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P-Value from Z-Score Calculator

Convert a z-score into a one-tailed or two-tailed p-value.

Result

P-Value
0.05

A p-value below your chosen significance level (commonly 0.05) is typically considered statistically significant.

About the P-Value from Z-Score

This calculator converts a z-score into a p-value, supporting one-tailed and two-tailed hypothesis tests. It is meant for anyone who already has a z-statistic, from a z-test, a proportion comparison, or a normal approximation, and needs the corresponding probability of seeing a result this extreme or more extreme by chance.

How It Works

You enter the z-score and choose whether you want a two-tailed test, a right-tailed one-tailed test, or a left-tailed one-tailed test. The calculator runs your z-score through the standard normal cumulative distribution function and combines the result differently depending on which tail (or both tails) you selected.

Two-tailed p = 2 * (1 - normalCdf(|z|)); Right-tailed p = 1 - normalCdf(z); Left-tailed p = normalCdf(z)

Formula & Methodology

The normal CDF is computed with the Zelen and Severo polynomial approximation, accurate to about 0.0000075. For a two-tailed test, the calculator takes the absolute value of z so the result reflects the combined probability of an equally extreme result in either direction, then doubles the tail area. For a one-tailed right test, it reports only the area beyond z in the positive direction; for a one-tailed left test, it reports the area below z directly, without taking an absolute value first, so a negative z on the left tail correctly returns a large probability and a positive z returns a small one.

Examples

Standard Two-Tailed Significance Check

A z-score of 1.96 tested two-tailed returns a p-value of almost exactly 0.05, which is why 1.96 is the conventional critical value for the common 5% significance threshold.

One-Tailed Right Test

The same z-score of 1.96 tested as one-tailed (right) returns a p-value of about 0.025, exactly half the two-tailed result, reflecting that only the upper tail of the distribution is being considered.

Advantages

  • Supports all three common tail configurations (two-tailed, one-tailed right, one-tailed left) from a single z-score input.
  • Uses a highly accurate normal CDF approximation, so results align closely with standard statistical software.
  • Returns an instant answer without needing a printed z-table or manual lookup.

Common Mistakes

  • Picking a one-tailed test to make a borderline result look significant, rather than deciding the tail direction before collecting data.
  • Confusing a right-tailed and left-tailed selection, which reports the wrong half of the distribution for a directional hypothesis.
  • Treating the returned p-value as the probability that the null hypothesis is true, when it actually measures the probability of the observed (or more extreme) result assuming the null hypothesis is true.

Edge Cases to Watch For

  • The result is capped at 1, so no tail selection can return a probability above 100%.
  • Choosing the tail direction after seeing the data, rather than deciding beforehand, will produce a valid-looking p-value that misrepresents the actual test being run.
  • A z-score of exactly 0 returns a p-value of 1 for the two-tailed test and 0.5 for either one-tailed test, correctly reflecting no deviation from the null hypothesis.

Common Use Cases

  • Students and researchers converting a z-statistic from other software output into a reportable p-value.
  • Analysts running quick significance checks on proportion or large-sample mean comparisons.
  • Anyone verifying by hand that a computed z-score crosses a standard significance threshold like 0.05 or 0.01.
Written & fact-checked by the Calculateus TeamLast updated August 5, 2026How we verify our formulas

Frequently asked questions

How do I choose between a one-tailed and two-tailed test?

Use a two-tailed test when you care about a difference in either direction (the far more common case). Use a one-tailed test only when you have a specific directional hypothesis decided BEFORE seeing the data (for example, testing only whether a value is higher, never lower) - choosing the tail after seeing results inflates your false-positive rate.

Related calculators

Confidence Interval CalculatorCalculate the confidence interval for a sample mean.Sample Size CalculatorCalculate the minimum sample size needed for a survey or study at a given confidence level and margin of error.Margin of Error CalculatorCalculate the margin of error for a survey result given sample size and confidence level.Correlation Coefficient CalculatorCalculate the Pearson correlation coefficient between two paired sets of numbers.

Conclusion

This calculator handles the mechanical step of turning a z-score into a p-value across all three common tail configurations, saving a table lookup. As always, the choice of tail direction should be settled before the data is examined, not after.