Score 85, mean 70, SD 10
z = 1.50, about the 93rd percentile (93.32%).
Z-score calculator
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The z-score is how many standard deviations a value sits from the mean. The percentile assumes the data follow a normal (bell-shaped) distribution; for skewed data it is only a guide.
z-score
1.50
85 is 1.50 standard deviations above the mean. On a normal curve about 93.32% of values fall below it, so it sits at roughly the 93th percentile.
- Percentile (below)
- 93.32%
- Above this value
- 6.68%
- Within ±|z| of the mean
- 86.64%
Common z-scores
| z | Percentile | Value here |
|---|---|---|
| -2 | 2.3% | 50 |
| -1 | 15.9% | 60 |
| 0 | 50.0% | 70 |
| 1 | 84.1% | 80 |
| 1.5 | 93.3% | 85 |
| 2 | 97.7% | 90 |
| 3 | 99.9% | 100 |
Z-scores in plain words
A z-score tells you how far a value is from the average, measured in standard deviations. It lets you compare things on different scales: a 72 in a hard test can be more impressive than an 85 in an easy one.
Say a test has a mean of 70 and a standard deviation of 10, and you scored 85. Your z-score is (85 − 70) ÷ 10 = 1.5. On a normal curve, about 93.32% of scores fall below 1.5, so you are close to the 93rd percentile.
A z-score of 0 is exactly average. About 68% of values sit between −1 and 1, about 95% between −2 and 2, and about 99.7% between −3 and 3. The colour bands on the meter follow those cut-offs.
The percentile here comes from the normal distribution formula, not from your actual data. If your data are skewed or have a few extreme values, count the values below yours instead.
Sources
What this calculator does not account for
- Percentile assumes a normal distribution.
- Not suited to strongly skewed data.
How this is calculated
Formula: Z-score
z =(x − μ) ÷ σWhat the symbols mean
- xYour value.
- μThe mean.
- σThe standard deviation.
Good to know
- The percentile is the share of a normal curve below z.
- To get the mean and SD from raw data, use the standard deviation calculator.
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Worked examples
Each example is worked out by the same code as the calculator above.
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<iframe src="/embed/z-score-calculator" title="Z-score" width="100%" height="720" style="border:0;max-width:42rem" loading="lazy"></iframe>Questions
How do you calculate a z-score?
Subtract the mean from your value and divide by the standard deviation. A score of 85 with a mean of 70 and SD of 10 has z = 15 ÷ 10 = 1.5.
What percentile is a z-score of 1.5?
About 93.32%. Roughly 93 out of 100 values on a normal curve fall below it.
Can a z-score be negative?
Yes. A negative z-score means the value is below the mean. z = −1 is about the 16th percentile.
What is a good z-score?
It depends on the context. Above 2 or below −2 is unusual, since only about 5% of values fall that far from the mean.
Does the percentile work for any data?
Only for data that is roughly normal (bell-shaped). For skewed data, such as incomes, the z-score is still valid but the percentile is only a guide.