P-Value Calculator

The p-value is the probability of observing results at least as extreme as your data, assuming the null hypothesis is true. It is the central output of statistical hypothesis testing and determines whether findings are considered statistically significant.

Conceptual formula: p-value = P(observing data this extreme | H₀ is true)

For a z-test (large samples, known standard deviation): z = (Sample Mean − Population Mean) ÷ (Standard Deviation ÷ √n) The p-value is then read from the standard normal distribution table for that z-score.

For a t-test (small samples, unknown population SD): t = (x̄ − μ₀) ÷ (s ÷ √n) where s = sample standard deviation, n = sample size, and degrees of freedom = n − 1.

What each variable means:

• H₀ (Null Hypothesis): the default assumption (e.g., “this drug has no effect”).
• H₁ (Alternative Hypothesis): what you are trying to prove (e.g., “this drug reduces blood pressure”).
• α (Alpha): your significance threshold, set before the experiment. Common values: 0.05 (5%) or 0.01 (1%).
• Degrees of Freedom: related to sample size; affects the shape of the t-distribution.

Interpreting p-values:

• p < 0.05: statistically significant at the 5% level. Reject H₀.
• p < 0.01: highly significant. Strong evidence against H₀.
• p > 0.05: fail to reject H₀ (does not prove H₀ is true, just insufficient evidence).
• p = 0.001: there is only a 0.1% chance of seeing this result if H₀ were true.

Worked example: A sample of 36 students averages 78 on a test. The population mean is 75, with SD = 12. z = (78 − 75) ÷ (12 ÷ √36) = 3 ÷ 2 = 1.5 p-value (two-tailed) ≈ 0.134 Since 0.134 > 0.05, we fail to reject H₀ — the difference is not statistically significant.

Critical misconception: p < 0.05 does NOT mean the effect is large or practically important. A study with 1,000,000 participants might find p = 0.001 for a trivially small difference. Always pair p-values with effect size (Cohen’s d, η², etc.).