One-Way ANOVA Calculator from Group Means, Standard Deviations and Sizes

Compare several group means with one test.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Enter the mean, standard deviation and size of each group. Leave the size at 0 for an unused group.

Enter the significance level.

The calculator gives the sums of squares, F, the p-value and a plain statement of the conclusion.

Formulas used

  • F = \frac{MSB}{MSW}
  • SSB = \sum n_i(\bar{x}_i - \bar{x})^2
  • SSW = \sum(n_i - 1)s_i^2
  • \eta^2 = \frac{SSB}{SSB + SSW}

Worked example

With the default numbers in the calculator above:

  1. Grand mean of all the values: 23.6667 (30 values in 3 groups)
  2. Sum of squares between the groups: 246.667
  3. Sum of squares within the groups: 551.25
  4. Degrees of freedom between and within: 2 and 27
  5. Mean squares between and within: 123.333 and 20.4167
  6. F statistic: 6.04082
  7. p-value: 0.006789
  8. Critical F value at the chosen significance level: 3.3541
  9. Effect size: the share of the variation explained by the groups (eta squared): 0.3091 (30.91 percent)

Change the numbers in the calculator to see the full working, every step, for your own case.

Common questions

What does ANOVA test?

Whether the means of three or more groups are all equal, against the idea that at least one is different.

Why not do many t-tests?

Each test has a chance of a false positive, and these add up. ANOVA keeps one overall error rate.

What do I do after a significant ANOVA?

Use a post-hoc test such as Tukey's to find which groups differ.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any first-year statistics text (analysis of variance).

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