Z-Score Calculator
Convert a value into a z-score and find its percentile on the normal curve.
93.32% of values fall below x · 6.68% above
A z-score counts how many standard deviations a value sits from the mean: z = (x − μ) ÷ σ. For a score of 85 with mean 70 and σ = 10, z = (85 − 70) ÷ 10 = 1.5 — meaning P(Z ≤ 1.5) = 0.9332, so the score beats about 93% of the group.
What a z-score tells you
A z-score measures how many standard deviations a value sits above or below the mean. A z of +1 is one standard deviation above the mean; −2 is two below. By rescaling every value to the same standard-normal ruler, z-scores let you compare results from completely different scales — a test score and a height, say — and read off where a value falls on the bell curve.
value minus the mean, divided by the standard deviation
Worked example
A score of 85 on a test with mean 70 and standard deviation 10.
- 1 Subtract the mean. x − μ = 85 − 70 = 15.
- 2 Divide by the standard deviation. z = 15 ÷ 10 = 1.5.
- 3 Read the percentile. P(Z ≤ 1.5) = 0.9332, so the score beats about 93% of the group — the top 6.7%.
z-scores and percentiles
Cumulative area to the left of each z on the standard normal curve.
| z-score | Percentile | Meaning |
|---|---|---|
| −2.0 | 2.3% | 2 SD below the mean |
| −1.0 | 15.9% | 1 SD below the mean |
| 0.0 | 50.0% | exactly at the mean |
| +1.0 | 84.1% | 1 SD above the mean |
| +1.96 | 97.5% | top 2.5% (two-tailed 5%) |
| +2.0 | 97.7% | 2 SD above the mean |
Interpreting and applying z-scores
Sign and size both matter. The sign says above or below the mean; the magnitude says how far in standard-deviation units. A z of −2.5 is more extreme than a z of +1.0.
The empirical rule. For roughly bell-shaped data, about 68% of values fall within ±1, 95% within ±2, and 99.7% within ±3 standard deviations of the mean — a fast sanity check before reaching for the table.
Percentile vs tail. The percentile is the area to the left of your z; the tail probability is what’s beyond it. A z of +1.5 sits at the 93rd percentile and leaves about 6.7% in the upper tail.
z vs t. Z-scores assume the standard normal model with a known σ. When σ is estimated from a small sample, use a t-score, which accounts for that extra uncertainty.