ANOVA Calculator
Run a one-way ANOVA across three or more groups to get the F statistic from between-group and within-group variance, with degrees of freedom.
ANOVA Calculator
Calculator
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Formula: F = MSB / MSW = (SSB/dfB) / (SSW/dfW)
Worked example — F = 18.75 (significant)
Formula
F = MSB / MSW = (SSB/dfB) / (SSW/dfW)
ANOVA tests whether means of 3+ groups differ significantly. F = between-group variance / within-group variance. High F → significant differences.
Worked Examples
Example 1: Treatment comparison
Problem:G1:[23,25,27], G2:[30,32,28], G3:[35,37,33]
Solution:Means: 25, 30, 35. SSB = 150 (df 2, MSB 75), SSW = 24 (df 6, MSW 4), F = 18.75
Result:F = 18.75 (significant)
Frequently Asked Questions
When do I use ANOVA vs t-test?
Use a t-test for comparing 2 groups and ANOVA for 3+ groups. ANOVA controls for inflated Type I error that occurs when running multiple t-tests.
What does the F-statistic mean in ANOVA?
The F-statistic is the ratio of between-group variance (MSB) to within-group variance (MSW). A large F value indicates that group means vary more than would be expected by chance. When F exceeds the critical value for your chosen significance level (typically alpha = 0.05) and degrees of freedom, you reject the null hypothesis that all group means are equal. The F distribution is always positive and right-skewed.
What are the assumptions of one-way ANOVA?
One-way ANOVA requires four key assumptions: (1) Independence: observations within and between groups must be independent of each other. (2) Normality: the data within each group should be approximately normally distributed, though ANOVA is fairly robust to mild violations with large samples. (3) Homogeneity of variances: group variances should be roughly equal (use Levene's test to check). (4) Continuous dependent variable: the outcome being measured must be on an interval or ratio scale.
What do I do after a significant ANOVA result?
A significant one-way ANOVA only tells you that at least two group means differ — it does not tell you which groups differ. You need post-hoc tests to identify specific pairwise differences while controlling for multiple comparisons. Common post-hoc methods include Tukey's HSD (most common for equal group sizes), Bonferroni correction (conservative but simple), Scheffe's test (most conservative, good for complex comparisons), and Games-Howell (when variances are unequal).
What is the difference between SSB and SSW in the output?
SSB, the between-groups sum of squares, measures how far each group mean sits from the grand mean, weighted by group size. SSW, the within-groups sum of squares, adds up how far individual observations sit from their own group mean, so it represents ordinary background noise. They partition the total variability exactly: SST = SSB + SSW. Dividing each by its degrees of freedom gives MSB and MSW, whose ratio is the F statistic.
How are the degrees of freedom for a one-way ANOVA determined?
With k groups and N observations in total, the between-groups degrees of freedom is dfB = k - 1 and the within-groups value is dfW = N - k; together they sum to the total N - 1. So three groups of five values give dfB = 2 and dfW = 12. Both numbers are needed to look up a critical value or p-value, because the F distribution is indexed by the numerator and denominator degrees of freedom as a pair, not by a single number.
What effect size should I report alongside the F statistic?
F and its p-value say whether a difference is detectable, not whether it is large. Eta squared, computed as SSB / (SSB + SSW), gives the proportion of total variability attributable to group membership and is computed from the SSB and SSW values shown in ANOVA Calculator's output. Omega squared is a less biased alternative for small samples. Reporting one of these alongside the group means keeps a statistically significant result from being mistaken for a practically important one.
How do unequal group sizes affect a one-way ANOVA?
Unbalanced designs are handled correctly by the sums-of-squares formulas, because each group mean is weighted by its own n when SSB is computed. The practical problem is that ANOVA's tolerance of unequal variances largely depends on balance: when the larger group also has the larger variance the test becomes conservative, and when the smaller group has the larger variance it becomes liberal and over-rejects. In that situation use Welch's ANOVA, which does not assume equal variances.
What is the difference between one-way and two-way ANOVA?
One-way ANOVA, which ANOVA Calculator performs, compares group means across a single categorical factor such as treatment. Two-way ANOVA uses two factors at once, for example treatment and sex, and partitions the between-groups variation into a main effect for each factor plus an interaction term that captures whether the effect of one factor depends on the level of the other. If your design crosses two factors, running separate one-way tests hides that interaction.
What does it mean if F comes out less than 1?
F below 1 means the between-groups mean square is smaller than the within-groups mean square, so the group means differ by less than the scatter inside the groups would lead you to expect. Such a result can never be statistically significant, because critical F values at conventional significance levels are always above 1. Treat it as no evidence of a difference; a persistently tiny F across studies can also hint that observations are not independent.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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