A z score restates a value as the number of standard deviations it sits from the mean. Subtract the mean and divide by the standard deviation, and any measurement in any units becomes a pure number. That is what makes heights, exam marks and blood pressures comparable on a single scale.
Under a normal distribution the z score maps directly to a percentile. A z of zero is the fiftieth percentile, one is the 84th, two is the 97.7th and three is the 99.87th. This page computes that mapping with a high-accuracy error function approximation, so the percentile is good to several decimal places rather than being read off a coarse table.
The formula
x- The raw value
μ- Mean of the distribution
σ- Standard deviation
z- The number of standard deviations from the mean
How to use this calculator
- Enter the raw value you are assessing.
- Enter the mean and standard deviation of the distribution.
- The gauge shows the z score.
- The percentile and tail probability appear below.
Worked examples
An IQ of 130
With a mean of 100 and a standard deviation of 15, the z score is (130 − 100) ÷ 15 = 2. That places the value at the 97.72nd percentile, so about one person in 44 scores at least this high.
Comparing across different scales
A mark of 78 where the class mean is 65 with a deviation of 8 gives a z of 1.625. A mark of 91 where the mean is 84 with a deviation of 5 gives 1.4, so the first result is actually the stronger performance despite the lower raw score.
How to read your score
Frequently asked questions
What does a negative z score mean?
The value is below the mean. A z of minus one sits one standard deviation below and corresponds to the 15.87th percentile.
Does the percentile require a normal distribution?
Yes. The z score itself is meaningful for any distribution, but converting it into a percentile assumes normality. For skewed data the mapping will be wrong.
What counts as an unusual z score?
Beyond plus or minus two is often called unusual, covering about 4.6% of a normal distribution. Beyond three is rare, at roughly 0.27%.
How does this relate to statistical significance?
A two-tailed z of 1.96 corresponds to a p-value of 0.05, which is the origin of that familiar threshold. The calculator reports the two-tailed probability directly.
Z scores and the normal distribution
| z score | Percentile | Above this value | Two-tailed |
|---|---|---|---|
| -3.00 | 0.1350% | 99.8650% | 0.2700% |
| -2.00 | 2.2750% | 97.7250% | 4.5500% |
| -1.96 | 2.4998% | 97.5002% | 4.9996% |
| -1.00 | 15.8655% | 84.1345% | 31.7311% |
| -0.50 | 30.8538% | 69.1462% | 61.7075% |
| +0.00 | 50.0000% | 50.0000% | 100.0000% |
| +0.50 | 69.1462% | 30.8538% | 61.7075% |
| +1.00 | 84.1345% | 15.8655% | 31.7311% |
| +1.65 | 95.0015% | 4.9985% | 9.9970% |
| +1.96 | 97.5002% | 2.4998% | 4.9996% |
| +2.00 | 97.7250% | 2.2750% | 4.5500% |
| +2.58 | 99.5060% | 0.4940% | 0.9880% |
| +3.00 | 99.8650% | 0.1350% | 0.2700% |
A two-tailed probability of 5% corresponds to a z of plus or minus 1.96.