What are degrees of freedom in an F-distribution?
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. See also our calculate Rayleigh Distribution Probability Density Function (PDF).
What is an F statistic?
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.
What is a cumulative (left-tail) probability for the F-distribution?
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.
What is a right-tail probability and when should I use it?
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. You might also find our Inverse Normal Distribution Calculator useful.
What is a critical F value?
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.
What significance level (α) should I use?
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.
How does the shape of the F-distribution change with degrees of freedom?
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.
Can I use this calculator for ANOVA tests?
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.