What a z-score measures
A z-score, or standard score, tells you how far a value is from the mean in units of standard deviation. It puts values from different scales on a common footing: a 85 on one test and a 640 on another can be compared once both are expressed as z-scores. When the data follow a normal distribution, the z-score also gives the percentile, the share of values below it.
The formula
- z = (x − μ) ÷ σ, where x is the value, μ the mean and σ the standard deviation
- Percentile = Φ(z), the cumulative normal probability
- Share above = 1 − Φ(z)
- Two-tailed share at least as far from the mean = 2 × (1 − Φ(|z|))
Worked example
A student scores 85 on an exam where the class average is 70 and the standard deviation is 10.
- z = (85 − 70) ÷ 10 = 1.5: the score is one and a half standard deviations above average.
- Φ(1.5) = 0.9332, so the student did better than about 93.3% of the class.
- Only 6.7% of students would be expected to score higher.
- 86.6% of scores lie between z = −1.5 and +1.5; the other 13.4% are at least as far from the mean, on either side.
Z-scores and percentiles
| Z-score | Percentile | Share above |
|---|---|---|
| −2.0 | 2.3% | 97.7% |
| −1.0 | 15.9% | 84.1% |
| 0 | 50.0% | 50.0% |
| 0.5 | 69.1% | 30.9% |
| 1.0 | 84.1% | 15.9% |
| 1.5 | 93.3% | 6.7% |
| 2.0 | 97.7% | 2.3% |
| 3.0 | 99.87% | 0.13% |
Where z-scores are used
- Standardized tests: scaled scores are built from z-scores so that results are comparable from one year to the next.
- Growth charts: pediatricians express a child’s height or weight as a z-score against CDC or WHO reference data.
- Bone density: a DEXA scan reports a T-score, which is a z-score against healthy young adults; −2.5 or lower indicates osteoporosis.
- Quality control: a measurement more than 3 standard deviations from the target signals a process out of control.
- Outlier detection: analysts often flag data points with |z| above 3.
Caveats
The percentile from a z-score is only accurate if the data are roughly normal. For skewed data such as incomes or house prices, the share of values above a given z-score can be very different. Check a histogram first, or use the actual ranks of the data.