How to Calculate a Z-Score (With Percentile)
calendar_monthPublished 2026-08-15verified_userReviewed by Calculopedia editorial
A z-score (also called a standard score) is one of statistics' most useful inventions because it answers a disarmingly simple question: how far above or below average is this value? Rather than describing the raw number, it measures distance in standard deviations — a common ruler that makes wildly different datasets comparable.
It's the trick behind grading curves, standardized-test scores, and manufacturing tolerances. Once you know a z-score, you instantly understand where a value sits relative to its own distribution of peers, without needing the original units at all.
The formula
Z = (x − mean) ÷ standard deviation
- x — the data point you're locating.
- mean (μ) — the average of the dataset.
- standard deviation (σ) — the typical spread of the data around the mean.
Subtract to get the distance from the mean, then divide by the spread to express that distance in standard-deviation units. A positive z means above average; a negative z means below.
Worked example
A class' exam mean is 70 with a standard deviation of 10. What's the z-score of a student who scored 85?
Z = (85 − 70) ÷ 10 = 1.5
An 85 is 1.5 standard deviations above the mean. Because the standard deviation is 10, every 10 points moves the student one full standard deviation — so 1.5 of them above 70 lands you at 85.
Try it going down: a score of 55 gives Z = (55 − 70) ÷ 10 = −1.5, a symmetric 1.5 below the mean.
Reading the percentile
In a normal distribution, each z-score maps to a percentile — the share of values at or below that point. Continuing the example, a z of 1.5 is around the 93rd percentile: about 93% of students scored at or below 85.
Common anchor points worth memorising:
| Z-score | Percentile |
|---|---|
| −2.00 | 2.3% |
| −1.00 | 15.9% |
| 0.00 | 50% |
| +1.00 | 84.1% |
| +1.50 | 93.3% |
| +2.00 | 97.7% |
The elegant pattern: the mean is the 50th percentile, one standard deviation up is roughly the 84th, two up the ~98th — and it's symmetric below.
Quick interpretation
| Z-score | Meaning |
|---|---|
| ≈ 0 | Average — right at the mean |
| +1 to +2 | Above average |
| +2 to +3 | Well above average |
| −1 to −2 | Below average |
| ±3 or beyond | Unusually far from the mean |
Anything beyond about ±3 standard deviations is rare in a normal distribution — the sort of outlier that flags a data problem or a truly exceptional value.
When to be careful
Z-scores are only meaningful when your data is reasonably normal (bell-shaped). For heavily skewed or multi-modal data, a z-score can mislead, because the "spread" doesn't describe the shape the same way. Always check the distribution before leaning on z-percentile tables.
Where z-scores shine
- Standardized tests — comparing scores across different test versions and years on one common scale.
- Quality control — flagging parts or measurements that drift beyond spec limits by monitoring their z-scores over time.
- Research — converting "apples and oranges" data (different units and scales) into a common scale so it can be averaged, compared or combined.
- Grades and rankings — seeing exactly where one score sits within its class distribution.
Key takeaways
- Z = (x − mean) ÷ standard deviation.
- Positive = above average; negative = below; zero = exactly average.
- In a normal distribution, z maps to a percentile (e.g., +1.5 ≈ 93rd).
- Only trust z-percentile tables when the data is roughly bell-shaped.
FAQ
What does a z-score of 0 mean? It means the value equals the mean — dead average. A score right at 70 in our example gives Z = 0.
Can a z-score be bigger than 3? Yes, but it's rare in a normal distribution. Values beyond about ±3 standard deviations are genuine outliers.
How does a z-score become a percentile? In a normal distribution there's a one-to-one mapping, used by z-tables or calculators, from the standard deviation distance to the cumulative percentage below that point.
Do I need normal data to use z-scores? You can always compute the number, but interpreting it as a percentile only holds for roughly normal data. For skewed data, use caution and check the distribution first.
Get your number instantly with the Z-Score Calculator, and when you're comparing two values directly, the Percentage Difference Calculator is a handy companion.