Z-Score Calculator with P-Value and Percentile
Compute z = (x − μ) / σ, then read the left-tail percentile, right-tail and two-tailed p-values. Edit z directly or enter a probability to work backwards to x.
Z-score formula
Φ is the cumulative distribution function (CDF) of the standard normal distribution. This calculator evaluates Φ with a double-precision rational approximation (accurate to about 1×10⁻¹⁴ across the range, including far tails such as z = −8) and inverts it with Acklam's algorithm plus a Halley refinement step, so reverse lookups are accurate to roughly nine significant digits.
Worked example
An exam has mean μ = 70 and standard deviation σ = 10. A student scores x = 85. The z-score is z = (85 − 70) / 10 = 1.5. Φ(1.5) = 0.93319, so the student is at the 93.3rd percentile: about 6.7% of scores are higher (right-tail p = 0.0668). The two-tailed p-value — the probability of a result at least 1.5 standard deviations from the mean in either direction — is 0.1336.
Going the other way: what score marks the top 5%? The left-tail probability is 0.95, so z = Φ⁻¹(0.95) = 1.645 and x = 70 + 1.645 × 10 = 86.4.
Z table (standard normal)
| z | Left tail P(Z| Right tail | Two-tailed p | |
|---|---|---|---|
| 0.00 | 0.50000 | 0.50000 | 1.00000 |
| 0.50 | 0.69146 | 0.30854 | 0.61708 |
| 1.00 | 0.84134 | 0.15866 | 0.31731 |
| 1.28 | 0.90008 | 0.09992 | 0.19984 |
| 1.50 | 0.93319 | 0.06681 | 0.13361 |
| 1.645 | 0.95002 | 0.04998 | 0.09997 |
| 1.96 | 0.97500 | 0.02500 | 0.05000 |
| 2.00 | 0.97725 | 0.02275 | 0.04550 |
| 2.33 | 0.98999 | 0.01001 | 0.02002 |
| 2.576 | 0.99500 | 0.00500 | 0.01000 |
| 3.00 | 0.99865 | 0.00135 | 0.00270 |
| 3.29 | 0.99950 | 0.00050 | 0.00100 |
One-tailed vs two-tailed p-values
Use a one-tailed test when your hypothesis has a direction ("the new process is faster"). Use a two-tailed test when any difference counts ("the process is different"). The classic critical values are z = 1.645 (one-tailed, α = 0.05), z = 1.96 (two-tailed, α = 0.05) and z = 2.576 (two-tailed, α = 0.01). For z = 1.96, the two-tailed p-value is 0.0500.
When is a z-score appropriate?
Z-scores assume the data are approximately normal and that σ is known (or the sample is large). With small samples and an estimated standard deviation, a t-distribution gives more accurate p-values. Z-scores are also used without normality simply to standardise values onto a common scale, but then the percentile interpretation does not hold.
Frequently asked questions
What does a z-score of 0 mean?
The value equals the mean. Half of a normal distribution lies below it, so it is the 50th percentile.
What is the p-value for z = 1.96?
Two-tailed p = 0.0500 and one-tailed (right) p = 0.0250. That is why 1.96 is the critical value for a 95% confidence interval.
Can a z-score be negative?
Yes. A negative z means the value is below the mean. z = −1 corresponds to the 15.87th percentile.
How do I convert a percentile to a z-score?
Enter the percentile as a probability (e.g. 0.90 for the 90th percentile) in the probability box. The calculator returns z = 1.2816 and the matching x.
How accurate is this normal CDF?
It uses a double-precision rational approximation with absolute error around 1e-14, far more precise than printed z tables, which give four or five decimals.