How to calculate a z-score
A z-score tells you how many standard deviations a value is above or below the mean. It lets you compare results on different scales and find a percentile. Here is the formula, a worked example and a fair comparison of two test scores.
The short version
- Find the mean and standard deviation. Of the group the score belongs to, such as the whole class.
- Subtract the mean from the raw score. x − μ.
- Divide by the standard deviation. z = (x − μ) ÷ σ.
- Read the percentile. Look up the area to the left of z on the standard normal curve.
Skip the arithmetic. The free Z-Score Calculator does this for you, shows the formula and the working with your own numbers, and runs in your browser - nothing you type is uploaded. Also useful: Standard Deviation Calculator.
The formula and an example
The score is 1.5 standard deviations above the average. On a normal curve the area to the left of z = 1.5 is about 0.9332, so the score is higher than roughly 93% of the group. A z-score of 0 is exactly average; −1 is one standard deviation below.
Going the other way
To find the raw score for a z-score: x = μ + z × σ. With a mean of 70, a standard deviation of 10 and z = 1.5: x = 70 + 15 = 85. This is how you find the score needed to reach a given percentile.
Comparing two tests fairly
Maths: you scored 78, the class mean was 65 and the standard deviation 8. English: you scored 84, the mean was 75 and the standard deviation 12.
The English mark is higher, but the maths result is further above that class's average, so it is the stronger performance relative to the group.
Common z-scores and percentiles
| z | Share below |
|---|---|
| −2 | 2.3% |
| −1 | 15.9% |
| 0 | 50% |
| 1 | 84.1% |
| 1.5 | 93.3% |
| 1.96 | 97.5% |
| 2 | 97.7% |
About 68% of a normal distribution lies within one standard deviation of the mean, 95% within two and 99.7% within three.
A caution about percentiles
The z-score itself is always correct. The percentile read from the normal curve is only accurate if your data is roughly bell-shaped. For strongly skewed data, count the actual share of values below the score instead.