Math
Z-Score Calculator
Enter a value with the mean and standard deviation of its distribution.
The formula
z = (value − mean) ÷ standard deviation
percentile = Φ(z)
Worked example
An IQ score of 130, where the mean is 100 and the standard deviation 15
- (130 − 100) ÷ 15 = 2
- Φ(2) = 97.72nd percentile
- 2.28% score higher
Where this goes wrong
Reading a one-sided percentile as a two-sided result
A z of 2 puts you at the 97.7th percentile, so 2.3% score higher — but 4.6% are at least that far from the mean in either direction. Significance tests almost always want the two-sided figure, and using the one-sided one makes a result look twice as unusual as it is.
Questions
- What does a z-score actually measure?
- How many standard deviations a value sits from the mean. It strips the units away, so a z of 2 means the same thing whether you started with millimetres, dollars or exam marks.
- Does this work for any data?
- The z-score itself does. The percentile does not — it assumes the data follow a normal distribution. For skewed data, such as incomes, the z-score is still computable but the percentile it implies will be wrong.
- What counts as an unusual z-score?
- Around 68% of a normal distribution lies within one standard deviation, 95% within two and 99.7% within three. Beyond three is genuinely rare — about one observation in 370.