Test Percentile to Score Calculator: Percentile ↔ Score on a Normal Curve

How it works

percentile → score: z = Φ⁻¹(percentile ÷ 100), score = mean + z × SD · top X%: percentile = 100 − X · score → percentile: z = (score − mean) ÷ SD, percentile = Φ(z) × 100 · Φ = standard normal cumulative distribution

A percentile rank says what share of test-takers scored below a score. If scores follow a normal (bell) curve, every percentile matches one z-score, the number of standard deviations above or below the mean. The calculator finds that z with the inverse normal function, then rescales it to the test with score = mean + z × standard deviation. Score to percentile reverses the steps with the normal cumulative distribution. Choose a preset for the common standardized scales: IQ-type standard scores (mean 100, SD 15), T-scores (50, 10), scaled subtest scores (10, 3), normal curve equivalents (50, 21.06) and z-scores (0, 1). The table shows the same rank on each scale, plus the stanine (1–9 bands of 4, 7, 12, 17, 20, 17, 12, 7 and 4%). Under More options you can cap the result to the real score range and round to the test's score steps. Limits: real tests are not perfectly normal, especially in the tails, and published percentile tables for tests such as the SAT or ACT are built from actual test-takers. Use this for tests that are scaled to a normal curve, or as an estimate.

Worked example

On a test with mean 500 and standard deviation 100, the 84th percentile is p = 0.84. The inverse normal gives z = Φ⁻¹(0.84) = 0.9945, so the score is 500 + 0.9945 × 100 = 599.4, which is 99.4 points above the mean. That rank is stanine 7, an IQ-type score of 114.9 and a T-score of 59.9. Going the other way, a score of 600 is z = 1.000 and the 84.13th percentile.

Sources: NIST/SEMATECH e-Handbook: normal distribution · Standardized score scales and percentile equivalents (z, standard, T, scaled, stanine, NCE) · Normal curve equivalent (mean 50, SD 21.06) · Statology: percentile to z-score

Frequently asked questions