Z-score, percentile, IQ and T-score converter

What these four numbers actually are

A Z-score, a percentile, a deviation IQ and a T-score are four ways of saying the same thing: where one value sits relative to everyone else in a normal distribution. They differ only in the scale they are printed on, which is why converting between them is arithmetic rather than estimation.

• Z-score — how many standard deviations the value is from the mean. Mean 0, standard deviation 1. This is the underlying quantity; the other three are rescalings of it.
• Percentile — the percentage of the distribution that falls below the value. Unlike the other three, it is not linear in the Z-score: the gap between the 50th and 60th percentile is far smaller in Z terms than the gap between the 88th and 98th.
• Deviation IQ — mean 100, standard deviation 15. Used by the Wechsler and Stanford–Binet scales.
• T-score — mean 50, standard deviation 10. Used across psychological instruments such as the MMPI, and identical to the Japanese hensachi used in university admissions.

Reading the result honestly

The most common misreading is treating percentile differences as if they were evenly spaced. Moving from the 50th to the 55th percentile is a Z change of about 0.13. Moving from the 94th to the 99th is a Z change of about 0.78 — six times as large, for the same five percentile points. Near the tails, small changes in raw score produce large changes in rank, and in the middle the opposite is true.

The second is forgetting that all of this assumes a normal distribution. If the underlying scores are strongly skewed — reaction times, income, most count data — then the Z-score is still a perfectly valid statement about standard deviations from the mean, but the percentile derived from it will be wrong, sometimes badly. Plot a histogram of your data before trusting a percentile conversion.

Common mistakes

• Using the sample SD when you need the population SD. If you are describing the sample you have, divide by n. If you are estimating the population it came from, divide by n − 1. For large samples the difference is negligible; for n below about 30 it is not.
• Assuming IQ always means SD 15. The Cattell scale uses 24, and some older tests use 16. A "132" means something quite different on each. The advanced settings above let you set the mean and SD explicitly.
• Confusing percentile with percentage correct. Scoring 90% on a test where everyone scored 95% puts you near the 1st percentile, not the 90th.
• Averaging percentiles. Percentiles are ranks, not quantities, so their mean is not meaningful. Convert to Z-scores, average those, then convert back.

The method used here

The normal CDF is computed with the Abramowitz–Stegun 7.1.26 rational approximation to the error function, which is accurate to about 1.5 × 10⁻⁷ — far beyond the two decimal places displayed. The inverse direction (percentile back to Z) uses Peter Acklam's algorithm, accurate to roughly 1.15 × 10⁻⁹ relative error across the full range. Neither introduces error that matters at the precision shown.

Percentile inputs are clamped to the range 0.01–99.99 because the normal distribution has infinite tails: a percentile of exactly 0 or 100 has no finite Z-score.

Where these scales are used

Deviation IQ and T-scores dominate clinical and educational psychology. Japan's university entrance system runs on hensachi, which is a T-score by another name, and Korea's CSAT standard score uses the same construction with a mean of 100 and a standard deviation of 20. If you are working with either of those, the conversions on this page are the underlying mathematics.

Sources

• Abramowitz, M. & Stegun, I. A. (1964). Handbook of Mathematical Functions, formula 7.1.26.
• Acklam, P. J. (2003). An algorithm for computing the inverse normal cumulative distribution function.
• Wechsler, D. (2008). Wechsler Adult Intelligence Scale — Fourth Edition. Pearson. (Source of the mean 100 / SD 15 convention.)