F-Distribution Calculator

Degrees of freedom represent the number of independent pieces of information used to estimate a parameter. In the F-distribution, df1 (numerator) is typically the number of groups minus one, and df2 (denominator) is the total sample size minus the number of groups. Together they define the shape of the F-distribution.

The F statistic is the ratio of two mean squares — typically the variance between groups divided by the variance within groups. It is used in ANOVA, regression analysis, and other tests to determine whether group means differ significantly. A larger F value suggests greater variance between groups relative to within groups.

The left-tail probability P(F ≤ f) is the area under the F-distribution curve to the left of your observed F value. It represents the probability of getting an F value less than or equal to the one you observed, assuming the null hypothesis is true.

The right-tail probability P(F ≥ f) is the area to the right of your observed F value. In most hypothesis tests (like ANOVA), you use the right-tail probability as your p-value, since large F values are evidence against the null hypothesis. If this probability is less than your significance level α, you reject the null hypothesis.

The critical F value is the threshold that your observed F statistic must exceed in order to reject the null hypothesis at a given significance level. For example, at α = 0.05 with df1 = 5 and df2 = 20, if your F statistic exceeds the critical value, you have statistically significant results.

The most common significance levels are 0.05 (5%) and 0.01 (1%). Using α = 0.05 means you accept a 5% chance of a false positive (Type I error). For stricter tests, use α = 0.01. The choice depends on the consequences of making an error in your specific analysis.

The F-distribution is right-skewed and always non-negative. With small degrees of freedom it is heavily skewed, but as df1 and df2 increase, it becomes more symmetric and bell-shaped. The mean of the distribution approaches 1 as df2 grows large, and the peak (mode) shifts accordingly.

Yes. After computing your F statistic from an ANOVA table, enter your numerator df (number of groups minus one) and denominator df (total observations minus number of groups), along with the F statistic. The calculator will return the p-value and critical F value so you can determine statistical significance.