One-Way ANOVA

Statistics · Axiom Academy

Analysis of Variance for comparing means across multiple groups In One-Way ANOVA, we examine one factor (independent variable) with k levels (groups). Each group has n i observations, with a total sample size of N . N = total sample size = n 1 + n 2 + ... + n k X ij = j-th observation in group i Example: Testing three teaching methods (k=3) with 5 students per method (n 1 =n 2 =n 3 =5, N=15) 2. Sum of Squares Decomposition The total variation in the data (SST) is partitioned into two components: variation between groups (SSB) and variation within groups (SSW). Mean Squares are the Sum of Squares divided by their respective degrees of freedom . These represent variance estimates. df between = k - 1 (number of groups minus 1) df within = N - k (total observations minus number of groups) MSB estimates variance between groups, while MSW estimates variance within groups (pooled error variance). The F-statistic is the ratio of between-group variance to within-group variance. A large F-value suggests group means are significantly different. F follows an F-distribution with (k-1, N-k) degrees of freedom Large F: Between-group variance >> Within-group variance → Significant difference Small F: Between-group variance ≈ Within-group variance → No significant difference Compare F to critical value or use p-value to test significance 5. Complete Calculation Example Example: Test scores for three teaching methods (k=3 groups, 4 students each, N=12) Method A: 85, 88, 82, 89 → mean = 86

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