Test Score Percentile Calculator
Convert a test score to a z-score and percentile rank using the score, the distribution mean, and the standard deviation.
How to use this tool
- Enter your test score.
- Enter the mean (average) score of the reference group.
- Enter the standard deviation of that group's scores.
- Read your z-score and percentile rank.
Turn a test score into a percentile. Enter your score, the group's mean, and the standard deviation to get your z-score and the percentage of test-takers you scored at or above.
Formula
Z-score = (Score − Mean) ÷ Standard deviation
Percentile = Φ(z) × 100%, where Φ is the standard normal cumulative distribution.
The percentile is the share of the reference group expected to score at or below your score, assuming scores follow a normal distribution.
How it works
A percentile rank tells you what fraction of test-takers scored at or below a given score. The calculator first standardizes the score into a z-score — how many standard deviations it lies above or below the mean — then maps that z to a percentile using the standard normal cumulative distribution function. We evaluate the CDF with the Abramowitz & Stegun 7.1.26 rational approximation, which is accurate to about one part in ten million, far tighter than the figures this tool reports.
This approach assumes the score distribution is approximately normal (bell-shaped), which holds well for many large standardized tests (IQ tests, the SAT sections, and similar) but less so for small samples or skewed grading. The percentile is a modeled estimate from the mean and standard deviation you supply, not a count from an actual roster; for the official percentile, use the testing organization's published norm tables. Make sure the mean and standard deviation come from the same reference group you want to compare against.
Reviewed by the AbraCalc Education Desk. This is an educational estimate; official percentile ranks are set by the test publisher using their full norming sample.
Worked example
Score 130, mean 100, standard deviation 15
- Z-score = (130 − 100) ÷ 15 = 30 ÷ 15 = 2.
- A z of 2 sits two standard deviations above the mean.
- Look up the normal CDF: Φ(2) ≈ 0.97725.
- Convert to a percentile: 0.97725 × 100% ≈ 97.73%.
Percentile rank = 97.72% (z-score 2.0000)
Z-score to percentile (standard normal)
| Z-score | Percentile |
|---|---|
| -2.0 | 2.28% |
| -1.0 | 15.87% |
| -0.5 | 30.85% |
| 0.0 | 50.00% |
| 0.5 | 69.15% |
| 1.0 | 84.13% |
| 2.0 | 97.72% |
Key terms
- Z-score
- The number of standard deviations a value lies above (positive) or below (negative) the mean.
- Percentile rank
- The percentage of a reference group that scored at or below a given score.
- Standard deviation
- A measure of how spread out scores are around the mean.
- Normal distribution
- The symmetric bell curve many standardized test scores approximately follow.
Frequently asked questions
- What does a percentile rank actually mean?
- If your percentile is 90, you scored at or above about 90% of the reference group. It is a relative measure of standing, not the percentage of questions you got right.
- Does this assume a normal distribution?
- Yes. The conversion uses the normal (bell) curve, which fits many large standardized tests well but can misstate percentiles for skewed or small-sample score sets.
- Where do I get the mean and standard deviation?
- From the test's published statistics or your class data. Many standardized tests publish a mean and standard deviation (for example, a mean of 100 with a standard deviation of 15).