Z-Score to Percentile Calculator
Convert a standard-normal z-score into a percentile, the area to its right and the two-tailed p-value.
How it works
percentile = Φ(z) × 100 · Φ = standard normal CDF · two-tailed p = 2 · (1 − Φ(|z|))
A z-score restates a value as the number of standard deviations it lies from the mean, and the standard normal cumulative distribution function Φ turns that into the proportion of the distribution falling below it. Multiply by 100 for a percentile. This calculator uses a high-accuracy polynomial approximation of Φ (error under 1.5×10⁻⁷), matching printed z-tables. The right-tail area gives a one-tailed p-value, and doubling the smaller tail gives the two-tailed p-value used in most hypothesis tests.
Worked example
For z = 1.00, Φ(1) = 0.8413, so the value is at the 84.13th percentile. The area to the right is 15.87%, the one-tailed p-value is 0.1587, and the two-tailed p-value is 2 × 0.1587 = 0.3173.
Sources: Standard normal table — Wikipedia
Frequently asked questions
What percentile is a z-score of 0?
Exactly the 50th percentile. A z-score of 0 is the mean of a symmetric normal distribution, with half the area on each side.
How do common z-scores map to percentiles?
z = 1 is the 84th percentile, z = 1.645 the 95th, z = 1.96 the 97.5th, and z = 2 the 97.7th. Negative z-scores mirror these below the 50th percentile — z = −1 is the 16th percentile.
When would I want the two-tailed p-value?
Use it when your hypothesis test does not specify a direction — for example testing whether a mean differs from a target in either direction. The two-tailed p-value counts extreme results in both tails, so it is twice the one-tailed value for a symmetric distribution.
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This calculator is for educational purposes. Double-check important results before acting on them.