--- title: "Z-score" description: "A z-score expresses a raw score as the number of standard deviations it sits above or below the mean of its norm group. A z of 0 is exactly average and +1 is one standard deviation above. It is the common currency into which most other standard scores are converted." canonical: https://www.assessall.com/guides/glossary/z/z-score updated: 2026-09-24 source: AssessAll --- # Z-score _Also known as: Standard score, Standardised score._ A z-score expresses a raw score as the number of standard deviations it sits above or below the mean of its norm group. A z of 0 is exactly average and +1 is one standard deviation above. It is the common currency into which most other standard scores are converted. ## Why almost every reported scale is a z-score in disguise Stanines, stens, T-scores and the familiar IQ-style scale are all linear transformations of z. A T-score is 50 plus ten times z; a stanine is five plus two times z, rounded; the IQ-style scale is 100 plus fifteen times z. Knowing this lets you move between any two reported scales without a conversion table. It also makes the shared dependency obvious: every one of them inherits the mean and standard deviation of whichever norm group was used, so all of them change if the norm group changes. ## The assumption to check Converting a z-score to a percentile assumes the distribution is approximately normal. For a well-behaved ability test on a large sample that is usually reasonable; for a short scale, a skewed one, or one with a ceiling that many people hit, it is not, and the percentile it produces will be wrong in a direction that flatters the top of the range. When a distribution is visibly non-normal the honest options are to report the empirical percentile directly from the data, or to say that the scale is not normally distributed and report bands. ## How wide is the band around one person's z? Everything above treats a z-score as a point. It is an estimate, and the arithmetic for its precision is short enough to do on the page — which is worth doing, because the answer is larger than almost anyone expects. In z units the standard error of measurement is the square root of one minus the reliability, because the scale's standard deviation is 1 by construction. At a reliability of .85 — an illustrative figure, since AssessAll publishes no reliability coefficient for any of its instruments — that is 0.387, and a 95 per cent interval is 1.96 times it, or plus or minus 0.76 z. So a candidate reported at z = +1.0, the 84th percentile, has a 95 per cent interval running from +0.24 to +1.76. On a T-score report that is 52 to 68. In percentile terms it is the 60th to the 96th. The number on the page is one point inside a band that covers more than a third of the distribution, and none of the derived scales — T, stanine, sten, the IQ-style scale — makes the band any narrower, because they are all the same estimate multiplied by a constant. Two practical consequences. A cut score expressed in z is a line drawn through that band, not a wall, so candidates within about 0.76 z of it are not reliably distinguishable from it. And comparing two candidates needs the wider standard error of the difference, not this one: at the same reliability it is 0.548 in z units, giving a 95 per cent band of 1.07 z before the instrument has said that two people differ at all. ## What a difference of one z actually looks like Because z is the common currency, group differences in the research literature are usually reported in the same unit under the name d — a standardised mean difference is a gap between two group means expressed in standard deviations. That makes the literature directly readable against a score report, which is the main reason to be comfortable with the unit. The most cited figure in employment testing is a worked example of exactly this. Roth, BeVier, Bobko, Switzer and Tyler (2001) report a Black-White difference of .99 for measures of general cognitive ability in industrial applicant samples, and .41 in incumbent samples — the same construct, the same unit, and a gap that more than halves once the population has already been through a selection process, which is range restriction visible in a single pair of numbers. Reading those as z-scores is the useful move: a difference of .99 means the average applicant in one group sits at roughly the 16th percentile of the other group's distribution. It also means that a difference of that size is smaller than the 95 per cent measurement band around any individual candidate computed above, which is the whole reason a group statistic can never be applied to a person in front of you. --- Source: https://www.assessall.com/guides/glossary/z/z-score