P-Value Calculator

About the P-Value Calculator

A p-value is the probability of getting a result at least as extreme as the one observed, assuming the null hypothesis — usually "no effect" or "no difference" — is true. A small p-value means the data would be surprising if there were really no effect, which counts as evidence against the null hypothesis. P-values appear in almost every scientific paper, clinical trial and A/B test report, and they are among the most misunderstood numbers in statistics.

This p-value calculator converts a test statistic into a p-value. It handles z scores from the standard normal distribution, t statistics with their degrees of freedom, and chi-square statistics with their degrees of freedom. For z and t you can choose a two-tailed, left-tailed or right-tailed test. The calculator compares the p-value with your chosen significance level and shows at a glance whether it is significant at the 10, 5, 1 and 0.1 percent levels.

How to Use the P-Value Calculator

Choose the kind of statistic: z, t or chi-square.

Enter the statistic and, for t or chi-square, the degrees of freedom.

For z and t, choose whether the test is two-tailed, left-tailed or right-tailed.

Choose the significance level, α — usually 0.05.

The Formulas

  two-tailed z:     p = 2 × P(Z > |z|)
  right-tailed z:   p = P(Z > z)
  left-tailed z:    p = P(Z < z)
  t:                the same, using the t distribution with df degrees of freedom
  chi-square:       p = P(χ² ≥ x)       (right tail only)

Step-by-Step Examples

A z score of 1.96, two-tailed.

  P(Z > 1.96) = 0.025
  p = 2 × 0.025 = 0.05

A z score of 2.5, right-tailed.

  p = P(Z > 2.5) = 0.0062

A t statistic of 2.1 with 15 degrees of freedom, two-tailed.

A chi-square of 3.84 with 1 degree of freedom.

  p = P(χ²₁ ≥ 3.84) = 0.050

The z of 1.96 and the chi-square of 3.84 sit exactly on the 5 percent boundary. The t of 2.1 just misses it, even though a z of 2.1 would give p = 0.036 — the t distribution's heavier tails demand stronger evidence from small samples.

Critical Values

The p-value approach and the critical-value approach always agree. A two-tailed z test is significant at 5 percent when |z| exceeds 1.96, at 1 percent when it exceeds 2.576, and at 0.1 percent when it exceeds 3.291. For t, the critical values are larger with few degrees of freedom — 2.571 with 5, 2.131 with 15, 2.042 with 30 — and approach 1.96 as the sample grows. For chi-square with 1 degree of freedom, the 5 percent critical value is 3.841, which is 1.96 squared.

One-Tailed or Two-Tailed?

A two-tailed test asks whether there is a difference in either direction; a one-tailed test asks only about one direction. The one-tailed p-value is half the two-tailed value when the effect goes the predicted way. That makes one-tailed tests tempting, but they are only legitimate when the direction was specified before the data was collected and an effect in the opposite direction would be treated the same as no effect. Choosing the tail after seeing the data doubles the real false-positive rate. When in doubt, use two tails.

What a P-Value Is Not

A p-value of 0.03 does not mean there is a 3 percent chance the null hypothesis is true, nor a 97 percent chance the effect is real. It does not measure the size or importance of an effect: with a huge sample, a trivially small difference can have a tiny p-value, and with a small sample, an important effect can miss significance. And p = 0.051 is not meaningfully different from p = 0.049. The American Statistical Association has warned against treating 0.05 as a bright line and recommends reporting effect sizes and confidence intervals alongside p-values.

Multiple Testing

If you run 20 independent tests at the 5 percent level when no real effects exist, you should expect about one "significant" result by chance alone. Testing many outcomes, subgroups or variables and reporting only the significant ones is a common way to produce misleading findings. Corrections such as Bonferroni's — dividing α by the number of tests — keep the overall false-positive rate under control.

Understanding Your Result

The headline is the p-value. Values below 0.0001 are shown as "< 0.0001".

The test line restates the statistic, degrees of freedom and tail.

The decision line compares p with your chosen α.

The significant at line shows the result at four common levels.

The worth knowing line restates what a p-value means.

When Should You Use This Calculator?

Use it to turn a z, t or chi-square statistic from a textbook or paper into a p-value.

Use it to check the output of statistical software.

Use it to find whether a result is significant at a chosen level.

Use it for statistics homework on hypothesis testing.

Common Mistakes

Reading p as the probability the null hypothesis is true. It is not.

Switching to a one-tailed test after seeing the data. Decide in advance.

Using z when the sample is small and σ is estimated. Use t.

Treating p = 0.05 as a cliff edge. Evidence changes gradually.

Ignoring effect size. A significant result may be too small to matter.

Running many tests without correction. With 20 tests, the chance of at least one false positive is about 64 percent.