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Z-Score Calculator: Z-Score, Percentile and P-Value

Find how many standard deviations a value lies from the mean, then read off the normal-distribution percentile, the tail areas and the two-tailed p-value. You can also start from a z-score or a percentile.

By Adzita Editorial Team · Updated 2026-11-03 · How we build and check our tools

What a z-score measures

A z-score re-expresses a value as the number of standard deviations it sits above or below the mean:

z = (x − μ) ÷ σ

A score of 85 on a test with mean 70 and standard deviation 10 has z = 1.5: it lies one and a half standard deviations above average. Because z-scores are unit-free, they let you compare values from different scales, such as an exam mark and a height. A z-score of 0 is exactly average, positive values are above the mean and negative values are below it.

For sample data, the mean and standard deviation come from the sample; the standard deviation calculator computes both.

From z-score to percentile and p-value

If the data follow a normal (bell-curve) distribution, each z-score maps to a cumulative probability, written Φ(z). Φ(1.5) is about 0.9332, meaning roughly 93.3% of values lie below a z-score of 1.5. The upper tail, P(Z > z), is 1 − Φ(z). The two-tailed p-value is twice the smaller tail and answers "how often would a value this far from the mean, in either direction, occur by chance?" The best-known benchmark is z = 1.96, where Φ is 0.9750 and the two-tailed p-value is 0.05.

How the probabilities are computed

The calculator does not use a lookup table. It evaluates the normal cumulative distribution with a convergent series, Φ(x) = ½ + φ(x) × (x + x³/3 + x⁵/(3·5) + x⁷/(3·5·7) + …), where φ is the bell-curve density, summing terms until they stop changing. For |z| above 6 the tail uses an asymptotic expansion instead. In testing against a reference implementation, probabilities agree to better than 12 decimal places for |z| up to 6, the relative error of the far tail is about one part in ten million at |z| = 6, and it is a few parts per million between 6 and 8. The reverse direction, from percentile to z, starts from a rational approximation and applies two Halley refinement steps. Displayed values are rounded to six decimals.

Common mistakes

• Assuming normality. A percentile from the normal curve is only meaningful if the data are roughly bell-shaped. Skewed data, such as incomes, can produce misleading percentiles.
• Mixing population and sample. If σ is estimated from a small sample, a t-distribution is more appropriate for inference than the normal curve.
• One tail or two. A p-value of 0.025 in one direction is 0.05 in both. State which one your test uses before comparing with a threshold.
• Reading a z-score as a probability. The z-score is a distance; the probability comes from the area under the curve.
• Using the wrong standard deviation. For the average of a sample, the relevant spread is the standard error σ ÷ √n, not σ.

Worked example

An exam has a mean score of 70 and a standard deviation of 10. A student scores 85. Where does that place the student?

Inputs

Result

Accuracy note: Results use standard double-precision arithmetic, so very large or very small values may be rounded. Check important results by another method.

Frequently asked questions

Related calculators

References and sources

1. NIST/SEMATECH e-Handbook of Statistical Methods: the normal distribution and standard normal tables. Link
2. Acklam, P. J., an algorithm for computing the inverse normal cumulative distribution function (rational approximation used as a starting point).

Check primary sources for anything you rely on.