Z-score Calculator
Calculate a z-score from a value, mean and standard deviation, plus its percentile rank in a normal distribution.
Well Above Average
A value of 85 sits at the 89.4th percentile — meaning it beats about 89% of a normally distributed population.
- Percentile89.44%
ƒShow your work
z = (x − μ) / σ
- 1z = (85 − 75) / 8 = 1.25
About the Z-score Calculator
A z-score converts a raw value into a standardized measure of how far it sits from the mean, expressed in standard deviations rather than in the original units — which makes it possible to compare a test score, a height, or any other measurement against the rest of its distribution, or even against a completely different data set. A z-score of 2 always means 'two standard deviations above average,' whether it's describing an SAT score or a factory part's tolerance.
This shows up constantly in statistics coursework, but also in practical score interpretation — a school reporting where a student's test result falls relative to the class, or a quality-control process flagging a measurement as unusually far from the expected value.
How it’s calculated
The formula is z = (x − μ) / σ, where x is the raw value, μ (mu) is the mean, and σ (sigma) is the standard deviation. Subtracting the mean centers the value at zero, and dividing by the standard deviation rescales it so that one full unit of z equals one standard deviation.
Once you have a z-score, the percentile — what fraction of a normally distributed population falls below that value — comes from the standard normal distribution's cumulative distribution function. A z-score of 0 sits at the 50th percentile (right at the mean); a z-score of +1 sits at roughly the 84th percentile; a z-score of −1 sits at roughly the 16th percentile.
This percentile conversion assumes the underlying data is normally distributed (the classic bell curve) — for data that doesn't follow that shape, the z-score is still meaningful as a distance-from-mean measure, but the percentile figure becomes less reliable.
Frequently asked questions
What does a negative z-score mean?
It means the value sits below the mean — the more negative, the further below. A z-score of −2 means the value is two standard deviations below average, which corresponds to a low percentile (roughly the 2nd-3rd percentile in a normal distribution).
What counts as a 'good' z-score?
It depends entirely on what's being measured and what direction is favorable — there's no universal good or bad. A z-score of +2 on a test is great; a z-score of +2 on a defect-rate measurement in manufacturing would be a serious problem.
How is a z-score related to percentile?
The z-score is converted to a percentile using the standard normal distribution — this calculator does that conversion automatically. A z-score of 0 always lands at the 50th percentile, and roughly 68% of a normal distribution falls between z-scores of −1 and +1.
Does the z-score formula work for non-normal data?
The z-score itself (how many standard deviations from the mean) is still valid for any distribution shape. But the percentile figure this calculator derives from it specifically relies on the data being normally distributed — for skewed data, that percentile can be noticeably off.
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