How this instrument works
A p-value measures how surprising an observed test statistic would be if the null hypothesis were actually true — smaller p-values mean the observed result would be rarer under the null hypothesis, usually taken as evidence against it. This calculator is scoped specifically to the p-value from a z-test statistic: given a z-score, it returns the left-tailed p-value P(Z ≤ z), the right-tailed p-value P(Z ≥ z), and the two-tailed p-value, which doubles whichever tail is smaller to account for testing in either direction.
Which of the three to use depends on your hypothesis. A one-tailed, left or right, test is appropriate when you have a specific directional prediction going in — 'the new method will increase the average' — and only cares about deviation in that one direction. A two-tailed test, the more common default, is appropriate when a result would count as significant regardless of which direction it deviates in, and it's stricter because it splits your significance threshold across both tails.
This calculator is deliberately scoped to a z-test statistic — it does not compute p-values from raw sample data or from a t-statistic. A t-based p-value calculation needs the t-distribution's cumulative distribution function, mathematically the incomplete beta function, a materially different and larger computation than the normal-distribution CDF used here, so it isn't offered on this page; use a dedicated t-test calculator for that case instead.
- Enter your z-test statistic into z.
- Read pLeft — the left-tailed p-value, P(Z ≤ z).
- Read pRight — the right-tailed p-value, P(Z ≥ z).
- Read pTwo — the two-tailed p-value, the figure most commonly compared against a significance level like 0.05.
Worked example — the famous z = 1.96 threshold
You have a z-test statistic of z = 1.96, perhaps from a hypothesis test comparing a sample result against an expected population value. Enter 1.96 into the field.
pLeft = Phi(1.96) ≈ 0.9750. pRight = 1 − 0.9750 = 0.0250. pTwo = 2 × 0.0250 = 0.0500 — the well-known result that a two-tailed z of 1.96 corresponds almost exactly to a p-value of 0.05, the most commonly used significance threshold in introductory statistics.
Questions
Should I use the one-tailed or two-tailed p-value?
Use a one-tailed p-value, pLeft or pRight, only if your hypothesis specifically predicted the direction of the effect before looking at the data — for example, testing whether a new process specifically increases output, not just changes it. Use the two-tailed p-value, pTwo, for the far more common situation where a result in either direction would count as evidence against the null hypothesis. Deciding which to use after seeing the data, rather than before, is a well-known way to inflate false-positive rates.
What does a p-value of 0.05 actually mean?
It means that if the null hypothesis were true, you'd expect to see a test statistic this extreme, or more extreme, about 5% of the time purely by chance. It does not mean there's a 5% probability the null hypothesis is true, and it does not measure the size or importance of an effect — a very small p-value can come from a tiny, practically meaningless effect if the sample size is large enough.
Why does this calculator only accept a z-statistic, not raw data or a t-statistic?
Computing a p-value from a t-statistic requires the t-distribution's cumulative distribution function, which involves the incomplete beta function — a materially more complex calculation than the normal distribution's CDF used here, and one this calculator doesn't yet implement. If your test statistic is a t-value rather than a z-value, common with small samples where the population standard deviation is unknown, a dedicated t-test p-value calculator is needed instead.
What's the difference between a p-value and a significance level (alpha)?
The significance level, alpha, is a threshold chosen before running the test, commonly 0.05, representing the false-positive rate you're willing to accept. The p-value is computed from your actual data after the test. You reject the null hypothesis when the p-value falls below alpha; the p-value measures the evidence in your specific sample, while alpha is a policy decision about how much risk of a false positive you're willing to tolerate.
Can the p-value be exactly 1.0?
Yes — at z = 0, meaning the observed test statistic exactly matches what the null hypothesis predicts, pLeft = pRight = 0.5 and pTwo = 2 × 0.5 = 1.0. This is the weakest possible evidence against the null hypothesis: a result that's exactly typical under the null hypothesis provides no reason at all to doubt it.