Triola Elementary Statistics · 11th Edition
Chapter 12: Analysis of Variance
Every key formula from Triola Elementary Statistics Chapter 12, in one searchable page. Click a card to study it — worked examples included.
12.2 · ONE-WAY ANOVA
Tests equality of three or more population means by comparing variances.
H0: μ1 = μ2 = … = μk
H1: at least one mean differs
Test statistic
F = variance between samplesvariance within samples
Right-tailed. P ≤ α ⇒ reject H0.
KEY NOTES
- Requirements: approx. normal populations, equal variances, random independent samples, one factor.
- Do NOT run pairwise t-tests — three tests at α = 0.05 drop overall confidence to ~0.857 (type I error inflates).
12.2 · WORKED EXAMPLE
WORKED EXAMPLE
Car-crash chest deceleration, small/medium/large cars:
H0: μ1 = μ2 = μ3; software gives P = 0.028.
P < 0.05 ⇒ reject H0: at least one mean differs.
H0: μ1 = μ2 = μ3; software gives P = 0.028.
P < 0.05 ⇒ reject H0: at least one mean differs.
Rejecting only says “not all equal” — it does NOT say which mean is different.
12.2 · THE TWO VARIANCES
Between samples
nsx̄², sx̄² = variance of the sample means
Within samples
sp² = mean of the sample variances
WORKED EXAMPLE
Three samples, n = 4 each: between = 4(0.0833) = 0.3332;
within = (3.0+2.0+2.0)/3 = 2.3333;
F = 0.3332/2.3333 = 0.1428 ⇒ fail to reject.
within = (3.0+2.0+2.0)/3 = 2.3333;
F = 0.3332/2.3333 = 0.1428 ⇒ fail to reject.
12.2 · SS AND MS FORMULAS
Sums of squares
SS(treat) = ∑ni(x̄i − x̄)²
SS(error) = ∑(ni − 1)si²
SS(total) = SS(treat) + SS(error)
Mean squares
MS(treat) = SS(treat)k − 1, MS(error) = SS(error)N − k
Test statistic
F = MS(treat)MS(error), df = k − 1, N − k
N = total number of values in all samples combined.
12.2 · THE ANOVA TABLE
Software shows: Source | SS | df | MS | F | P-value.
KEY NOTES
- Numerator df = k − 1, denominator df = N − k.
- Large F ⇒ small P-value ⇒ reject H0.
- If a population is far from normal, use the Kruskal-Wallis test (13-5).
- Equal sample sizes make ANOVA robust: variances up to 9× apart still work.
WATCH OUT!
Mistakes that cost points
- ANOVA is RIGHT-tailed — F can never be negative
- Rejecting H0 does NOT identify which pair differs
- Don’t use ANOVA on matched/paired samples
- MS = SS/df — divide by the RIGHT df (k−1 vs N−k)
- Far-from-normal populations ⇒ Kruskal-Wallis, not ANOVA
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