F Distribution Table

When you use an F table

The F distribution turns up in three common places. A one-way ANOVA compares the variation between group means with the variation inside groups, and assumes independent groups, roughly normal data and similar spread in each group. A regression F test checks the null hypothesis that all slope coefficients are zero. An F test for equal variances compares two sample variances; it is sensitive to non-normal data, so check normality first or use a more robust test such as Levene’s. In each case you compute an F statistic from your data, then compare it with the critical value from this table, or get an exact p-value from the calculator above.

How to read the F table

• Pick the tab for your significance level α, such as the F table 0.05. Find the column for df1, the numerator degrees of freedom, and the row for df2, the denominator degrees of freedom. The cell where they meet is the right-tail critical value: an F statistic above it is significant at level α.
• Take a one-way ANOVA with 3 groups of 10 values each. There are 3 − 1 = 2 degrees of freedom between groups and 30 − 3 = 27 within groups. At α = 0.05 the cell for df1 = 2 and df2 = 27 is 3.354. Suppose your F statistic is 4.5. That is larger, so the groups differ significantly, and the calculator gives the exact p-value 0.0206.

Rows and columns that are missing

Printed F tables show df1 from 1 to 10, then 12, 15, 20, 24, 30, 40, 60, 120 and infinity, and df2 from 1 to 30, then 40, 60, 120 and infinity. If your df is not listed, the cautious habit is to use the next smaller value, which gives a slightly larger critical value. The calculator removes the guesswork: type any whole df from 1 to 1000, or inf, and it returns the exact value. For example F(0.05; 1, ∞) = 3.841, which is the square of the normal critical value 1.96.

Left-tail values and swapped degrees of freedom

A left-tail critical value is not printed, because it can be found from the right-tail table: F(1 − α; df1, df2) = 1 / F(α; df2, df1). To find the value that leaves 5% in the left tail for df1 = 10 and df2 = 20, look up F(0.05; 20, 10) = 2.774 and take its reciprocal. The result is 0.426. The most common mistake is swapping df1 and df2: F(0.05; 10, 20) = 2.348, but F(0.05; 20, 10) = 2.774. Always put the numerator df on the columns and the denominator df on the rows.

Which alpha to choose

The 0.05 table is the default for most coursework. Use 0.01 when a false positive is costly, and 0.10 for exploratory work. The 0.025 table is useful when you run a two-sided variance-ratio test at the 5% level, because each tail then holds 0.025. Put the larger sample variance on top, then compare the ratio with the right-tail value from the 0.025 table; if you keep the original order, also check the left-tail value 1 / F(0.025; df2, df1). As a check, F(0.01; 5, 10) = 5.636 and F(0.10; 3, 20) = 2.380.