One-way ANOVA | Finocturne

Question, try, then observe

Three different-looking means: different populations?

Illustrative scores from three distinct student groups. Change center spacing and within-group spread; read observations, mean squares and the F right tail. Shapes distinguish the groups.

01 · Make a predictionDo unequal sample means establish that every group differs?

Adjacent group mean gap (points) 4

Baseline within-group SD (points) 6

Observations per group 8

Three groups: observations and centers

Grand mean (dashed) = 50 · Group means (solid marks)

Read all observations

Observation	A	B	C
1	37.43	41.43	45.43
2	39.88	43.88	47.88
3	42.33	46.33	50.33
4	44.78	48.78	52.78
5	47.22	51.22	55.22
6	49.67	53.67	57.67
7	52.12	56.12	60.12
8	54.57	58.57	62.57

Test method

Significance level α

Classical assumes equal population variances; Welch allows unequal variances. Both require independent observations; Welch does not handle paired or repeated measurements.

Compare unequal spread: B has twice, C three times the SD

H₀: μA=μB=μC · H₁: at least one mean differs

Divide between- and within-group sums of squares by their degrees of freedom

MSB = \frac{256}{2} \approx 128; MSW = \frac{756}{21} \approx 36

Classical F: between-group mean square divided by within-group mean square

F = \frac{128}{36} \approx 3.5556

F uses the right tail. A significant omnibus test supports at least one different population mean, not every pair.

Between versus within: mean squares

Between MS

128

Within MS

MS = SS / df. Compare variation per degree of freedom, rather than comparing raw SS with different df.

Source	SS	df	MS
Between	256	2	128
Within	756	21	36
Total	1,012	23	—

F · reference distribution

df₁=2, df₂=21 · α=0.05 · p=0.0468

Observed tail (solid)α rejection region (hatched)Statistic: dotted · Critical: dashed

Statistic: 3.556Critical value(s): 3.467The drawing truncates long tails. Probabilities use the complete distribution, not the displayed area.

F3.556df₁=2, df₂=21

p-value0.0468

η²0.253Between-group share of sample variation

Reject H₀ at the selected α; assumptions and practical magnitude still matter.

No automatic post-hoc pairwise decisions are provided. See the discussion of Tukey/Tukey–Kramer, Games–Howell and multiplicity below. Strategies sharing market months are not automatically independent groups.

What did you just see?

Means five points apart tell a different story when students within a group vary by one point versus fifteen points. Read separation together with within-group dispersion.

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.

Another 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.

Show formal definition and formula

One-way ANOVA tests population mean equality across independent groups defined by one factor. It is named analysis of variance because a decomposition of variation supplies the test of means.

Classical sums of squares and F

SSB = \sum nⱼ {(x̄ⱼ - x̄)}^{2}; SSW = \sum \sum {(xᵢⱼ - x̄ⱼ)}^{2}; F = \frac{\frac{SSB}{k - 1}}{\frac{SSW}{N - k}}

Welch weights and correction

wⱼ = \frac{nⱼ}{{sⱼ}^{2}}; W = \sum wⱼ; C = \sum \frac{{(1 - wⱼ / W)}^{2}}{nⱼ - 1}; F = \frac{\frac{\sum wⱼ {(x̄ⱼ - x̄w)}^{2}}{k - 1}}{1 + \frac{2 (k - 2) C}{k^{2} - 1}}

Welch df and classical effect size

ν₁ = k - 1; ν₂ = \frac{k^{2} - 1}{3 C}; η^{2} = \frac{SSB}{SSB + SSW}

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.

Common mistakes

A significant omnibus ANOVA supports at least one population mean difference, not differences in every pair. η² describes the sample's between-group share of variation, not causation. Welch F must not be placed in a classical SS/MS decomposition.

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.

You can now explore…

Separate between and within variation
Distinguish significance, effect size and post hoc testing