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Z-score to p-value calculator

Enter a Z-score to get its exact p-value from the standard normal distribution. Works for one-tailed and two-tailed tests, and stays precise far into the tail where most calculators return zero.

Hypothesis direction

Testing for a difference in either direction

Results update as you type — there is no submit button.

Enter a test statistic to begin

Your p-value, a plain-English reading, and a shaded distribution chart appear here instantly.

Where this statistic comes from

You will typically be holding one of these:

  • A one-sample or two-sample Z-test
  • A two-proportion test (A/B testing, conversion rates)
  • Standardised regression coefficients in large samples
  • Any value you have standardised as (x − μ) / σ

When a Z-test is the right choice

A Z-test assumes you know the population standard deviation, or that your sample is large enough for the distinction not to matter. In practice that means one of two situations:

  • You are testing proportions. Conversion rates, click-through rates, pass/fail counts. The standard error comes from the proportion itself, so there is nothing left to estimate.
  • Your sample is large. Above roughly n = 30, the t-distribution is close enough to normal that the two tests agree to three decimal places.

If your sample is small and you estimated the standard deviation from the data, use the t-score calculator instead. Using Z there produces a p-value that is too small, which is the error that matters — it makes results look more significant than they are.

One-tailed and two-tailed Z-tests

The standard normal distribution is symmetric, so a two-tailed p-value is exactly twice the one-tailed value. A Z-score of 1.96 gives a one-tailed p of 0.025 and a two-tailed p of 0.05 — which is where the familiar 1.96 comes from.

The sign of your Z-score does not change a two-tailed p-value: −2.5 and +2.5 are equally extreme. It does matter for a one-tailed test, where a Z-score pointing the wrong way gives a p-value above 0.5. If you see that, you have almost certainly picked the wrong tail direction.

Why extreme Z-scores break other calculators

Past about Z = 8, the lower tail of the normal distribution is so close to 1 that double-precision arithmetic cannot tell it apart. A calculator that computes the upper tail as 1 − CDF(z) therefore returns exactly zero — and reports p = 0, which is never a true p-value.

This calculator evaluates the upper tail directly through the complementary error function, so Z = 8 correctly returns 6.22 × 10⁻¹⁶ and Z = 10 keeps going. See the methodology page for the numerical detail.

Frequently asked questions

How do you find the p-value from a Z-score?

The p-value is the area of the standard normal distribution beyond your Z-score — one tail for a one-tailed test, both tails for a two-tailed test. There is nothing else to supply: unlike t, chi-square and F, the normal distribution has no degrees of freedom. Enter the Z above and the exact tail area is computed directly, so it stays accurate far past the point where subtracting from 1 would round to zero.

What p-value does a Z-score of 1.96 give?

A Z-score of 1.96 gives a two-tailed p-value of 0.0500 and a one-tailed p-value of 0.0250. This is the reason 1.96 is the critical value quoted for a 95% confidence level: it marks the point where exactly 5% of the normal distribution lies in the two tails combined.

Can a Z-score be negative?

Yes. A negative Z-score simply means your value falls below the mean. For a two-tailed test the sign is irrelevant, since −2.1 and +2.1 are equally far from the centre. For a one-tailed test the sign decides which tail you are testing, so it matters a great deal.

What is the difference between a Z-score and a p-value?

A Z-score measures distance: how many standard deviations your result sits from the mean. A p-value converts that distance into a probability: how often a result at least that far out would occur if the null hypothesis were true. The Z-score is the input; the p-value is the output.

Do I need degrees of freedom for a Z-test?

No. The standard normal distribution has no parameters to estimate, so there are no degrees of freedom. If you were asked for degrees of freedom, you are probably running a t-test, chi-square test, or F-test instead.