Z-score calculator
A z-score (or standard score) tells you how many standard deviations a single value sits above or below the mean. It is the common way to compare scores from different scales — exam marks, lab measurements, test results — on one standard footing, and to read off where a value lands on a bell curve.
How it works
The core formula is z = (x − μ) / σ, where x is your value, μ is the population mean and σ is the standard deviation. A z-score of 0 sits exactly at the mean; positive scores are above it, negative scores below.
The calculator also reports the percentile — the percentage of a normal distribution lying below your value — by evaluating the standard normal cumulative distribution function (computed here with the Abramowitz & Stegun erf approximation). Switch to the reverse mode to recover a raw value from a z-score using x = μ + z × σ.
Example
A test has mean μ = 70 and standard deviation σ = 10. A score of x = 85 gives z = (85 − 70) / 10 = 1.5, meaning the score is 1.5 standard deviations above average — about 93.3% of results fall below it.
| Z-score | Approx. percentile below |
|---|---|
| -2 | 2.3% |
| -1 | 15.9% |
| 0 | 50% |
| 1 | 84.1% |
| 2 | 97.7% |
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