Z-score
Also known as: standard score, standardised value.
How many standard deviations a value lies above or below the average of its group. It puts different measures on one scale.
Synthetic dataExamples on this page use synthetic data created for illustration.
Definition
A z-score says how many standard deviations a value lies above or below the average of its group: subtract the mean, then divide by the standard deviation. It puts values measured in different units on one scale and makes unusual values easy to spot.
In plain terms
Is a price of 45 high? Is a rating of 4.0 low? It depends on the group. A z-score answers by measuring distance from the group’s average in units of its usual spread. A z of 0 is average, +1 is one standard deviation above, −2 is two below.
For roughly bell-shaped data, about 95% of values fall within ±2. For skewed or heavy-tailed data that rule fails and large z-scores are common, so treat them as a prompt to look, not as proof of an error (see Outlier).
Example
| Quantity | Value | Group mean | Group standard deviation | Z-score |
|---|---|---|---|---|
| A price | 45 | 40 | 5 | +1.0 |
| A rating | 4.0 | 4.3 | 0.2 | −1.5 |
| Stories on a day (see Event Detection) | 48 | 10.3 | 1.1 | +34 |
The first two can be compared directly: the rating is further below its group than the price is above its own. The third is so far out that it is certainly not routine variation.
Common misunderstandings
- “A z-score above 2 means the value is wrong.” It means unusual relative to the group; the value may be right.
- “The 95% rule holds for any data.” It is a property of bell-shaped data; skewed data break it.
- “A z-score is a probability.” It is a distance; a probability follows only with an assumption about the distribution.
References
- NIST/SEMATECH e-Handbook of Statistical Methods . NIST.
- David S. Moore, George P. McCabe, Bruce A. Craig. Introduction to the Practice of Statistics . W. H. Freeman, 2021. ISBN 978-1319013387.