How the table fits together
A one-way ANOVA splits the total spread of the data (SS total) into the part between the group means and the part within the groups. Every cell of the table follows from a few identities, so a table with gaps can be filled one cell at a time.
| Source | SS | df | MS | F |
|---|---|---|---|---|
| Between groups | SS between | k − 1 | SS between ÷ df between | MS between ÷ MS within |
| Within groups | SS within | N − k | SS within ÷ df within | |
| Total | SS between + SS within | N − 1 |
The df column comes from the design alone, so it is usually the place to start (a common slip is writing N − 1 for the within row). The SS and df columns both add up; the MS column does not, and the total row has no MS or F. When a problem gives MS and F instead of SS, work backwards: MS within = MS between ÷ F, then SS = MS × df.
Reading the F table
Each problem prints the part of the F table it needs. Go across to the column for df₁, the numerator df (between groups), and down to the row group for df₂, the denominator df (within groups). The three rows in each group are α = 0.10, 0.05 and 0.01. Reject H₀ when F is greater than the critical value at the question's α. Comparing F with all three rows gives the p-value region: above the 0.05 value but below the 0.01 value means 0.01 < p < 0.05.
The values are computed from the F distribution and checked against the NIST table.
An example from set EX9V2K
A lab tests 4 battery brands with 5 batteries each. The table gives SS between = 94.56 and F = 3.94. With k = 4 and N = 20, df between = 3, df within = 16 and df total = 19. MS between = 94.56 ÷ 3 = 31.52. Working backwards, MS within = 31.52 ÷ 3.94 = 8, so SS within = 8 × 16 = 128 and SS total = 94.56 + 128 = 222.56. From the F table, F(3, 16) at α = 0.05 is 3.24 and at 0.01 is 5.29, so 0.01 < p < 0.05: reject H₀ at the 5% level. Open set EX9V2K
Questions
How do you calculate F in ANOVA?
F = MS between ÷ MS within, where each MS is its SS divided by its own df. The total row has no MS and no F. If you get an F below 1 with a large MS between, check that it is not upside down.
Are between groups and treatment the same row?
Yes. Textbooks and software call the same rows by different names: between groups, treatment or factor for the first, and within groups, error or residual for the second. The arithmetic is the same. The Row names choice under Formulas, rounding and row names changes every label on this page.
What does a significant F tell you?
That there is evidence the group means are not all equal: at least two differ. It does not say which ones. A follow-up such as Tukey's test compares the groups in pairs.
Sources, checked 2 November 2026, all from the NIST/SEMATECH e-Handbook of Statistical Methods: 1.3.6.7.3 Upper critical values of the F distribution, 7.4.3.3 The ANOVA table and tests of hypotheses about means and 7.4.3.4 One-way ANOVA calculations.