What is the F-distribution and when is it used?
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). See also our Frequency Table Calculator.
What are degrees of freedom df1 and df2 in an F-test?
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.
What is a critical F-value?
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.
What is the p-value in an F-test?
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.
What does P(F ≤ f) mean?
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.
How do I use this F-table calculator?
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.
What significance level (α) should I use?
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.
Why does the F-distribution only take positive values?
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.