One subtraction, one division
A z-score locates a value on its own distribution: z = (x − μ) ÷ σ. Subtract the mean to centre the value at zero, divide by the standard deviation to measure the gap in SD units. The sign is direction (above or below the mean), the size is distance. A z of 1.5 says "one and a half standard deviations above average", a sentence that means the same thing for exam marks, blood pressure and daily temperatures, which is the entire point of the standardisation.
A mark of 85 where the mean is 70 and the SD is 10: z = (85 − 70) ÷ 10 = 1.5. If scores run roughly normal, the z table turns that into a percentile: Φ(1.5) = 0.9332, so the mark beats about 93% of the group. A mark of 62 on the same test: z = −0.8, percentile about 21. Same formula both directions; the sign does the talking.
Comparing across different scales
Standardisation is the z-score's day job. Two students, one scoring 85 in a class averaging 70 (SD 10), the other 610 in a cohort averaging 520 (SD 80): raw marks cannot be compared, but z-scores can: 1.5 against 1.125, so the first performance is the stronger relative showing. Admissions tutors, sports scientists comparing athletes across events, and data pipelines that standardise variables before modelling all run this exact conversion, because it strips the units and the difficulty of the instrument out of the comparison.
Sample version and percentile reading
With sample statistics the same shape holds: z = (x − x̄) ÷ s. Two cautions follow. First, z-scores computed inside a small sample are bounded and slightly shrunken (the largest possible |z| is (n − 1)/√n), so a z of 2 in eight observations is less exceptional than a z of 2 in a population. Second, the percentile reading leans on normality: z = 1.5 is the 93.32nd percentile of a normal distribution, but in a skewed income distribution the same z sits elsewhere. Quote the z freely; quote its percentile with the distribution's shape in mind.
Dividing by the variance instead of the standard deviation. The units give it away: (x − μ) is in marks, the variance is in marks squared, and the quotient would not be a pure count of SDs. If your z comes out implausibly small on spread-out data, this is the first suspect.
Frequently asked questions
What does a z-score of 0 mean?
The value equals the mean exactly. Positive z is above the mean, negative below, and the size counts standard deviations.
How high can a z-score go?
In a population, unbounded. In a sample of n, the largest possible absolute z is (n - 1)/sqrt(n), because the value is part of the mean and SD it is judged against.
What z-score is the 95th percentile?
1.645 under normality (one-sided). The 97.5th percentile, used in 95% confidence intervals, is 1.96. The inverse normal calculator converts either direction.