One-way ANOVA Calculator
Compare independent groups with classical or Welch one-way ANOVA, including F inference, effect size and method assumptions.
Independent groups · classical or Welch · overall test, not pairwise conclusions
Results
Review or adjust your inputs, then calculate to see your results and interpretation.
Understand this calculation
The usual ANOVA null is equality of all population means. Classical ANOVA partitions variation around the grand mean into between- and within-group components and compares variation per degree of freedom. A large F means separation is large relative to within-group noise, interpreted using the matching F distribution.
See calculation method and assumptions
Formula
Classical sums of squares and F
Classical sums of squares and F
Classical effect size
Welch weights and correction
Welch weights and correction
Welch degrees of freedom
Variables, rates and periods
- k, N, nⱼ
- Group count, total count and group j count.
- x̄ⱼ, x̄
- Group mean and count-weighted grand mean.
- SSB, SSW, MS
- Between/within sums of squares; MS=SS/df.
- wⱼ, W, x̄w, C
- Welch weights, total weight, weighted mean and correction.
- ν₁, ν₂, η²
- F numerator/denominator df; classical sample effect size.
- H₀
- All group population means are equal.
Read the observations, not only the means
- Circles, squares and triangles represent the three groups.
- Short solid lines show group means and a dashed line shows the grand mean.
- Separating centers raises between-group variation;
- spreading observations raises within-group variation.
- Counts affect precision and df as well.
F compares two mean squares
- Between SS weights group-center distances from the grand mean by group counts.
- Within SS sums observation distances from their own group centers.
- Dividing by k−1 and N−k gives mean squares.
- Classical F is their ratio;
- its right-tail area is the model-based p-value.
Choose classical or Welch
- Classical ANOVA assumes a common error variance, approximately normal within-group errors and independent observations.
- Welch weights groups using their estimated variances and adjusts F and df;
- it does not use the same SS decomposition.
- Unequal variances combined with unequal counts make method choice especially important.
Post-hoc comparisons and multiplicity
- Repeating unadjusted pairwise t-tests inflates the chance of at least one false rejection across the family.
- Tukey HSD addresses all pairwise mean comparisons with common variance;
- Tukey–Kramer allows unequal counts.
- Games–Howell addresses unequal variances and counts.
- Choose a method for the planned question and assumptions, not the smallest p-value.
Significance, effect size and value
- η² is Between SS divided by Total SS: the between-group share of sample variation.
- The p-value describes compatibility with H₀;
- η² describes a variance share.
- Practical value also needs differences in original units and a cost judgment.
- These measures answer different questions.
Compare independent groups with classical or Welch one-way ANOVA, including F inference, effect size and method assumptions.
Worked example
- Enter at least three named groups of observations; unequal counts are supported.
- Select classical or Welch according to the variance assumption, and set α.
- Read group summaries, F and p-value; classical ANOVA also supplies SS/MS and η². Consider adjusted follow-up comparisons afterward.
Frequently asked questions
Which groups differ after a significant ANOVA?
Use planned contrasts or properly adjusted post-hoc comparisons. This version provides omnibus inference and method guidance; it does not compute automatic Tukey or Games–Howell pairwise decisions.
Does Welch fix every assumption problem?
No. Welch addresses unequal variances in mean comparisons with unequal counts, but still needs independent, suitably distributed observations and can be sensitive to outliers.
Can I directly compare three return strategies?
Strategies measured over the same market months may share shocks and serial dependence. Check the data structure first. Independent one-way ANOVA does not model repeated measures or time series.
Related calculators
Student’s t-test Calculator
Compare mean differences, uncertainty and test direction.
Statistical Distributions & Critical Values Calculator
Explore normal, t, χ² and F tails and critical values.
Mean & Standard Deviation Calculator
Separate mean return from dispersion.