mean. Formula: z = (x − mean) ÷ standard_deviation.

A value with z = 0 is the mean. z = 1 is one standard deviation above the mean. z = −2 is two standard deviations below. For a roughly normal distribution, |z| > 2 covers about 5% of values; |z| > 3 covers about 0.3%.

The point of converting to z-scores: it puts different scales on the same yardstick. A 75 on a math test where the class mean is 70 with SD 10 gives z = 0.5. A 68 on a French test where the mean is 65 with SD 5 gives z = 0.6. The French score is relatively better, which raw scores can’t tell you.

Z-scores feature in standardised tests (SAT, GMAT), clinical reference ranges (bone density, growth curves), quality control (process capability indices), and any context where you need to compare across populations with different means and spreads.

Our statistics calculator computes mean and standard deviation; the z-score is a one-line subtraction-then-division on the result.