A Common Scale for Any Distribution
A z-score expresses a value as the number of standard deviations it sits from the mean:
z = (x − μ) / σ
This standardisation is what makes different measurements comparable. A test score of 128 on a scale with mean 100 and standard deviation 15 has z = 1.87, and so does a height, a rainfall total or a share price the same distance from its own mean.
Z-score Chart
| Z-score | Percentile | Below | Above | Within ±z |
|---|---|---|---|---|
| −3.0 | 0.13 | 0.0013 | 0.9987 | 99.73% |
| −2.0 | 2.28 | 0.0228 | 0.9772 | 95.45% |
| −1.0 | 15.87 | 0.1587 | 0.8413 | 68.27% |
| 0 | 50.00 | 0.5000 | 0.5000 | 0% |
| 1.0 | 84.13 | 0.8413 | 0.1587 | 68.27% |
| 1.96 | 97.50 | 0.9750 | 0.0250 | 95.00% |
| 2.0 | 97.72 | 0.9772 | 0.0228 | 95.45% |
| 3.0 | 99.87 | 0.9987 | 0.0013 | 99.73% |
The Empirical Rule
For a normal distribution, about 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. The precise figure behind the familiar "95%" is z = 1.96, which is why that number appears throughout statistics.
Where Z-scores Are Used
- Standardised tests — converting raw scores to percentiles.
- Quality control — a process is out of control when measurements exceed ±3 sigma. Six Sigma targets a defect rate corresponding to roughly 3.4 per million.
- Medicine — bone density is reported as a T-score, which is a z-score against a young adult reference.
- Finance — the Altman Z-score for bankruptcy prediction; risk models expressed in standard deviations.
- Anomaly detection — flagging observations beyond a threshold z.
The Normality Assumption
Converting a z-score to a percentile requires the distribution to be approximately normal. The z-score itself is always computable, but the percentile is meaningless for a strongly skewed distribution. Income has a well-defined mean and standard deviation, and a z-score of 2 on income does not correspond to the 97.7th percentile.
Frequently Asked Questions
Can a z-score be negative?
Yes — it simply means the value is below the mean. The sign carries direction, the magnitude carries distance.
What counts as an unusual z-score?
Beyond ±2 is conventionally unusual, covering about 5% of a normal distribution. Beyond ±3 is rare, about 0.3%.
What is the difference between z and t?
The z-score uses a known population standard deviation. The t-statistic uses the sample standard deviation and has heavier tails to account for that extra uncertainty. Above roughly 30 observations the two nearly coincide.