Critical Value Calculator

A critical value is a threshold on the test statistic scale that defines the boundary of the rejection region. If your computed test statistic exceeds the critical value (in absolute terms for two-tailed tests, or directionally for one-tailed tests), you reject the null hypothesis. It is determined by the chosen significance level α and the distribution of the test statistic.

Your choice depends on your alternative hypothesis (H₁). If H₁ states a difference in a specific direction (e.g. μ > μ₀ or μ < μ₀), use a one-tailed test. If H₁ simply states that two values differ without specifying direction (μ ≠ μ₀), use a two-tailed test. Two-tailed tests split α across both tails (α/2 each side), resulting in more extreme critical values.

For a two-tailed test at α = 0.05 (95% confidence), the Z critical values are ±1.96. For a one-tailed right test at α = 0.05, the critical value is 1.645. For a one-tailed left test, it is −1.645. These are among the most commonly used values in statistics.

Not exactly. The t-distribution has heavier tails than the standard normal (Z) distribution, so t critical values are larger in magnitude, especially for small degrees of freedom. As the degrees of freedom increase (generally above 30–120), the t-distribution converges toward the Z distribution, and the critical values become nearly identical.

Set your degrees of freedom to n − 1, where n is your sample size. For example, with a sample of 15 observations, df = 14. Then enter your desired significance level α and tail type. The calculator will return the corresponding t critical value from the inverse of the t-distribution CDF.

The chi-square distribution is used in goodness-of-fit tests, tests of independence in contingency tables, and tests about a population variance. You need to supply the degrees of freedom (typically number of categories minus 1, or (rows−1)×(cols−1) for contingency tables). Chi-square critical values are always positive since the distribution is right-skewed.

The F critical value is used in analysis of variance (ANOVA) and tests comparing two population variances. It requires two degrees of freedom: df1 (numerator, related to the number of groups minus 1) and df2 (denominator, related to the total observations minus the number of groups). The F-distribution is right-skewed and always positive.

The significance level α represents the probability of rejecting the null hypothesis when it is actually true (a Type I error). Common choices are 0.10, 0.05, and 0.01. A smaller α means you require stronger evidence to reject H₀, which pushes the critical value further into the tail — making it harder to achieve statistical significance.