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ANOVA

One-Way ANOVA — Three Teaching Methods

Compare exam scores across Traditional, Blended, and Online teaching methods. Full SS decomposition, F-ratio, η²=.86, Tukey post-hoc.

📊 F(2,12) = 36.24, p < .001, η² = .86
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📊 3 · ANOVA

3a · One-Way ANOVA: Three Teaching Methods
ANOVA
Research Question: Do three teaching methods (Traditional, Blended, Online) produce different exam scores?
TraditionalBlendedOnline
708575
728878
688272
749080
718776
M=71M=86.4M=76.2
ANOVA Decomposition
$$SS_{Total} = SS_{Between} + SS_{Within}$$ $$SS_B = \sum_{j=1}^{k} n_j(\bar{x}_j - \bar{x}_{..})^2, \quad SS_W = \sum_{j=1}^{k}\sum_{i=1}^{n_j}(x_{ij}-\bar{x}_j)^2$$ $$F = \frac{MS_B}{MS_W} = \frac{SS_B/(k-1)}{SS_W/(N-k)}$$
  1. 1
    Grand Mean:
    $$\bar{x}_{..} = \frac{71(5)+86.4(5)+76.2(5)}{15} = \frac{1178}{15} = 77.87$$
  2. 2
    SSBetween:
    $$SS_B = 5(71-77.87)^2 + 5(86.4-77.87)^2 + 5(76.2-77.87)^2$$ $$= 5(47.20) + 5(72.76) + 5(2.79) = 236.0 + 363.8 + 13.9 = 613.7$$
  3. 3
    SSWithin:
    $$SS_W = [(70{-}71)^2{+}(72{-}71)^2{+}\cdots] + \cdots = 20.0 + 38.8 + 42.8 = 101.6$$
  4. 4
    F-statistic:
    $$MS_B = \frac{613.7}{3-1} = 306.85, \quad MS_W = \frac{101.6}{15-3} = 8.467$$ $$F(2, 12) = \frac{306.85}{8.467} = 36.24$$
  5. 5
    Effect size η²:
    $$\eta^2 = \frac{SS_B}{SS_T} = \frac{613.7}{613.7+101.6} = \frac{613.7}{715.3} = 0.858 \quad (\text{large, } \eta^2 > .14)$$
36.24
F(2,12)
<.001
p
0.858
η²
306.85
MSB
8.467
MSW
🔴 Reject H₀ — Teaching method has a significant effect, F(2,12)=36.24, p<.001, η²=.86 (large). Post-hoc Tukey: B>C>A, all pairs significant.
APA-7
A one-way ANOVA revealed a significant effect of teaching method on exam scores, F(2, 12) = 36.24, p < .001, η² = .86. Post-hoc Tukey tests indicated that Blended (M = 86.40) outperformed Online (M = 76.20) and Traditional (M = 71.00), all ps < .001.

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