IQ Percentile Calculator
Convert an IQ score into a percentile, z-score, and rarity. Works with whichever standard deviation your test uses — 15 (Wechsler and most modern tests), 16 (older Stanford-Binet), or 24 (Cattell).
Example: with IQ score 120 · Test scale (standard deviation) SD 15 — Wechsler / most modern tests → Percentile (0–100): 90.9 (higher than about 90.9% of people).
- RarityTop 9.1% — about 1 in 11 people score 120 or higher
- z-scorez = +1.33 (1.33 SD above the mean)
Computed by the calculator below using its default values. Change any input to see your own numbers.
Percentile = Φ((IQ − 100) ÷ SD), the cumulative normal distribution. IQ tests are normed to mean 100; most modern tests use SD 15.
How an IQ score becomes a percentile
IQ scores are not raw test results — they are deviation scores. Test makers give the exam to a large norming sample, then rescale results so the population average is exactly 100 and the spread (standard deviation) is a fixed number, usually 15. That construction means the bell curve is baked in: your percentile is simply the share of the normal distribution sitting below your score.
One standard deviation covers a lot of people. Between 85 and 115 sits about 68% of the population; between 70 and 130, about 95%. That is why moving from 130 to 145 changes rarity far more (1 in 44 to 1 in 741) than moving from 100 to 115 does.
The same score means different things on different scales
A few tests spread scores differently. The older Stanford-Binet used SD 16, and the Cattell scale used SD 24 — so a Cattell 148 equals a Wechsler 130: both are z = +2, the 97.7th percentile. Always check which scale produced a score before comparing it to a chart, a cutoff, or someone else's number.
Scores also carry measurement error of roughly ±3–5 points, and norms thin out badly above about 160 — there are too few people that far out to norm against. Treat extreme percentiles as illustrative, not as precision instruments.
How it’s calculated
Percentile = Φ((IQ − 100) ÷ SD) × 100, where Φ is the standard normal CDF, computed with the Abramowitz–Stegun 7.1.26 approximation (max error ≈ 1.5×10⁻⁷). Mean is 100 by construction; SD is 15 (Wechsler-style), 16 (older Stanford-Binet), or 24 (Cattell). Rarity = 1 ÷ tail probability, rounded.
Assumes a perfectly normal distribution and an error-free score; real tests carry a ±3–5 point standard error of measurement, and norms beyond about 160 are extrapolation.
IQ to percentile and rarity (SD 15 scale)
| IQ score | Percentile | About 1 in N score this high or higher |
|---|---|---|
| 100 | 50 | 1 in 2 |
| 110 | 74.8 | 1 in 4 |
| 115 | 84.1 | 1 in 6 |
| 120 | 90.9 | 1 in 11 |
| 130 | 97.7 | 1 in 44 |
| 145 | 99.87 | 1 in 741 |
| 160 | 99.997 | 1 in 31,600 |
Computed from the standard normal distribution with mean 100, SD 15 (Wechsler-style scoring); rarity rounded.
Common mistakes
- Comparing scores from tests with different SDs — 148 on a Cattell-scaled test equals about 130 Wechsler, not the 99.9th percentile.
- Reading the percentile as percent of questions answered correctly — it is the share of the population scoring below you.
- Treating a single point as exact: with a typical ±4-point measurement error, a 120 really spans roughly the 86th to 95th percentile.
- Using old ratio IQs (mental age ÷ chronological age × 100) as if they were modern deviation scores — they are not comparable.
Frequently asked questions
What percentile is an IQ of 120?
About the 91st percentile on an SD-15 test. Roughly 9% of people score 120 or higher — about 1 in 11. The calculator shows the exact tail for any score.
What is the IQ percentile formula?
Percentile = Φ((IQ − 100) ÷ SD) × 100, where Φ is the cumulative standard normal distribution. With SD 15, an IQ of 115 is one standard deviation up: Φ(1) ≈ 0.841, the 84th percentile.
What IQ do you need for Mensa?
The 98th percentile on a supervised, accepted test. That works out to roughly 131 on an SD-15 scale, 132 on SD-16, or 148 on the Cattell SD-24 scale — the commonly published cutoffs.
Is 100 an average IQ?
Yes, by definition. Tests are renormed periodically so the population median stays pinned at 100 (rising raw performance — the Flynn effect — is absorbed into each renorming). Half of test-takers land below 100.
How rare is an IQ of 145?
On an SD-15 scale that is z = +3: the 99.87th percentile, about 1 in 740 people. Scores that far out are hard to measure reliably because norming samples contain so few people there.