MATH2089 Chap.14 Analysis of Variance
Analysis of Variance
One way ANOVA compares several population means through a single decomposition of variability. Between group variation reflects separation among sample means, while within group variation reflects residual noise. Their mean square ratio forms the F statistic under the equal means null.
This chapter reads sums of squares, degrees of freedom and mean squares, checks independence, normal errors and variance structure, and limits the global conclusion before adjusted follow up comparisons. The global null states that all population group means are equal. The alternative is that not all are equal, which means at least one supported difference exists. Rejection does not prove that every pair differs.
This scope matters because the F test compresses a multi-group pattern into one broad comparison before any follow-up question is addressed. Total variation around the grand mean is separated into between-group and within-group components. Between-group sums of squares grow when group means are separated. Within-group sums of squares measure residual variation around each group mean.
Each sum is divided by its own degrees of freedom to form a mean square. The F statistic divides the between mean square by the within mean square, so numerator and denominator roles must not be swapped. An ANOVA table should be read as a linked calculation. Verify degrees of freedom, compute mean squares and check the standard total-equals-between-plus-within identity when that decomposition applies.
The F reference distribution uses numerator and denominator degrees of freedom. Software may supply the p-value, but the construction should remain visible so a row or division error can be detected. Assumptions concern independent observations, approximately normal errors for small samples and a suitable variance structure. Independence comes from the sampling or assignment design and cannot be repaired by a plot.
Side-by-side group displays and residual plots reveal skew, outliers and unequal spread. Severe imbalance combined with unequal variance deserves particular caution because the pooled residual scale may represent groups unevenly. After a global rejection, follow the question. A planned contrast encodes a prespecified comparison through coefficients that sum to zero.
All-pairs exploration needs a multiplicity-aware procedure and adjusted intervals or p-values. A collection of unadjusted t tests does not preserve the error control of the original global question. Report group estimates and uncertainty alongside significance so practical magnitude remains visible.
ANOVA is closely related to regression: group indicators create a linear model and the same fitted-versus-residual variation logic reappears.
A complete conclusion names the factor and response, gives F with both degrees of freedom and p-value, states only the global claim licensed by the test, reports relevant group differences and checks, and explains any adjusted follow-up comparison without overstating equality or causation.
What this chapter covers
- 01
Equal means null model
- 02
Between group variation
- 03
Within group variation
- 04
Mean squares and F ratio
- 05
ANOVA assumptions
- 06
Adjusted follow up comparisons
Worked example · free
Compute an F statistic
- +1Between mean square is 24 divided by 2, which equals 12.
- +1Within mean square is 36 divided by 12, which equals 3.
- +1F is 12 divided by 3, which equals 4.
- +1Use an F distribution with numerator 2 and denominator 12 degrees of freedom for the p value.
Key terms
- Between variation
- Variation explained by separation among group means in a one way ANOVA.
- Within variation
- Residual variation among observations around their own group means.
- F statistic
- The ratio of between mean square to within mean square in the global ANOVA test.
Analysis of Variance FAQ
What does ANOVA rejection establish?
It establishes evidence that not all population means are equal. It does not identify the differing groups and does not imply that every pair differs.
Why not run many ordinary t tests?
Multiple unadjusted tests inflate the probability of at least one false positive. ANOVA provides a global test, followed by planned contrasts or multiplicity adjusted comparisons.
Which assumptions need checking?
Check independent observations, approximately normal within group errors for small samples and a suitable variance structure. Unequal spread plus severe imbalance can be especially problematic.
Exam move
Reconstruct an ANOVA table from sums of squares and degrees of freedom, then read a software table in reverse. Draw side by side group plots before interpreting F. Practise a global conclusion and an adjusted follow up statement so the scope of each stage remains clear. Verify the total, between and within variation identity and label the numerator and denominator degrees of freedom before calculating F.
Build one example where only one group differs so you can see why global rejection does not imply that all pairs differ. Write a planned contrast with coefficients summing to zero and compare it with an all-pairs question requiring multiplicity control. Add residual plots for unequal spread and an outlier under balanced and unbalanced group sizes. Conclude with group estimates and effect size, not only the global p value.
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