Quick answer
A z-score tells you how many standard deviations a value sits above or below the mean: z = (X − μ) / σ. Enter a raw score with the mean and standard deviation to get the z-score, or work backward from a known z-score to find the raw score, percentile, and probability under the normal curve.
Key takeaways
- A z-score of 0 means the value equals the mean exactly; positive z is above the mean, negative z is below it.
- Z-scores standardize any normally distributed data set to a common scale, letting you compare values measured in completely different units.
- The empirical rule states roughly 68% of values fall within ±1 SD, 95% within ±2 SD, and 99.7% within ±3 SD of the mean.
- Percentile and z-score are related but different — a z-score of 1 corresponds to about the 84th percentile, not the 1st.
The z-score formula, worked example
z = (X − μ) / σ
Example: raw score X = 75, mean μ = 70, standard deviation σ = 10
- z = (75 − 70) / 10 = 5 / 10 = 0.5
- A z-score of 0.5 means the score sits half a standard deviation above the mean.
- Looking up z = 0.5 on the standard normal distribution gives a percentile of about 69.1% — the score beats roughly 69% of the distribution.
Z-scores, percentiles, and the empirical rule
For any normal distribution, the empirical rule (68-95-99.7 rule) describes how much data falls within each band of standard deviations:
| Range | Approximate % of data |
|---|---|
| μ ± 1σ (−1 ≤ z ≤ 1) | ~68% |
| μ ± 2σ (−2 ≤ z ≤ 2) | ~95% |
| μ ± 3σ (−3 ≤ z ≤ 3) | ~99.7% |
Comparing scores across different scales
Z-scores let you compare performance on tests with entirely different scoring scales. Suppose Student A scores 1350 on the SAT (mean 1050, SD 200) and Student B scores 30 on the ACT (mean 21, SD 5):
- Student A: z = (1350 − 1050) / 200 = 1.5
- Student B: z = (30 − 21) / 5 = 1.8
Even though the raw scores use unrelated scales, Student B's higher z-score means they performed better relative to their own test's distribution of scores.
Common mistakes
- Treating a z-score as if it were a percentile directly — a z-score of 1 is roughly the 84th percentile, not the 1st.
- Mixing up left-tail, right-tail, and two-tailed probability — each answers a different question about where a value falls.
- Applying z-scores and the empirical rule to data that isn't approximately normally distributed, where the percentages no longer hold.
- Dropping the sign of a negative z-score, which flips whether a value is above or below the mean.
Related calculators
- Standard Deviation Calculator — for computing σ from a raw data set
- Confidence Interval Calculator — for estimating a range around a sample statistic
- P-Value Calculator — for hypothesis testing significance from a test statistic
- Statistics Calculator — for a full set of descriptive statistics on your data