Z-Score Formula
Where x = observed value, μ (mu) = population mean, σ (sigma) = population standard deviation. The z-score tells you how many standard deviations the value is from the mean.
Interpreting Z-Scores
| Z-Score | Meaning | Percentile (approx) |
|---|---|---|
| −3.0 | 3 SD below mean | 0.13% |
| −2.0 | 2 SD below mean | 2.28% |
| −1.0 | 1 SD below mean | 15.87% |
| 0.0 | At the mean | 50% |
| +1.0 | 1 SD above mean | 84.13% |
| +2.0 | 2 SD above mean | 97.72% |
| +3.0 | 3 SD above mean | 99.87% |
The 68-95-99.7 Rule (Empirical Rule)
- 68% of data falls within ±1 standard deviation of the mean (z between −1 and +1)
- 95% of data falls within ±2 standard deviations (z between −2 and +2)
- 99.7% of data falls within ±3 standard deviations (z between −3 and +3)
Example
Test score = 78, Class mean = 70, SD = 8
z = (78 − 70) / 8 = 8 / 8 = 1.0
A score of 78 is 1 standard deviation above the mean → approximately 84th percentile
Frequently Asked Questions
What is a good z-score?
There is no universally "good" z-score — it depends on context. In hypothesis testing, |z| > 1.96 typically indicates statistical significance at the 5% level. In academic testing, z > 1.0 means above average. In quality control, z-scores outside ±3 often trigger investigation.
What is the difference between z-score and percentile?
The z-score is the raw standardized distance from the mean in standard deviation units. The percentile converts this to a probability — the percentage of values in a normal distribution that fall below this z-score. A z-score of 1.0 corresponds to the 84.13th percentile.
Can I use z-scores with non-normal distributions?
You can calculate a z-score for any distribution, but the percentile/probability interpretation only applies to normal (bell-curve) distributions. For skewed or non-normal data, z-scores still show relative position but don't directly map to normal distribution probabilities.
What is the difference between z-score and t-score?
Z-scores use the known population standard deviation (σ). T-scores (t-statistics) use the sample standard deviation (s) and are used when the population SD is unknown — which is almost always the case in practice. T-distributions have heavier tails than normal distributions, especially with small samples.