Z-Score / Normal Distribution Calculator
Convert between z-scores, percentiles, and areas under the standard normal curve — with a shaded diagram of exactly which region you're looking at.
Last updated
Z-score ↔ percentile
Percentile here means the percentage of the distribution that falls at or below that z-score. Edit either field — the other updates automatically.
Area under the curve
Quick reference: common critical values
| Confidence level | α (two-tailed) | One-tailed z* | α (one-tailed) | Two-tailed z* |
|---|---|---|---|---|
| 90% | 0.10 | 1.282 | 0.05 | 1.645 |
| 95% | 0.05 | 1.645 | 0.025 | 1.960 |
| 99% | 0.01 | 2.326 | 0.005 | 2.576 |
"One-tailed z*" is the z-score with that much probability in a single tail (used for one-sided tests / upper or lower confidence bounds). "Two-tailed z*" is the z-score used when the same total probability α is split evenly between both tails (the usual confidence-interval and two-sided-test value) — e.g. the familiar 1.96 for a 95% confidence interval.
How to use the z-score calculator
- Type a z-score to see its percentile, or a percentile to see its z-score.
- For areas, choose left tail, right tail, between or outside two z-scores, and enter the values.
- Switch to I have an area / confidence level to find critical values.
About this tool
A z-score measures how many standard deviations a value sits from the mean of a standard normal distribution (mean 0, standard deviation 1). The top converter turns a z-score directly into its percentile — the proportion of the distribution lying at or below it — using the exact relationship Percentile = Φ(z) × 100, where Φ is the standard normal cumulative distribution function.
The area calculator handles the four region shapes you'll actually need: a single left or right tail (one-tailed queries) or a region between/outside two z-scores (two-tailed queries). Switch "I have z-score(s) → find the area" to work forward from a z-score to a probability, or flip to "I have an area / confidence level → find z-score(s)" to work backward — this is how critical values for confidence intervals and hypothesis tests are derived: give it a confidence level like 95% and it solves for the ±z boundary that captures exactly that much probability symmetrically around the mean.
Under the hood, Φ(z) is computed with a high-accuracy rational approximation to the error function (max error around 1.5×10⁻⁷), and its inverse uses Peter Acklam's rational approximation for the normal quantile function, refined with one step of Newton's method — accurate to roughly 9 decimal places, well beyond what any real-world stats problem needs.
Everything runs as JavaScript in your browser tab — nothing you enter is sent anywhere.
Questions about z-scores
What is a z-score?
How many standard deviations a value is above or below the mean. A z-score of 2 means two standard deviations above.
How do I calculate a z-score?
Subtract the mean and divide by the standard deviation. A score of 130 with mean 100 and standard deviation 15 has z = 2.
What z-score is the 95th percentile?
About 1.645. For a two-tailed 95% confidence interval the critical value is 1.96.
What percentage is within one standard deviation?
About 68%. About 95% is within two and 99.7% within three, for a normal distribution.