F-Table Calculator

The F-distribution is a continuous probability distribution used primarily in hypothesis tests that compare variances or group means. It arises most commonly in ANOVA (Analysis of Variance), regression F-tests, and tests for equality of variances. The distribution is defined by two degrees of freedom parameters: numerator df (df1) and denominator df (df2).

In a one-way ANOVA, df1 (numerator degrees of freedom) equals the number of groups minus 1 (k − 1), and df2 (denominator degrees of freedom) equals the total number of observations minus the number of groups (N − k). In a regression F-test, df1 is the number of predictors and df2 is n minus the number of parameters.

The critical F-value is the threshold from the F-distribution table at a given significance level (α) and specific degrees of freedom. If your computed F-statistic exceeds this critical value, you reject the null hypothesis. For example, with df1=3, df2=20, and α=0.05, the critical F-value is approximately 3.0984.

The p-value P(F ≥ f) gives the probability of obtaining an F-statistic as extreme as or more extreme than your observed value, assuming the null hypothesis is true. A p-value smaller than your chosen significance level (e.g. 0.05) means you reject the null hypothesis.

P(F ≤ f) is the cumulative (left-tail) probability — the probability that a random variable from the F-distribution is less than or equal to your observed F-statistic. It equals 1 minus the right-tail p-value P(F ≥ f). This is also called the CDF (cumulative distribution function) value at f.

Enter your numerator degrees of freedom (df1) and denominator degrees of freedom (df2), then select your significance level (α) to get the critical F-value. Optionally, enter your computed F-statistic to also receive the right-tail and left-tail p-values. Compare your F-statistic to the critical value to make a rejection decision.

The most common significance level in social and natural sciences is α = 0.05 (5%), meaning you accept a 5% chance of a Type I error (false positive). For more stringent tests, α = 0.01 or α = 0.001 is used. The choice of α should be made before data collection based on the consequences of a false positive in your field.

The F-statistic is the ratio of two chi-squared distributed quantities (each divided by their degrees of freedom), both of which represent sums of squared values. Since squared values are always non-negative and the ratio of non-negative numbers is non-negative, the F-distribution is defined only for values ≥ 0.