From a percentile to a value
Percentiles describe position in a group. Being at the 90th percentile means scoring higher than 90% of people. When a measurement follows a normal distribution, you can go straight from a percentile to the matching value using only the mean and the standard deviation. That is how cut-off scores, reference ranges and “top 2%” thresholds are set. This calculator does the conversion and gives the z-score, the share above and the quartiles.
The formula
- Find the z-score whose cumulative probability equals the percentile: z = Φ⁻¹(percentile)
- Value = mean + z × standard deviation
- Rarity = 1 in 100 ÷ (100 − percentile)
The inverse normal function Φ⁻¹ has no closed form; the calculator evaluates it to full double precision.
Worked example: an IQ threshold
IQ tests are scaled to a mean of 100 and a standard deviation of 15. High-IQ societies typically admit people at or above the 98th percentile.
- z = Φ⁻¹(0.98) = 2.054.
- Value = 100 + 2.054 × 15 = 130.8.
- Only 2% of people score higher, or 1 person in 50.
- The middle half of the population scores between 89.9 and 110.1, the first and third quartiles.
Common percentiles and z-scores
| Percentile | Z-score | IQ equivalent |
|---|---|---|
| 50th (median) | 0.000 | 100 |
| 75th (Q3) | 0.674 | 110 |
| 84th | 0.994 | 115 |
| 90th | 1.282 | 119 |
| 95th | 1.645 | 125 |
| 97.5th | 1.960 | 129 |
| 98th | 2.054 | 131 |
| 99th | 2.326 | 135 |
| 99.9th | 3.090 | 146 |
Where percentiles are used
- Children’s growth: a child at the 75th percentile for height is taller than 75% of children of the same age and sex on the CDC growth charts.
- Standardized tests: score reports show the percentile rank next to the scaled score.
- Medicine: many lab reference ranges run from the 2.5th to the 97.5th percentile of healthy people.
- Pay and performance: salary surveys often report the 25th, 50th and 75th percentiles of pay for a job.
Percentile, percentage and percentile rank
A percentage is a share of a total (75% of questions answered correctly). A percentile is a position relative to other people (better than 75% of test takers). The two can be very different: on a hard test, 60% correct might be the 90th percentile. This calculator assumes normally distributed data; for skewed data such as income, compute percentiles from the actual values instead. To go the other way, from a value to its percentile, use the z-score calculator.