Math
Z-score
Standardize an observation using a population mean and standard deviation.
Calculators / Math / Two-sided z-test p-value
Two-sided z-test p-value calculator
Calculate a two-sided standard-normal tail probability from a z statistic.
On this page
Absolute z1.96
Two-sided p-value0.05probability
Two-sided tail area5%
Calculator controls
How to calculate
Calculate a two-sided standard-normal tail probability from a z statistic.
p = 2 × integral from |z| to infinity of exp(−t²/2)/√(2π) dt. Computed with composite Simpson integration to 12, using 4096 intervals; supported |z| ≤ 10.
Worked example
Z statistic: 1.96.
Absolute z: 1.96 ; Two-sided p-value: 0.05 probability; Two-sided tail area: 5 %.
Assumptions and limits
A p-value is a tail probability under the specified null model, not the probability that a hypothesis is true. This assumes a valid standard-normal z statistic and a prespecified two-sided test. It does not replace a t test for an estimated small-sample standard deviation. Values are approximate, not exactly zero.
Results are rounded for display; calculations use unrounded values. Read our calculation methodology.
Understanding the result
The result measures how much standard-normal probability lies at least as far from zero as the entered z statistic, in either direction.
When this tool is useful
Use only after defining a valid two-sided z test and its assumptions. Statistical significance is not an effect size or practical importance.
Understanding your inputs
| Input | What to enter |
|---|---|
| Z statistic | Enter a number of at least -10 and no more than 10. |
Frequently asked questions
Is p the chance that the null hypothesis is true?
No. It is a probability of data this extreme or more extreme under the stated null model, not a probability assigned to the hypothesis.