What a one-way ANOVA is deciding
Three or more group means need comparing, and a stack of pairwise t-tests is the wrong tool: each extra test spends more of the error budget, until some pair crosses p < 0.05 by luck alone (three groups make three pairs; at 0.05 each, the family risk is already near 14%). ANOVA asks the joint question first: is the variation between group means large compared with the variation within groups? The F statistic is that ratio of mean squares, and a single test at a single α decides whether anything differs at all.
Five students per method, group means 84.6, 75.6 and 90.0. Between groups: SSB = 529.2 on 2 df, MSB = 264.6. Within groups: SSW = 186.4 on 12 df, MSW = 15.533. F = 264.6 ÷ 15.533 = 17.034, p = 0.0003. The methods do not all perform alike. Eta squared, SSB/SST = 0.740, says group membership accounts for about 74% of the score variation in this sample: a large effect by any convention.
After the omnibus test
A significant F licenses pairwise questions; it does not answer them. Post-hoc procedures (Tukey's HSD is the standard) compare pairs while holding the family error rate at α, which the raw t-tests refused to do. Report the pattern they find: which groups differ, by how much, with intervals. And keep the ANOVA's assumptions in view: independent observations, roughly normal residuals, similar variances across groups (Levene's test checks the last; Welch's ANOVA relaxes it). The one-way ANOVA calculator on this site runs the same numbers and prints the table in this format.
Reading a significant F as "all groups differ". It says at least one contrast does. Two groups can be identical while the third drives the whole result; the post-hoc step exists to find out which story your data tell.
Frequently asked questions
Why not just run t-tests between each pair?
Error inflation: each test adds its own 5% false-alarm risk, so three pairs risk about 14% overall. ANOVA tests jointly at one alpha, and post-hoc methods keep the pairwise step honest.
What does eta squared add to the F test?
Magnitude. F and p say the group differences are unlikely under the null; eta squared (SSB/SST) says how much of the variation group membership explains: 0.740 in the example is large.
What if my variances differ across groups?
Welch's ANOVA adjusts the denominator degrees of freedom for unequal variances and is the usual fallback; Levene's test is how you check whether you need it.