Math · Statistics
One-way ANOVA calculator.
Compare the means of three or more groups. Paste each group on its own line, or enter n, mean and standard deviation. Your data stays in this browser.
For NIST’s resistor example (three temperatures, five resistors each), the group means are 5.34, 7.72 and 8.56. The ANOVA gives SST = 27.897, SSE = 17.452, F = 9.59 on 2 and 12 degrees of freedom and p = 0.00325, so the means are not all equal at the 5% level.
How it works
One-way ANOVA splits the total variation into variation between group means and variation within groups: SS(Total) = SST + SSE, where SST = Σ ni(ȳi − ȳ)² and SSE = ΣΣ (yij − ȳi)². With k groups and N observations, MST = SST / (k − 1), MSE = SSE / (N − k) and F = MST / MSE. The p-value is the upper tail of the F distribution with k − 1 and N − k degrees of freedom. The null hypothesis is that all group means are equal.
Worked example
NIST measured the change in resistance of resistors held at three temperatures, five per temperature: 6.9, 5.4, 5.8, 4.6, 4.0 / 8.3, 6.8, 7.8, 9.2, 6.5 / 8.0, 10.5, 8.1, 6.9, 9.3. The group means are 5.34, 7.72 and 8.56 and the grand mean is 7.2067. SST = 27.897 (2 df), SSE = 17.452 (12 df), MST = 13.949, MSE = 1.454, so F = 9.59. NIST gives the 5% critical value F(0.05; 2, 12) = 3.89 and a p-value of 0.00325; this calculator shows 3.885 and 0.0032, the same values to the precision NIST prints. Because F is far above the critical value, the null hypothesis of equal means is rejected. Summary mode needs rounded standard deviations, so it matches to about three digits.
Limits
NIST states the test assumes the sampled populations are normally distributed. Independent observations and similar spread across groups also matter. A significant F says that not all means are equal, not which ones differ; follow-up comparisons are needed for that, and running many pairwise t-tests inflates the false-positive rate.
Sources, opened 8 October 2026: NIST/SEMATECH e-Handbook, 7.4.3 Are the means equal? (one-way ANOVA hypotheses and normality assumption); NIST/SEMATECH e-Handbook, 7.4.3.1 One-way ANOVA overview (SS(Total) = SST + SSE decomposition); NIST/SEMATECH e-Handbook, 7.4.3.4 One-way ANOVA calculations (the formulas for SST, SSE, MST, MSE and F applied to the resistor data); NIST/SEMATECH e-Handbook, 7.4.3.3 The ANOVA table and tests of hypotheses about means (ANOVA table, critical value 3.89 and p-value 0.00325).
Frequently asked questions
How do you calculate a one-way ANOVA?
Compute the between-group sum of squares SST and the within-group sum of squares SSE, divide each by its degrees of freedom (k - 1 and N - k) to get MST and MSE, then F = MST / MSE. For the NIST resistor data F = 9.59.
Can I run ANOVA from mean and standard deviation?
Yes. Switch to summary mode and enter n, mean and sample standard deviation for each group. Results match raw-data mode up to the rounding of the standard deviations.
What does the p-value in ANOVA mean?
It is the chance of an F statistic at least as large as yours if all group means were really equal. A small p-value (below 0.05 at the usual level) is evidence that at least one mean differs.
Which groups are different?
ANOVA does not say. NIST lists follow-up techniques such as confidence intervals for differences between means and multiple comparisons. This calculator reports the overall F test only.
What are the assumptions of one-way ANOVA?
NIST states that the sampled populations are assumed to be normally distributed. Observations should be independent, and the groups should have similar spread.
How is this different from a t-test?
A t-test compares two means. ANOVA compares three or more at once with a single test. With exactly two groups, ANOVA gives F equal to the pooled t statistic squared.
Learning aid, not for decisions
Educational calculator. Results assume the conditions listed on this page, which real data often only roughly meet. Do not use it for clinical, regulatory, legal or financial decisions.