Z-score formulas
| To find | Formula |
|---|---|
| z-score | z = (x − μ) ÷ σ |
| Raw score | x = μ + z·σ |
| z-score in a sample | z = (x − x̄) ÷ s |
Worked example
A test has a mean of 70 and a standard deviation of 10. Maya scores 85. Her z-score is (85 − 70) ÷ 10 = 1.5: one and a half standard deviations above the mean. The area to the left of z = 1.5 is 0.9332, so she scored higher than about 93% of students (if the scores are normally distributed).
Common z-scores
| z | Percentile | Used for |
|---|---|---|
| ±1.000 | 15.87% / 84.13% | 68% of values lie within ±1σ |
| ±1.645 | 5% / 95% | 90% confidence intervals, one-tailed 5% tests |
| ±1.960 | 2.5% / 97.5% | 95% confidence intervals |
| ±2.576 | 0.5% / 99.5% | 99% confidence intervals |
| ±3.000 | 0.13% / 99.87% | 99.7% of values lie within ±3σ |
Comparing scores from different tests
Z-scores put different scales on a common footing. A 78 on a test with mean 65 and SD 6 (z ≈ 2.17) is a stronger result than an 88 on a test with mean 80 and SD 8 (z = 1.0), even though 88 is the bigger number.
For probabilities between two values, use the normal distribution calculator. To find the mean and standard deviation of your data first, use the standard deviation calculator.
What a z-score looks like
On a bell curve, each z-score is a position measured in standard deviations from the mean. The shaded bands show how much of a normal distribution lies within 1, 2 and 3 standard deviations.
How to read your z-score
- The sign tells you the direction: positive is above the mean, negative is below.
- The size tells you how unusual: |z| under 1 is ordinary, 1–2 is somewhat unusual, above 2 is unusual and above 3 is rare.
- The percentile (area to the left) tells you the share of values below it, assuming a normal distribution. z = 1.75 is at the 96th percentile.
Worked example: which result is better?
Sam scores 82 in biology (class mean 75, SD 4) and 88 in history (mean 80, SD 8). The biology z-score is (82 − 75) ÷ 4 = 1.75; the history z-score is (88 − 80) ÷ 8 = 1.0. Relative to classmates, the biology result is stronger, about the 96th percentile against the 84th, even though 88 is the higher raw score.
Common mistakes to avoid
- Using the wrong standard deviation. Use the SD of the group the value is compared with, not of a different group.
- Reading percentiles for skewed data. Converting z to a percentile assumes a normal distribution. For skewed data, the percentile can be well off.
- Sign errors. Always subtract the mean from the value (x − μ), not the other way round.
- Confusing a z-score with a percentile. z = 1 is not the 1st percentile. It is about the 84th.
Z-scores in Excel, Google Sheets, TI-84 and Python
| Software | How |
|---|---|
| Excel / Google Sheets | =STANDARDIZE(x, mean, sd) for z; =NORM.S.DIST(z, TRUE) for the percentile; =NORM.S.INV(p) for the reverse. |
| TI-84 | Percentile: 2nd › DISTR › normalcdf(−1E99, z). Reverse: invNorm(p). |
| Python | scipy.stats.zscore(data) for every value; scipy.stats.norm.cdf(z) and norm.ppf(p) for percentiles. |
Related tools: Normal Distribution Calculator, Standard Deviation Calculator, IQR & Outlier Calculator, Histogram Maker and Descriptive Statistics Calculator.