Last updated: 2026-07-02

What This Calculator Does

This calculator runs a one-way analysis of variance (ANOVA) across three groups. Paste the raw values for each group and it reports the full ANOVA table — the between- and within-groups sums of squares, their degrees of freedom, and mean squares — together with the F statistic, its p-value, and a plain-language significance decision.

What one-way ANOVA answers

Three groups produce three averages, and those averages will almost never be identical even when the groups come from the same underlying process — random sampling alone guarantees some spread. One-way ANOVA asks a precise question: if all three population means were truly equal, how surprising would differences among the sample means this large be? The answer is the p-value.

The idea behind the test is to compare two sources of variation. Some of the spread in the pooled data comes from the groups sitting at different levels; the rest comes from the natural scatter of observations around their own group mean. If the first source is large relative to the second, the group means are probably not all equal.

The sums of squares

ANOVA partitions the total variation into a between-groups part and a within-groups part.

The between-groups sum of squares measures how far each group mean sits from the grand mean $\bar{x}$ (the mean of every observation pooled together), weighted by group size $n_j$:

The within-groups sum of squares measures how far each observation $x_{ij}$ sits from its own group mean $\bar{x}_j$:

Degrees of freedom and mean squares

Each sum of squares is turned into a mean square by dividing by its degrees of freedom. With $k$ groups and $N$ total observations:

The within-groups mean square $MS_W$ is a pooled estimate of the common error variance. The between-groups mean square $MS_B$ estimates that same variance only when the group means are equal; when they differ, it is inflated.

The F statistic

The test statistic is the ratio of the two mean squares:

When every population mean is equal, both mean squares estimate the same variance and $F$ hovers near 1. When the group means genuinely differ, the numerator grows and $F$ rises above 1. The p-value is the upper-tail area of the F distribution with $(df_B, df_W)$ degrees of freedom beyond the observed $F$:

Worked example

Suppose three teaching methods are tested with five students each:

• Group 1: 85, 90, 88, 92, 87 (mean 88.4)
• Group 2: 78, 82, 80, 85, 79 (mean 80.8)
• Group 3: 91, 95, 93, 96, 94 (mean 93.8)

The grand mean is $\bar{x} \approx 87.67$, with $N = 15$ and $k = 3$.

With $p \approx 0.00001$, far below 0.05, there is strong evidence that at least one teaching method produces a different mean score.

After a significant result

A significant ANOVA says at least one group mean differs — not which one. To locate the differences, follow up with a post-hoc procedure such as Tukey's HSD or pairwise Two-Sample t-Test Calculator (Welch) comparisons with a correction for multiple testing. Pair the verdict with an effect-size measure, such as eta-squared, to describe how much of the total variation the group differences explain.

Assumptions

One-way ANOVA rests on a few conditions:

• Independence — the three groups are independent, and observations within each group are independent.
• Approximate normality — the values in each group are roughly normally distributed, or the samples are large enough for the central limit theorem to apply.
• Homogeneity of variance — the groups share a similar spread. When variances differ markedly, a Welch-adjusted ANOVA is more reliable than the classical form used here.

When only two groups are involved, the classical ANOVA reduces exactly to a One-Sample t-Test Calculator style comparison of means; the Two-Sample t-Test Calculator (Welch) is the direct two-group counterpart. To inspect the spread within a single group first, the Variance and Standard Deviation Calculator and Descriptive Statistics Calculator tools summarize its center and dispersion.