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.
Converting between scales
- 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)
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.
PercentileRelated terms
Check a selection process against the four-fifths rule
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