P-value to Z-score Calculator

What this calculator does

A p-value says how surprising a result is under a null hypothesis. A Z-score says the same thing in a different currency: how many standard deviations the result sits from the centre of a standard normal distribution. The two carry the same information, and for a test statistic that follows a normal curve you can move freely from one to the other.

This tool goes from the p-value to the Z-score, which is the inverse of what a table or a Z-score to p-value calculator does. It is handy for finding the critical Z behind a significance level, for reconstructing a test statistic from a published p-value, and for meta-analysis, where results are often combined on the Z scale.

How to use it

1. Type the p-value in P-value as a decimal, for example 0.0231 (not 2.31%).
2. Choose One-tailed if the p-value measures only one side of the distribution, or Two-tailed if it counts extreme results in both directions.
3. Press Calculate Z-score. The result appears in the Z-score row, rounded to six decimals.

• The tool returns the Z for which the upper tail has area equal to your p-value. A p-value below 0.5 therefore gives a positive Z, a p-value of exactly 0.5 gives 0, and a one-tailed p-value above 0.5 gives a negative Z.
• For a two-tailed p-value the tool uses half of it for each tail, so the Z-score is always the positive cut-off point whose mirror image is −Z.
• A p-value of zero or less, or a one-tailed p-value of 1 or more, has no usable Z-score on the normal curve. The calculator shows 0 for those inputs.
• Exponent notation such as 1e-5 is not accepted; type 0.00001 instead.

How the conversion works

Let Φ be the cumulative distribution function of the standard normal distribution and Φ⁻¹ its inverse, the quantile function (also called the probit). For a one-tailed p-value in the upper tail:

Z = Φ⁻¹(1 − p)One-tailed p-value.

A two-tailed p-value splits the probability equally between the two tails, so each tail holds p ÷ 2:

Z = Φ⁻¹(1 − p ÷ 2)Two-tailed p-value; the other cut-off is −Z.

There is no closed form for Φ⁻¹, so software finds it numerically. This calculator takes a rational first estimate (Acklam's algorithm) and then applies one Halley correction step against the exact tail area, which leaves a result accurate to about 15 significant digits across the whole range, even for tiny p-values. The tail is measured directly from p rather than from 1 − p, so very small probabilities lose nothing to rounding.

P-value to Z-score conversion table

The familiar cut-offs are all here. A two-tailed test at the 5% level rejects when |Z| exceeds 1.96, and a one-tailed test at the same level needs Z above 1.6449. At 1% the two-tailed cut-off is 2.5758 and the one-tailed one is 2.3263.

Worked examples

A published one-tailed p-value. A paper reports p = 0.0231 for a one-sided test. Then Z = Φ⁻¹(1 − 0.0231) = 1.9936. The result is just under 2, consistent with a p-value a little below 0.025.

The same number, two-tailed. If that 0.0231 had been a two-tailed p-value, each tail would hold 0.01155, so Z = Φ⁻¹(1 − 0.01155) = 2.2718. The two-tailed Z is larger because the same total probability is split across two ends of the curve and each end is smaller.

A weak result. For p = 0.3 (one-tailed) the Z-score is 0.5244, barely half a standard deviation from the centre. A one-tailed p-value of 0.8 gives −0.8416: the observed result lies on the opposite side of the null from the direction being tested.

Related ideas

• To go the other way, from a Z-score to a p-value, use the Z-score to p-value calculator. For a full significance test of two proportions or means, use the p-value calculator.
• To explore the bell curve itself, see the normal distribution calculator, or turn a raw observation into a standardised score with the Z-score calculator.
• Critical values for other distributions are in the critical value calculator.

A Z-score is only the right yardstick if the test statistic is approximately normal. For small samples with an estimated variance, a t-distribution is the usual choice, and its p-values do not convert to the same Z-scores.