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One-Way ANOVA worksheets

Compute between and within sums of squares, complete ANOVA tables, and interpret omnibus F tests.

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Show group means, grand mean, both sums of squares, all degrees of freedom, mean squares, and F.

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Student pages print the assumptions, tail convention, distribution reference, and workspace needed to complete each inference problem. Separate worked keys reproduce the decision from the same validated document.

Learning objective

What this activity builds

Partition variation into a complete one-way ANOVA table and state the limited omnibus conclusion.

Worked example

Four groups contain five observations each. What are the F-test df?

df=(3,16).

Between df=k-1 and within df=N-k.

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Make the practice count

  • Check SS_total=SS_between+SS_within.
  • Divide each SS by its own df.
  • Phrase rejection as at least one mean differs.

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Math SheetNameScoreTeacherDateOne-Way ANOVAShow group and grand means, between and within sums of squares, the complete ANOVA table, and the omnibus conclusion.1.Groups are A=[10,11,12]; B=[11,12,13]; C=[12,13,14]. Compute each group mean, the grand mean, SS_between, and SS_within.Assumptions: Samples are independent; each population is approximately normal with a common variance.Reference: Use SS_between=sum n_j(x-bar_j-x-double-bar)^2 and SS_within=sum sum(x_ij-x-bar_j)^2.Name the conditioning distribution, show the statistic and df, then state a contextual decision.math-sheet.comPage 1 of 8
Math SheetNameScoreTeacherDateOne-Way ANOVAShow group and grand means, between and within sums of squares, the complete ANOVA table, and the omnibus conclusion.2.Groups are A=[12,13,14,15]; B=[14,15,16,17]; C=[16,17,18,19]; D=[19,20,21,22]. Complete the between, within, and total rows of the one-way ANOVAtable, then test equal population means at alpha=0.1.Assumptions: Samples are independent; each population is approximately normal with a common variance.Reference: Table columns are source, SS, df, MS, and F (F only in the between row). F reference: P(F_3,12>=21.400000000)=0.0000416005; critical F=2.605524921.Name the conditioning distribution, show the statistic and df, then state a contextual decision.math-sheet.comPage 2 of 8
Math SheetNameScoreTeacherDateOne-Way ANOVAShow group and grand means, between and within sums of squares, the complete ANOVA table, and the omnibus conclusion.3.Groups are A=[14,15,16,17,18]; B=[17,18,19,20,21]; C=[20,21,22,23,24]. Given MS_between=45 and MS_within=2.5, perform the omnibus F test of equalpopulation means at alpha=0.01; report F, its degrees of freedom, the p-value, and a contextual interpretation.Assumptions: Samples are independent; each population is approximately normal with a common variance.Reference: F=MS_between/MS_within with df=(2,12). F reference: P(F_2,12>=18.000000000)=0.0002441406; critical F=6.926608140.Name the conditioning distribution, show the statistic and df, then state a contextual decision.math-sheet.comPage 3 of 8
Math SheetNameScoreTeacherDateOne-Way ANOVAShow group and grand means, between and within sums of squares, the complete ANOVA table, and the omnibus conclusion.4.Groups are A=[16,17,18]; B=[17,18,19]; C=[18,19,20]; D=[20,21,22]. Compute each group mean, the grand mean, SS_between, and SS_within.Assumptions: Samples are independent; each population is approximately normal with a common variance.Reference: Use SS_between=sum n_j(x-bar_j-x-double-bar)^2 and SS_within=sum sum(x_ij-x-bar_j)^2.Name the conditioning distribution, show the statistic and df, then state a contextual decision.math-sheet.comPage 4 of 8
Math SheetNameScoreTeacherDateOne-Way ANOVAShow group and grand means, between and within sums of squares, the complete ANOVA table, and the omnibus conclusion.5.Groups are A=[18,19,20,21]; B=[20,21,22,23]; C=[22,23,24,25]. Given MS_between=16 and MS_within=1.6667, perform the omnibus F test of equalpopulation means at alpha=0.1; report F, its degrees of freedom, the p-value, and a contextual interpretation.Assumptions: Samples are independent; each population is approximately normal with a common variance.Reference: F=MS_between/MS_within with df=(2,9). F reference: P(F_2,9>=9.600000000)=0.0058609809; critical F=3.006452417.Name the conditioning distribution, show the statistic and df, then state a contextual decision.math-sheet.comPage 5 of 8
Math SheetNameScoreTeacherDateOne-Way ANOVAShow group and grand means, between and within sums of squares, the complete ANOVA table, and the omnibus conclusion.6.Groups are A=[20,21,22,23,24]; B=[23,24,25,26,27]; C=[26,27,28,29,30]; D=[30,31,32,33,34]. Complete the between, within, and total rows of the one-wayANOVA table, then test equal population means at alpha=0.01.Assumptions: Samples are independent; each population is approximately normal with a common variance.Reference: Table columns are source, SS, df, MS, and F (F only in the between row). F reference: P(F_3,16>=36.500000000)=0.0000002194; critical F=5.292214046.Name the conditioning distribution, show the statistic and df, then state a contextual decision.math-sheet.comPage 6 of 8
Math SheetNameScoreTeacherDateOne-Way ANOVAShow group and grand means, between and within sums of squares, the complete ANOVA table, and the omnibus conclusion.7.Groups are A=[22,23,24]; B=[23,24,25]; C=[24,25,26]. Complete the between, within, and total rows of the one-way ANOVA table, then test equalpopulation means at alpha=0.05.Assumptions: Samples are independent; each population is approximately normal with a common variance.Reference: Table columns are source, SS, df, MS, and F (F only in the between row). F reference: P(F_2,6>=3.000000000)=0.1250000000; critical F=5.143252850.Name the conditioning distribution, show the statistic and df, then state a contextual decision.math-sheet.comPage 7 of 8
Math SheetNameScoreTeacherDateOne-Way ANOVAShow group and grand means, between and within sums of squares, the complete ANOVA table, and the omnibus conclusion.8.Groups are A=[24,25,26,27]; B=[26,27,28,29]; C=[28,29,30,31]; D=[31,32,33,34]. Compute each group mean, the grand mean, SS_between, and SS_within.Assumptions: Samples are independent; each population is approximately normal with a common variance.Reference: Use SS_between=sum n_j(x-bar_j-x-double-bar)^2 and SS_within=sum sum(x_ij-x-bar_j)^2.Name the conditioning distribution, show the statistic and df, then state a contextual decision.math-sheet.comPage 8 of 8

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