Z-Score Calculator
Putting Any Value on a Common Scale
A raw score by itself doesn't say much — scoring 700 means something very different on a test with a mean of 500 than on one with a mean of 690. The z-score fixes this by converting any value into a standardized unit: the number of standard deviations it sits above or below the mean. Once converted, values from entirely different scales become directly comparable.
The Formula
x is the raw value, μ (mu) is the mean of the distribution, and σ (sigma) is its standard deviation. A positive z means the value is above the mean; a negative z means it's below.
Reading a Z-Score
For data that's approximately normally distributed, the z-score converts directly into a percentile via the standard normal distribution:
| Z-score | Percentile (cumulative probability) |
|---|---|
| -2.00 | 2.28% |
| -1.00 | 15.87% |
| 0.00 | 50.00% |
| +1.00 | 84.13% |
| +1.645 | 95.00% |
| +1.96 | 97.50% |
| +2.00 | 97.72% |
Use the Standard Normal Distribution Calculator to get the exact cumulative probability for any z-score, not just these common reference points.
Where Z-Scores Are Used
- Standardized testing — SAT, GRE, and IQ scores are all built from z-scores scaled to a chosen mean and standard deviation.
- Outlier detection — values with |z| above roughly 2 or 3 are commonly flagged as unusual.
- Quality control — measuring how far a manufactured part's dimension falls from the process average, in standardized units.
- Growth charts — pediatric height and weight percentiles are computed from z-scores relative to age-based norms.
How to Use This Calculator
- Enter the value you want to standardize (x).
- Enter the mean (μ) of the distribution it belongs to.
- Enter the standard deviation (σ) of that distribution.
- Select Calculate to get the z-score.
Related Calculations
Once you have a z-score, convert it to an exact probability with the Standard Normal Distribution Calculator. To find the standard deviation your data actually has, use the Standard Deviation Calculator.