P-value CalculatorFree online math calculator — instant results, no sign-up
P-value from a z-score.
About the P-value Calculator
This calculator computes the one-tailed and two-tailed p-values associated with a standard normal test statistic (z-score). It is commonly used by students, researchers, and data analysts performing z-tests for hypothesis testing. It also displays the standard critical z-value for a two-tailed 95% confidence level.
- Z-score
- The standardized test statistic measuring the number of standard deviations an observation is from the mean; accepts any positive or negative real number.
Formula and method
One-tailed p = 1 − Φ(|z|) Two-tailed p = 2 × (1 − Φ(|z|)) Critical z (95%) = Φ⁻¹(0.975)
- zStandard normal test statistic (z-score)
- |z|Absolute value of the input z-score
- Φ(x)Cumulative distribution function (CDF) of the standard normal distribution evaluated at x
- Φ⁻¹(p)Inverse cumulative distribution function (quantile function) of the standard normal distribution
The tool takes the absolute value of the provided z-score and computes the upper-tail probability using the standard normal cumulative distribution function Φ(x). The one-tailed p-value is 1 − Φ(|z|) and the two-tailed p-value is 2 × (1 − Φ(|z|)), both rounded to 6 decimal places. The critical z-value is fixed at the two-tailed 95% threshold (0.975 quantile) and rounded to 3 decimal places.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Z-score1.96
Results
- Two-tailed p0.049996
- One-tailed p0.024998
- Critical z (95%)1.96
How to use the P-value Calculator
- 1Convert the input z-score to its absolute value |z|.
- 2Evaluate the standard normal CDF, Φ(|z|), to determine the probability to the left of |z|.
- 3Subtract Φ(|z|) from 1 to calculate the one-tailed p-value (upper tail).
- 4Multiply the one-tailed p-value by 2 to obtain the two-tailed p-value, and output the 95% critical value.
Reading your result
A smaller p-value provides stronger evidence against the null hypothesis; if the two-tailed p-value is less than your significance level (typically α = 0.05), you reject the null hypothesis. The one-tailed p represents the probability in a single extreme tail, whereas the two-tailed p accounts for extreme deviations in either direction. The critical z-value of 1.96 indicates the threshold required to achieve significance at α = 0.05 in a two-tailed test.
Limitations
This calculator assumes a standard normal distribution (Z-distribution) and does not adjust for small sample sizes requiring a Student's t-distribution with degrees of freedom. It always calculates the tail probability based on |z|, assuming the test is testing against the extreme tail, rather than calculating a specific directional lower tail (Φ(z)) for negative scores.
Common questions
Why does entering a negative z-score give the same p-value as a positive one?+
The calculator evaluates the absolute value of the z-score (|z|). In standard two-tailed hypothesis testing, deviation magnitude matters equally in both directions, and for a one-tailed test it returns the upper tail probability corresponding to that magnitude.
When should I use the two-tailed p-value versus the one-tailed p-value?+
Use the two-tailed p-value when your alternative hypothesis tests for any difference (either greater than or less than the null value). Use the one-tailed p-value only if your directional hypothesis was specified before analyzing data (e.g., strictly greater).
When should I use a t-distribution instead of a z-score?+
Use a t-test instead of a z-test when the population standard deviation is unknown and the sample size is small (typically n < 30). For large sample sizes, the t-distribution closely approximates the normal z-distribution.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
Further reading
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Results are estimates for informational purposes and are not financial, medical or legal advice.