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Z-Score & Percentile Calculator

Calculate a z-score and its percentile in a normal distribution from a value, mean, and standard deviation.

Z-score
1
Percentile (share of distribution ≤ this value)
84.13%
Share of distribution above this value
15.87%

How it works

A z-score measures how many standard deviations a value sits from the mean of a normal distribution: z = (x − μ) ÷ σ. A z-score of 1 means the value is one standard deviation above the mean; −1 means one standard deviation below.

The percentile comes from the cumulative distribution function (CDF) of the standard normal curve evaluated at that z-score — the share of the distribution that falls at or below the value. This calculator computes it with the Abramowitz & Stegun polynomial approximation of the error function (accurate to within about 0.00002 percentage points), rather than a printed z-table, so it works for any input, not just the values a table happens to list.

Standard deviation must be greater than zero — a distribution with no spread has no meaningful z-score, since every value would be identical to the mean.

FAQ

What does a z-score of 0 mean?

A z-score of 0 means the value equals the mean exactly, which always lands at the 50th percentile of a normal distribution — half the distribution falls below it, half above.

Does this assume the data is normally distributed?

Yes — the percentile calculation specifically uses the normal (bell curve) distribution's CDF. The z-score itself (how many standard deviations from the mean) is meaningful for any distribution, but the percentile figure only matches reality if the underlying data is approximately normal.

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