Z-Score to Percentile Calculator
Enter a z-score to get its percentile: the share of a normal distribution at or below it. The calculator also gives the area to the right, the two-tailed p-value and the area between −z and z, with the area shaded on a normal curve.
How many standard deviations the value is from the mean
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Convert values to z-scores and estimate probability within a normal model.
What a z-score percentile tells you
A z-score says how many standard deviations a value lies from the mean, and its percentile says what share of a normal distribution falls at or below it. The calculator converts one into the other with the standard normal cumulative distribution function Φ, so a z-score of 1.96 is the 97.5th percentile: 97.5% of values are lower and 2.5% are higher. If you have a raw score instead, the z-score calculator turns it into z first.
z = (x − μ) / σ
Percentile = Φ(z) × 100%
Area to the right = P(Z ≥ z) = Φ(−z)
Two-tailed p-value = 2 × Φ(−|z|)
Area between −|z| and |z| = 2Φ(|z|) − 1
Z-score to percentile table
| Z-score | Percentile | Area to the right |
|---|---|---|
| −3 | 0.13% | 0.99865 |
| −2 | 2.28% | 0.97725 |
| −1.96 | 2.5% | 0.975002 |
| −1.645 | 5% | 0.950015 |
| −1 | 15.87% | 0.841345 |
| −0.5 | 30.85% | 0.691462 |
| 0 | 50% | 0.5 |
| 0.5 | 69.15% | 0.308538 |
| 1 | 84.13% | 0.158655 |
| 1.28 | 89.97% | 0.100273 |
| 1.645 | 95% | 0.049985 |
| 1.96 | 97.5% | 0.024998 |
| 2 | 97.72% | 0.02275 |
| 2.326 | 99% | 0.010009 |
| 2.576 | 99.5% | 0.004998 |
| 3 | 99.87% | 0.00135 |
Printed z tables usually stop at a z-score between 3.5 and 4. This calculator accepts any z-score from −37 to 37 and writes very small areas in scientific notation instead of rounding them to zero. To read a table by hand see the z table and the guide on how to read a z-table.
Reading the answer
- Percentile: the share of values at or below the score. A z-score above 0 is above the mean and gives a percentile above 50%.
- Area to the right: the one-tailed p-value for a right-tailed z test; use the area to the left for a left-tailed test. See the z test calculator and the p-value calculator.
- Two-tailed p-value: the chance of a z-score at least this far from 0 in either direction, which is the p-value of a two-sided z test.
- Area between −|z| and |z|: the coverage of the symmetric interval; for z = 1.96 it is 95%, the basis of the 95% confidence interval and the empirical rule.
Worked example
An exam has a mean of 75 and a standard deviation of 5, so a score of 85 has z = (85 − 75)/5 = 2. Its percentile is 97.72%: 97.72% of scores are lower and 2.28% are higher. Load example uses z = 1.96 instead, the value behind the familiar 95% interval: the percentile is 97.5%, the area to the right is 0.024998, the two-tailed p-value is 0.049996 (about 0.05) and the area between −1.96 and 1.96 is 0.950004. To go the other way, from a percentile back to z, use the percentile to z-score calculator.
Software equivalents
| Software | Percentile (area to the left) | Area to the right | Two-tailed p-value |
|---|---|---|---|
| Excel / Sheets | NORM.S.DIST(z, TRUE) | NORM.S.DIST(-z, TRUE) | 2*NORM.S.DIST(-ABS(z), TRUE) |
| R | pnorm(z) | pnorm(z, lower.tail = FALSE) | 2 * pnorm(-abs(z)) |
| Python (SciPy) | scipy.stats.norm.cdf(z) | scipy.stats.norm.sf(z) | 2 * scipy.stats.norm.sf(abs(z)) |
| TI-84 | normalcdf(-1E99, z) | normalcdf(z, 1E99) | 2*normalcdf(abs(z), 1E99) |
The TI-84 function is explained on the normalcdf page. For a normal distribution with any mean and standard deviation use the normal distribution calculator.
Frequently Asked Questions
How do I convert a z-score to a percentile?
Find the area under the standard normal curve to the left of the z-score, Φ(z), and multiply it by 100. In Excel that is =NORM.S.DIST(z, TRUE)*100, in R pnorm(z)*100, and in Python scipy.stats.norm.cdf(z)*100. A z-table gives the same area to four digits.
What percentile is a z-score of 1.96?
A z-score of 1.96 is the 97.5th percentile. The area to its right is 2.5%, so the two-tailed p-value is 5% and 95% of the distribution lies between −1.96 and 1.96.
What percentile are z-scores of 0, 1, −1 and 2?
A z-score of 0 is the 50th percentile, 1 is the 84.13th, −1 is the 15.87th and 2 is the 97.72nd. These follow from the areas under the normal curve and match the 68-95-99.7 rule.
What is the difference between the percentile and the two-tailed p-value?
The percentile counts only the area to the left of z. The two-tailed p-value counts the area beyond |z| on both sides and is used for two-sided hypothesis tests, so it is twice the smaller of the two one-sided areas.
Can I use a z-score percentile for data that is not normal?
The percentile assumes the values follow a normal distribution. For skewed or discrete data the true percentile can differ a lot, and it is better to rank the actual data with the percentile calculator instead.
Why does the percentile show as 100% or 0% for large z-scores?
The percentile is rounded to two decimals, so a z-score such as 6 reads 100%. The area to the right, which the calculator computes from the upper tail, shows the tiny remainder exactly, for example 9.8659e-10 for z = 6.
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