Percentile to Z-Score Calculator
Enter a percentile to get the z-score that has that share of a normal distribution below it. You can also enter a top percent or a middle percent, and add a mean and standard deviation to get the score itself.
For example 90 for the 90th percentile
With the standard deviation, gives the raw score
Must be positive
Related Calculators
Z-Score to Percentile Calculator
Convert a z-score to a percentile with the areas to the left and right, the two-tailed p-value and a shaded normal curve.
invNorm Calculator
Find the value or z-score for a given area with invNorm(area, μ, σ) — left, right, or center tail.
Critical Value Calculator
Find Z, t, chi-square, and F critical values for any significance level and tail choice.
What the calculator does
A percentile says what share of a distribution lies at or below a score. Given that share, the calculator finds the z-score with exactly that share of the standard normal distribution below it, which is the inverse of the cumulative distribution function Φ. The same z-score can be read from the top (a right-tail percentage) or for the middle of the distribution, which is how two-sided critical values such as 1.96 arise. It reverses the z-score to percentile calculator.
Percentile p: z = Φ⁻¹(p / 100)
Top q percent: z = Φ⁻¹(1 − q / 100)
Middle c percent: z = Φ⁻¹(1 − (1 − c / 100) / 2), with the interval from −z to z
Score: x = μ + zσ
Percentile to z-score table
| Percentile | Z-score |
|---|---|
| 50th | 0 |
| 75th | 0.6745 |
| 80th | 0.8416 |
| 85th | 1.0364 |
| 90th | 1.2816 |
| 95th | 1.6449 |
| 97.5th | 1.96 |
| 99th | 2.3263 |
| 99.5th | 2.5758 |
| 99.9th | 3.0902 |
For percentiles below the 50th the z-score is the negative of the one at 100 minus the percentile, for example −1.2816 at the 10th percentile.
| Middle percent (confidence level) | Z-score of each limit |
|---|---|
| 68.27% | 1 |
| 80% | 1.2816 |
| 90% | 1.6449 |
| 95% | 1.96 |
| 98% | 2.3263 |
| 99% | 2.5758 |
| 99.9% | 3.2905 |
The middle-percent values are the critical values of confidence intervals; the critical value calculator and the confidence interval calculator use them.
Worked example
A test is scored with a mean of 500 and a standard deviation of 100. The 90th percentile has z = 1.281552, so the score needed is 500 + 1.281552 × 100 = 628.1552. The middle 95% of scores lie between 500 − 1.959964 × 100 = 304.0036 and 500 + 1.959964 × 100 = 695.9964. Load example fills in the first case; switch the selector to the middle percent and enter 95 for the second.
Software equivalents
| Software | z-score for a percentile p | Score for mean μ and standard deviation σ |
|---|---|---|
| Excel / Sheets | NORM.S.INV(p) | NORM.INV(p, μ, σ) |
| R | qnorm(p) | qnorm(p, mean = μ, sd = σ) |
| Python (SciPy) | scipy.stats.norm.ppf(p) | scipy.stats.norm.ppf(p, loc=μ, scale=σ) |
| TI-84 | invNorm(p) | invNorm(p, μ, σ) |
Here p is a fraction between 0 and 1, so 0.9 for the 90th percentile. The TI-84 function is explained on the invNorm page, and the areas behind these values are in the z table. The percentile calculator ranks a data set instead of assuming a normal distribution.
Frequently Asked Questions
How do I convert a percentile to a z-score?
Find the z-score that has that share of the standard normal distribution below it, which is the inverse normal function Φ⁻¹(p). In Excel use =NORM.S.INV(p) with p as a fraction, in R qnorm(p), and in Python scipy.stats.norm.ppf(p). The 90th percentile gives 1.281552.
What z-score is the 95th percentile?
The 95th percentile has a z-score of 1.6449. Do not confuse it with 1.96, which is the 97.5th percentile: it leaves 2.5% in each tail, so it bounds the middle 95%.
Why is 1.96 the critical value for 95% but the 97.5th percentile?
A 95% two-sided interval leaves 5% outside, split as 2.5% in each tail. The upper limit therefore has 97.5% of the distribution below it, so it sits at the 97.5th percentile. Choose the middle percent option and enter 95 to see the interval from −1.96 to 1.96.
How do I find the z-score for the top 10 percent?
Choose the top percent option and enter 10. The z-score has 10% of the distribution at or above it, which is the same as the 90th percentile, 1.281552.
How do I get the actual score for a percentile?
Multiply the z-score by the standard deviation and add the mean: x = μ + zσ. Enter the mean and standard deviation in the optional fields and the calculator does it for you; leave both blank if you only want the z-score.
Can I use this for data that is not normally distributed?
Only if the data are close to normal. The z-score comes from the normal distribution, so for skewed data the score at a given percentile can be far off. Rank the actual values with the percentile calculator in that case.
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