Z-Score Calculator
How many standard deviations a value sits from the mean, and the percentile and tail probabilities it implies under a normal distribution.
z = (x − μ) ÷ σ standardises any normal variable.
How the z-score calculator works
z = (x − μ) ÷ σ standardises any normal variable. The calculator then evaluates the standard normal cumulative distribution to give the percentile (share below x), the upper tail (share above) and the two-tailed probability of a result at least this extreme.
Formula: z = (x − μ) / σ; percentile = Φ(z)
Worked examples
| Inputs | Z-score | Note |
|---|---|---|
| IQ 130 (mean 100, SD 15) | 2 | z = 2, 97.7th percentile |
| x = 85 | -1 | z = −1, 15.9th percentile |
| x = 100 | 0 | z = 0, 50th percentile |
FAQFrequently asked questions
What is a z-score?
A value re-expressed in standard deviations from the mean: z = 2 means two SDs above.
What percentile is z = 1.96?
About the 97.5th — 2.5% of a normal distribution lies above it, which is why ±1.96 bounds a 95% interval.
Can z-scores be negative?
Yes — below the mean.
Does this assume a normal distribution?
The z-score itself does not; the percentile and tail probabilities do.
How accurate is the percentile?
The normal CDF is evaluated with a standard approximation accurate to about 1.5 × 10⁻⁷.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Common Core State Standards — Mathematics — the terms and methods taught in US schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.