What this calculator does
A z-score expresses how far a value sits from the mean, measured in standard deviations. It strips away the original units entirely, which is what makes it possible to compare a test score against a height measurement, or this year’s results against last year’s on a different scale.
Converting the z-score into a percentile answers the more intuitive question: what proportion of the distribution falls below this value. That step assumes the data is roughly normally distributed, which is a real assumption and not always a safe one.
The formula
Subtracting the mean centres the value at zero, and dividing by the standard deviation rescales it so one unit equals one standard deviation. The percentile comes from the standard normal cumulative distribution function, computed here with a standard numerical approximation accurate to about seven decimal places.
| Term | Meaning |
|---|---|
| z | (x − μ) ÷ σ, the number of standard deviations from the mean. |
| μ | The population mean. |
| σ | The population standard deviation. |
| Percentile | The proportion of the distribution below this value. |
| p-value | The probability of a value at least this extreme in either direction. |
The inputs explained
| Field | What to enter |
|---|---|
| Value | The value you are assessing. |
| Mean (μ) | The mean of the distribution. |
| Standard deviation (σ) | The standard deviation. Must be positive. |
When to use it
Interpreting a test result
IQ tests are scaled to a mean of 100 and a standard deviation of 15. A score of 130 is two standard deviations above the mean, placing it around the 97.7th percentile.
Comparing across different scales
A score of 85 on one exam and 62 on another cannot be compared directly. Converting both to z-scores puts them on a common footing.
Quality control
Manufacturing tolerances are frequently expressed in standard deviations. A value beyond three z is unusual enough that it typically triggers investigation.
Judging whether a result is surprising
The two-tailed p-value gives the probability of a result at least this extreme in either direction. Values below 0.05 are conventionally treated as significant, though that threshold is a convention rather than a law.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
Z-scores on an IQ-style scale
A distribution with a mean of 100 and a standard deviation of 15, at a range of values.
| Value | Z-score | Percentile | Proportion above |
|---|---|---|---|
| 70 | -2.000 | 2.28% | 97.7% |
| 85 | -1.000 | 15.9% | 84.1% |
| 100 | 0 | 50.0% | 50.0% |
| 115 | 1.000 | 84.1% | 15.9% |
| 130 | 2.000 | 97.7% | 2.28% |
| 145 | 3.000 | 99.9% | 0.135% |
How the standard deviation changes the reading
The same value and mean with different standard deviations, showing how spread determines whether a result is remarkable.
| Standard deviation | Z-score | Percentile | Two-tailed p-value | Interpretation |
|---|---|---|---|---|
| 5 | 6.000 | 100.0% | 1.9802e-9 | Extreme (beyond 99%) |
| 10 | 3.000 | 99.9% | 0.0027 | Extreme (beyond 99%) |
| 15 | 2.000 | 97.7% | 0.0455 | Unusual (beyond 95%) |
| 20 | 1.500 | 93.3% | 0.1336 | Within the typical range |
| 30 | 1.000 | 84.1% | 0.3173 | Within the typical range |
| 50 | 0.6000 | 72.6% | 0.5485 | Within the typical range |
Questions
What is a good z-score?
It depends entirely on context. Roughly 68 per cent of a normal distribution falls between −1 and +1, and 95 per cent between −2 and +2. A z beyond ±2 is unusual, and beyond ±3 is rare.
Can a z-score be negative?
Yes: it simply means the value is below the mean. The sign carries the direction, and the magnitude carries how far.
Does this require the data to be normally distributed?
The z-score itself does not: it is just a rescaling. But converting it into a percentile does, since that step uses the normal distribution. For heavily skewed data the percentile will be misleading.
What is the difference between a z-score and a t-score?
Z-scores use a known population standard deviation. When it is estimated from a small sample, the t-distribution is more appropriate, having heavier tails to reflect the extra uncertainty. Above about 30 observations the two converge.
How is the percentile calculated?
From the standard normal cumulative distribution function. There is no closed-form expression for it, so this calculator uses a well-established numerical approximation.
For probabilities between two bounds, see normal distribution probability. For summarising a data set, use descriptive statistics.