Math
Z-Score Calculator
Enter a value with the mean and standard deviation of its distribution.
The link carries your figures, so it reopens on exactly these numbers.
The z values worth knowing
On a standard normal distribution — mean 0, standard deviation 1. The last two columns are the one-sided and two-sided readings of the same z, and they differ by a factor of two.
| Z-score | Percentile | Share above this value | Share at least this far from the mean |
|---|---|---|---|
| 1Two thirds of values sit within ±1. | 84.1345% | 15.8655% | 31.7311% |
| 1.645The 5% one-sided threshold. | 95.0015% | 4.9985% | 9.997% |
| 1.96The 5% two-sided threshold — the one behind “p < 0.05”. | 97.5002% | 2.4998% | 4.9996% |
| 2 | 97.725% | 2.275% | 4.55% |
| 2.576The 1% two-sided threshold. | 99.5002% | 0.4998% | 0.9995% |
| 3About 1 in 370 in either direction. | 99.865% | 0.135% | 0.27% |
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.