Transparent method
The formula
Worked example
Example inputs
How to interpret the result
The percentile that comes out of a z-score is only as good as the assumption that the underlying data is normally distributed, and plenty of real data isn't. A z-score computed from actual income or wait-time data, both famously skewed rather than bell-shaped, converts to a percentile that misrepresents where a given value actually sits within that specific dataset.
Before you rely on it
What to check
Look at whether your list of values is plausibly bell-shaped before trusting the percentile conversion. The z-score itself is always valid arithmetic; the percentile reading depends on that shape.
The common error
Where people go wrong with z-score calculator
Treating the percentile output as exact for any dataset. It's an approximation that assumes normality, and on a skewed distribution the real percentile can differ noticeably from what this conversion reports.
Sensitivity evidence
How value changes the z-score
| Value | Z-score | Approximate percentile |
|---|---|---|
| 68 | -0.875 | 19.1% |
| 77 | 0.25 | 59.9% |
| 85worked example | 1.3 | 89.4% |
| 94 | 2.4 | 99.1% |
| 102 | 3.4 | 100.0% |
Every input, tested
Which input moves the z-score most
| Input | Tested from | To | Z-score at each end | Swing |
|---|---|---|---|---|
| Value | 77 | 94 | 0.25 to 2.4 | 2.1 (170%) |
| Mean | 68 | 83 | 2.1 to 0.25 | 1.9 (150%) |
| Standard deviation | 7 | 9 | 1.4 to 1.1 | 0.317 (25%) |
Two variables at once
Z-score by value and mean
| Value \ Mean | 60 | 75 | 90 |
|---|---|---|---|
| 68 | 1 | -0.875 | -2.8 |
| 77 | 2.1 | 0.25 | -1.6 |
| 85 | 3.1 | 1.3 | -0.625 |
| 94 | 4.3 | 2.4 | 0.5 |
| 102 | 5.3 | 3.4 | 1.5 |
Step by step
The worked example, input by input
| Input | Value used |
|---|---|
| Value | 85 |
| Mean | 75 |
| Standard deviation | 8 |
| Z-score | 1.3 |
| Approximate percentile | 89.4% |
Where the inputs come from
Enter the value you want to standardize, the mean of the distribution and its standard deviation. The standard deviation must be positive. A z-score measures how many standard deviations the value is above or below the mean. Use the population mean and deviation when known; for a sample, use the sample statistics. Make sure all three numbers are in the same units.
When the result stops applying
The percentile shown assumes a normal distribution, so it is unreliable for skewed data. A z-score depends on the reference group. Sample statistics estimate the population, so small samples give shakier z-scores. Outliers inflate the standard deviation and hide themselves.
What to compare it with
See the probabilities on the normal probability and p-value calculators, the spread on the standard deviation page, and the center on the mean and median calculator. The confidence interval calculator uses z values, and the sample size page depends on them.
People also ask
Frequently asked questions
What does a z-score of 1.25 mean?
The value is 1.25 standard deviations above the mean, which for normal data is around the 89th percentile.
What is a typical range for z-scores?
Most values fall between minus 2 and plus 2, and values beyond 3 in either direction are rare.
Can I compare scores from different tests with z-scores?
Yes, standardizing each score against its own mean and deviation makes them comparable, assuming similar distributions.