About this z-score calculator
A z-score tells you how many standard deviations a value sits above or below the mean. A z-score of 0 is exactly average, +1 is one standard deviation above, and −2 is two below. This calculator works in four directions: it finds the z-score for a raw value, converts a z-score into the probability and percentile under the standard normal curve, converts a percentile back into a z-score, and turns a z-score back into a raw value on your own scale.
Everything runs in your browser using an accurate normal distribution function, so there is no table lookup and no rounding to two decimals.
The formulas
| Goal | Formula |
|---|---|
| Z-score of a value | z = (x − μ) / σ |
| Raw value from z | x = μ + z × σ |
| Percentile | P(Z ≤ z) = Φ(z), the standard normal CDF |
| Sample version | Use the sample mean and sample standard deviation in place of μ and σ |
Worked example
A test has a mean of 70 and a standard deviation of 10. A student scores 85. The z-score is (85 − 70) / 10 = 1.5. The area to the left of z = 1.5 is about 0.9332, so the student scored higher than roughly 93.3% of test takers, assuming scores are approximately normal.
Common z-score reference values
| Z-score | Percentile (below) | Common meaning |
|---|---|---|
| −2.576 | 0.5% | 99% confidence lower bound |
| −1.96 | 2.5% | 95% confidence lower bound |
| −1 | 15.87% | One SD below the mean |
| 0 | 50% | The mean |
| 1 | 84.13% | One SD above the mean |
| 1.645 | 95% | One-tailed 5% cutoff |
| 1.96 | 97.5% | 95% confidence upper bound |
| 2.576 | 99.5% | 99% confidence upper bound |
When z-scores are useful
- Comparing different scales. A z-score puts an exam mark and a timed run on the same footing so you can say which result was further from average.
- Spotting outliers. Values beyond about ±3 are unusual in roughly bell-shaped data.
- Hypothesis tests. The z statistic feeds directly into a p-value.
The normal-curve percentile is only trustworthy when the data is approximately bell-shaped. For heavily skewed data, rely on the actual ranks instead.
Related tools: find your mean and spread with the Standard Deviation Calculator, turn a z into a test result with the P-Value Calculator, or build a range with the Confidence Interval Calculator.
Frequently asked questions
Subtract the mean from the value and divide by the standard deviation: z = (x - mean) / standard deviation. A positive result means the value is above the mean, a negative result means it is below.
Look up the area under the standard normal curve to the left of the z-score. For example, z = 1 gives about 0.8413, which is the 84.1st percentile. This calculator does that lookup for you with full precision.
It depends on context. A z-score near 0 is typical. Scores above +2 or below -2 are unusual in normal data, and beyond 3 in either direction are very rare. Whether high or low is good depends on what you are measuring.
Yes. A negative z-score means the value is below the mean. For example, z = -1.5 is one and a half standard deviations under average.
A z-score uses the known population standard deviation, or a large sample. A t statistic uses the sample standard deviation and the t distribution, which has heavier tails for small samples. In statistics, a "T-score" in testing contexts can also mean a standardized score with a mean of 50 and an SD of 10.
Multiply the z-score by the standard deviation and add the mean: x = mean + z times standard deviation. Choose the "Raw value from a z-score" option to do this automatically.