Z-Score to Percentile Calculator
Find the percentile a z-score corresponds to.
Result
How it is calculated
The percentile is the area under the standard normal curve to the left of z: Φ(z) = ½ × (1 + erf(z ÷ √2)), shown as Φ(z) × 100. The error function is evaluated with the Abramowitz and Stegun 7.1.26 polynomial approximation, so results are accurate to roughly a few parts in ten million before rounding to two decimals. The tool also gives the area to the right, 100 − percentile. It assumes the data follow a normal distribution; z is unitless, and negative z gives percentiles below 50.
Worked example
Enter z = 1.28, a value 1.28 standard deviations above the mean. Compute z ÷ √2 = 1.28 ÷ 1.4142 = 0.9051, apply the error function (about 0.7994), then ½ × (1 + 0.7994) = 0.8997. As a percentage this is 89.97%. The area to the right is 100 − 89.97 = 10.03%. So the value sits at about the 90th percentile.
Common mistakes
- Using it on data that are not roughly normal. The percentile is only meaningful if the distribution is bell-shaped; skewed data will give misleading ranks.
- Entering the raw score instead of the z-score. Subtract the mean and divide by the standard deviation first:
z = (x − mean) ÷ SD. - Confusing the left tail with the right tail. The main figure is the share below z; for a p-value of a one-sided test above z, use the right-hand figure.
- Reading a percentile as a percentage score. The 90th percentile means better than about 90% of the group, not 90% correct.
About this calculator
This calculator converts a z-score — how many standard deviations a value sits from the mean — into a percentile using the cumulative distribution function (CDF) of the standard normal distribution, Φ(z). The percentile equals Φ(z)×100, found by integrating the standard normal probability density function from −∞ to z; a z-score of 0 gives the 50th percentile, +1.96 gives about the 97.5th, and −1.96 gives about the 2.5th.
Statisticians, researchers, teachers grading on a curve, and students use this to interpret standardized test scores (SAT, IQ tests), lab results in clinical medicine, and hypothesis-testing output where a z-statistic needs to be translated into a probability or percentile rank. Enter a z-score to get its percentile, or use it alongside a raw score, mean, and standard deviation.
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