How to find a z-score
To find a z-score, subtract the mean from the value and divide by the standard deviation. A positive z is above the mean, a negative z below it, and z = 0 is exactly the mean.[1]
z = (x − μ) ÷ σx is the value, μ the mean, σ the standard deviation. For a sample, use x̄ and s.
Worked examples
Test score 85, class mean 70, SD 10: z = (85 − 70) ÷ 10 = 1.5. The area to the left is 0.9332, so the score beats about 93.3% of a normal distribution.
Value 42, mean 50, SD 5: z = (42 − 50) ÷ 5 = −1.6, with 5.48% of the distribution below it.
Need the mean and standard deviation first? Paste your data into the statistics calculator or the standard deviation calculator.
Z-score to percentile: the z-table
A z-score becomes a percentile through the standard normal cumulative distribution: the area under the bell curve to the left of z.[2][3] The table gives the common values; the calculator handles any z to about 15 significant digits and switches to scientific notation for tiny tails instead of printing 0.
| z | Area left P(Z < z) | Area right P(Z > z) | Share below |
|---|---|---|---|
| −3 | 0.0013 | 0.9987 | 0.13% |
| −2.5 | 0.0062 | 0.9938 | 0.62% |
| −2 | 0.0228 | 0.9772 | 2.28% |
| −1.5 | 0.0668 | 0.9332 | 6.68% |
| −1 | 0.1587 | 0.8413 | 15.87% |
| −0.5 | 0.3085 | 0.6915 | 30.85% |
| 0 | 0.5 | 0.5 | 50% |
| 0.5 | 0.6915 | 0.3085 | 69.15% |
| 1 | 0.8413 | 0.1587 | 84.13% |
| 1.5 | 0.9332 | 0.0668 | 93.32% |
| 1.96 | 0.975 | 0.025 | 97.5% |
| 2 | 0.9772 | 0.0228 | 97.72% |
| 2.5 | 0.9938 | 0.0062 | 99.38% |
| 3 | 0.9987 | 0.0013 | 99.87% |
Percentile to z-score
Going the other way, the z-score for a percentile is the point with that much area to its left. The 90th percentile is z = 1.2816, the 95th is 1.6449 and the 97.5th is 1.96, the 1.96 used in 95% confidence intervals. Add a mean and SD in the calculator and it also returns the raw value: x = μ + z × σ.
| Percentile | z-score |
|---|---|
| 50th | 0 |
| 75th | 0.6745 |
| 80th | 0.8416 |
| 90th | 1.2816 |
| 95th | 1.6449 |
| 97.5th | 1.96 |
| 99th | 2.3263 |
| 99.9th | 3.0902 |
Probability between two values
The probability between two values is the area left of the upper z minus the area left of the lower z. About 68% of a normal distribution lies within 1 standard deviation of the mean, 95% within 2 and 99.7% within 3 (exactly 68.27%, 95.45% and 99.73%).[1] Choose “Probability between two values” in the calculator, enter both values with the mean and SD, and the shaded curve shows the region.
Only for bell-shaped dataA z-score can be computed for any data, but turning it into a percentile assumes a normal distribution. For strongly skewed data (incomes, waiting times), use the data’s own percentile rank instead.
Common z-score mistakes
- Dividing by the variance. The denominator is the standard deviation, not its square.
- Reading the wrong tail. “Greater than” questions need the area to the right: 1 − P(Z < z).
- Using a z-score with a small sample and unknown σ when a test is involved. Use t instead: see the t-test calculator and the p-value calculator.
How we check this calculator
The normal areas come from the complementary error function, computed with the regularized incomplete gamma function, and the inverse uses a rational approximation refined by one Halley step. Tests check Φ(1.96) = 0.9750021048517795, Φ(−3) = 0.0013498980316301, the 97.5th and 99.5th percentiles (1.959963984540054 and 2.5758293035489) and a round trip at six percentiles, all against the NIST normal table and its published values.[3]
- 14 November 2026: page and calculator published.
More statistics calculators
- Statistics calculatorMean, median, mode, SD, quartiles and outliers
- Standard deviation calculatorSample and population SD, step by step
- P-value calculatorFrom z, t, chi-square or F
- Confidence interval calculatorMeans and proportions, t or z
- t-test calculatorOne-sample, Welch, pooled and paired
- Binomial distribution calculatorExactly, at most, at least x successes
Frequently asked questions
How do I calculate a z-score?
Subtract the mean from your value and divide by the standard deviation: z = (x − μ) ÷ σ. A score of 85 with mean 70 and SD 10 gives z = 1.5.
What percentile is a z-score of 1?
About the 84th percentile: 84.13% of a normal distribution lies below z = 1.
Can a z-score be negative?
Yes. A negative z-score means the value is below the mean. z = −2 is two standard deviations below the mean, around the 2.3rd percentile.
What is a good z-score?
It depends on the question. For a test score, a higher z is better; z = 2 is better than about 97.7% of a normal group. In quality control, values beyond ±3 are usually treated as unusual.
What z-score is the 95th percentile?
z = 1.6449. The 97.5th percentile is z = 1.96, which is why 95% two-sided intervals use 1.96.