Calculate p-values from test statistics for hypothesis testing. Supports z-score, t-score, chi-square, and F-test with one-tailed and two-tailed options.
Standard normal distribution (large samples, n ≥ 30)
H₁: μ ≠ μ₀ (different from)
Enter your test statistic to calculate the p-value
Select test type (z, t, χ², F) and tail type
A p-value (probability value) is a statistical measure that helps you determine the strength of your evidence against the null hypothesis. It represents the probability of obtaining results at least as extreme as the observed results, assuming the null hypothesis is true.
The p-value is not the probability that the null hypothesis is true. It's the probability of seeing data as extreme as yours if the null hypothesis were true. This subtle distinction is crucial for proper statistical interpretation.
For data from a normal distribution with known population standard deviation (σ):
Left-tailed (H₁: μ < μ₀)
p = Φ(Z)
Right-tailed (H₁: μ > μ₀)
p = 1 − Φ(Z)
Two-tailed (H₁: μ ≠ μ₀)
p = 2 × Φ(−|Z|)
Where Φ(Z) is the cumulative distribution function (CDF) of the standard normal distribution
For small samples (n < 30) or unknown population standard deviation:
Left-tailed
p = cdft,df(t)
Right-tailed
p = 1 − cdft,df(t)
Two-tailed
p = 2 × cdft,df(−|t|)
Where df = n − 1 (degrees of freedom)
For goodness of fit tests and independence tests:
Right-tailed (typical)
p = 1 − CDFχ²(χ², df)
Chi-square tests are inherently one-tailed (right-tailed)
For ANOVA and comparing variances:
Right-tailed (typical)
p = 1 − CDFF(F, df₁, df₂)
Where df₁ = numerator degrees of freedom, df₂ = denominator degrees of freedom
| P-Value Range | Evidence Strength | Interpretation |
|---|---|---|
| p < 0.001 | Extremely Strong | Very convincing evidence to reject H₀ |
| 0.001 ≤ p < 0.01 | Very Strong | Strong evidence to reject H₀ |
| 0.01 ≤ p < 0.05 | Strong | Moderate evidence to reject H₀ (commonly used threshold) |
| 0.05 ≤ p < 0.10 | Weak | Marginal evidence; results are suggestive but not conclusive |
| p ≥ 0.10 | None/Minimal | No evidence to reject H₀; results are consistent with H₀ |
Tests if the parameter is less than a value.
H₀: μ ≥ μ₀
H₁: μ < μ₀
Example: Testing if a new drug lowers blood pressure
Tests if the parameter is greater than a value.
H₀: μ ≤ μ₀
H₁: μ > μ₀
Example: Testing if a new teaching method increases scores
Tests if the parameter is different from a value.
H₀: μ = μ₀
H₁: μ ≠ μ₀
Example: Testing if coin is fair (P(heads) ≠ 0.5)
A researcher tests whether the average height of a population differs from 170 cm. They calculate a z-score of 2.15 at α = 0.05.
p = 2 × Φ(−|2.15|)
p = 2 × Φ(−2.15)
p = 2 × 0.0158 = 0.0316
Result: Since p-value (0.0316) < α (0.05), we reject the null hypothesis. There is statistically significant evidence that the population mean differs from 170 cm.
A company tests whether a new training program increases productivity. With a sample of 25 employees, they calculate t = 1.85 at α = 0.05.
p = 1 − cdft,24(1.85)
p = 1 − 0.9617
p = 0.0383
Result: Since p-value (0.0383) < α (0.05), we reject the null hypothesis. The training program significantly increases productivity.
A researcher tests if dice rolls follow a uniform distribution. With df = 5 and χ² = 11.07 at α = 0.05.
p = 1 − CDFχ²(11.07, 5)
p = 1 − 0.9500
p = 0.0500
Result: Since p-value (0.05) = α (0.05), this is exactly at the boundary. By convention, we typically fail to reject H₀ when p = α, but this is a borderline case.
1% significance level
5% significance level
10% significance level
A p-value of 0.05 means there is a 5% probability of obtaining results at least as extreme as the observed results, assuming the null hypothesis is true. In other words, if H₀ were true, you would expect to see such extreme results only 5% of the time by random chance.
No. P-values represent probabilities and must be between 0 and 1 (inclusive). A p-value of 0 indicates extremely strong evidence against H₀, while a p-value of 1 indicates the observed result is exactly what H₀ predicts.
Statistical significance (p < α) only indicates that an effect is unlikely due to chance. Practical significance considers whether the effect size is meaningful in real-world terms. A very large sample can detect tiny differences that are statistically significant but practically meaningless.
Use a z-test when: (1) sample size is large (n ≥ 30), or (2) population standard deviation (σ) is known. Use a t-test when: (1) sample size is small (n < 30), and (2) population standard deviation is unknown (you use sample standard deviation s instead).
They are related! For a two-tailed test at significance level α: if p < α, the (1-α)% confidence interval will NOT contain the null hypothesis value. For example, if p < 0.05, the 95% confidence interval excludes H₀. Both methods lead to the same conclusion about significance.
No! Failing to reject H₀ only means there is insufficient evidence to reject it - it does NOT prove H₀ is true. The data simply aren't conclusive enough. This is why we say "fail to reject" rather than "accept" the null hypothesis.
Type I error (α): Rejecting H₀ when it's actually true (false positive). The significance level α is the maximum acceptable probability of this error. Type II error (β): Failing to reject H₀ when it's actually false (false negative). Power = 1 − β is the probability of correctly rejecting a false H₀.
Higher degrees of freedom (larger sample size) makes the t-distribution approach the normal distribution, resulting in smaller p-values for the same test statistic. With more data, you have more confidence in your results. This is why larger studies have more statistical power.
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Understanding p-values is fundamental to making data-driven decisions. Whether you're conducting research, analyzing business metrics, or completing coursework, this calculator helps you determine statistical significance quickly and accurately.
Remember: p-values are just one piece of the statistical puzzle. Always consider effect size, confidence intervals, sample size, and practical significance when interpreting your results. Use this tool as part of a comprehensive statistical analysis.