What is Z-score?
Also called 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.
Converting between scales, and the band around the result
- Raw 62, norm-group mean 50, standard deviation 8 → z = (62 − 50) ÷ 8 = +1.5
- T-score = 50 + 10 × 1.5 = 65
- Stanine = 5 + 2 × 1.5 = 8
- Percentile ≈ 93rd (assuming an approximately normal distribution)
- Precision, at an illustrative reliability of .85: SEM in z units = √(1 − .85) = 0.387
- 95 per cent interval = 1.5 ± 1.96 × 0.387 = +0.74 to +2.26 → T-score 57 to 73 → 77th to 99th percentile
The reliability used here is an illustration. AssessAll publishes no reliability coefficient for any instrument, so the band above cannot currently be computed for an AssessAll score — by a buyer or by us.
Not the same as percentile
A z-score is on an equal-interval scale, so differences mean the same thing anywhere along it and it can be averaged. A percentile is ordinal and compressed in the middle, and averaging percentiles is not meaningful.
PercentileSources
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