Z-Score Calculator
How many standard deviations a value, or a sample mean, lies from the mean, with the probability below, above and in both tails.
Result
Z-score: 1.5
Exact arithmetic; the probabilities assume a normal distribution.
- Probability below, P(Z < z)
- 93.3193%
- Probability above, P(Z > z)
- 6.6807%
- Two-tailed probability
- 13.3614%
https://calcgallery.com/statistics/z-score-calculator/
Results follow the mathematical operation and values shown. Check that you selected the intended operation, convention and dataset type before relying on the result. Read the full disclaimer
How to use the Z-Score Calculator
- Choose what you have. One value against a known population, a sample mean against a known population, or a value within your own data.
- Enter the numbers.
- Read the z-score, then the probabilities of lying below it, above it, and at least as far from the mean in either direction.
The formula
one value z = (x − μ) ÷ σa sample mean z = (x̄ − μ) ÷ (σ ÷ √n)your own data z = (x − x̄) ÷ sμ and σ are the population mean and standard deviation. For your own data, the page uses the data’s own mean x̄ and its sample standard deviation s.
From z to a probability
The z-score locates a value on the standard normal curve, whose mean is 0 and whose standard deviation is 1. The shaded area in the diagram is the share of the curve below that point. The page works this out exactly rather than reading a printed z-table, so a z of 1.2345 needs no rounding to the nearest row.
About 68% of a normal distribution lies within one standard deviation of the mean, about 95% within two and about 99.7% within three. A z-score beyond ±2 is therefore unusual, and beyond ±3 rare.
Percentiles and z-scores
P(Z < z) is the value’s percentile if the data are normal. A z of 1.5 is at about the 93rd percentile: roughly 93% of the distribution lies below it.
With your numbers
- z = (85 − 70) ÷ 10 = 1.5
Quick reference
Exact arithmetic; the probabilities assume a normal distribution.
| Example | Z-score |
|---|---|
| 85 against mean 70, SD 10 | 1.5 |
| 55 against mean 70, SD 10 | -1.5 |
| IQ 130, mean 100, SD 15 | 2 |
| Sample mean 72 of 25, mean 70, SD 10 | 1 |
| Sample mean 102 of 36, mean 100, SD 15 | 0.8 |
| 90 within eight scores | 1.2065 |
Common questions about the Z-Score Calculator
- What does a z-score mean?
- How many standard deviations a value is from the mean. A z of 1.5 is one and a half standard deviations above it; a z of −2 is two below it. Because it is measured in standard deviations, a z-score lets you compare values from different scales, such as a result on one test with a result on another.
- Why does a sample mean use σ ÷ √n?
- Because averages vary less than single values. The mean of 25 people’s heights is far more stable than any one person’s height, and its standard deviation, the standard error, is σ ÷ √n. Using σ itself for a sample mean is the most common mistake with z-scores: with n = 25 it makes the z five times too small, and an unusual sample mean looks ordinary.
- Which probability do I want?
- P(Z < z) is the share of the distribution below the value, which is its percentile. P(Z > z) is the share above it. The two-tailed probability is the chance of landing at least this far from the mean on either side, which is the p-value for a two-sided z test.
- Do the probabilities need normal data?
- Yes. The z-score itself is just arithmetic, but turning it into a probability assumes the values follow a normal distribution. For a sample mean that holds approximately even for non-normal data once the sample is reasonably large, by the central limit theorem. For a single value from skewed data it does not.