How this calculator works
One-way ANOVA compares variation between sample means with pooled variation within the groups. The p-value is the upper tail of an F distribution. The model assumes independent observations, normally distributed errors within groups and equal population variances. A small p-value supports a difference somewhere among the means, without identifying which pairs differ. Summary mode requires sample standard deviations, calculated with n - 1 in the denominator.
Formula
Grand mean = sum(n_i mean_i) / N. SS_between = sum[n_i(mean_i - grand mean)^2]. SS_within = sum[(n_i - 1)s_i^2]. F = [SS_between/(k - 1)] / [SS_within/(N - k)].
Worked example
The groups 1,2,3 and 3,4,5 have means 2 and 4. Between-group SS is 6 and within-group SS is 4. The F statistic is 6 with 1 and 4 degrees of freedom, giving p about 0.070484. At 5%, there is insufficient evidence to reject equal means.
Advertisement
Frequently asked questions
Can group sizes differ?
Yes. Each group must contain at least two observations. Summary mode uses a separate size for every group.
What if every value within a group is identical?
If all groups have zero internal variance, the finite F test used here is undefined and the calculator reports an error.
Does a non-significant result prove equal means?
No. The sample may have limited power, and the assumptions still need assessment. This tool does not perform post-hoc pairwise tests or repeated-measures ANOVA.