Z-Score & Percentile Calculator
Convert between a value, its z-score and its percentile on a normal distribution with your mean and standard deviation, with tail probabilities and a ladder.
The z-score and percentile of a value (or the value at a percentile), the probability above, below and in both tails, and where the 1st to 99th percentiles fall.
Example: A score of 130 on a scale with mean 100 and standard deviation 15 has z = 2: the 97.72nd percentile, with 2.28 % above it. The 90th percentile of the same scale is 119.22.
How many
standard deviations.
What a z-score is, how it turns into a percentile, and when the normal model is not the right one.
z and the percentile
A z-score is a value's distance from the mean in standard deviations: z = (x − μ)/σ. On a normal distribution the fraction of the population below that point is Φ(z), the standard normal cumulative distribution — the percentile. The page computes Φ from the complementary error function rather than a table, so 1.96 gives 97.500 % and not a rounded 97.5. The inverse Φ⁻¹ (Acklam's algorithm refined by one Newton step) turns a percentile back into a z and then into a value, x = μ + zσ.
Tails
“Below” is Φ(z), “above” is 1 − Φ(z), and the two-tailed figure 2(1 − Φ(|z|)) is the probability of being at least this far from the mean on either side — the quantity a two-sided test compares with α. “Within” is its complement, the familiar 68 / 95 / 99.7 % at z = 1, 2, 3. The ladder lists the value at the 1st to 99th percentiles for the same μ and σ, useful for setting cut-offs.
When not to use it
Everything here assumes the population is normal with the μ and σ you give. Skewed or bounded quantities (incomes, waiting times, counts) are not, and a percentile from this page would be wrong for them; use the empirical percentile from the data instead (the Descriptive Statistics page). Very large |z| values are flagged for the same reason — the far tails of real data rarely follow the model. Nothing leaves the browser; the same four anonymous usage counts as the rest of the site apply.
SOURCES
Last reviewed 19 September 2026. How results are checked: How we verify.