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Statistics for the Behavioral Sciences, 4/e

One-Way Analysis of Variance With Post Hoc Comparisons

# Symbols and Formulas

SYMBOLS

 Symbol Stands For (0.0K) total mean or grand mean (GM) (0.0K) score within a group (0.0K) mean of a group SStot total sum of squares SSw within-groups sum of squares SSb between-groups sum of squares SSsubj subjects sum of squares SSerror error sum of squares Ng number of subjects within a group N total number of subjects or total number of scores in a repeated measures ANOVA (0.0K) sum over or across groups MSb mean square between groups MSw mean square within groups dfb between-groups degrees of freedom K number of groups or number of trials in a repeated measures ANOVA S number of participants (subjects) dfw within-groups degrees of freedom dftot total degrees of freedom dfsubj subjects degrees of freedom dferror error degrees of freedom F F ratio, ANOVA test Fcomp your computed F ratio Fcrit the critical value of F from Table C LSD least significant difference HSD honestly significant difference q studentized range statistic LSDa, HSDa LSD and HSD mean difference values required for significance at a particular α level (.05 or .01, usually)

FORMULAS

Before solving any of the formulas introduced, the following values need to be computed for the data: ∑Xg, ∑X2g, Ng, ∑X, ∑X2, and N. ∑Xg is the sum of the scores within each group; ∑X2g is the sum of the squared scores within each group; ∑Ng is the number of observations within each group; ∑X is the sum of all the scores; ∑X2 is the sum of all the squared scores; and N is the total number of observations. In addition, for one-way repeated measures ANOVA, ∑Xm, (∑Xm)2, S, and K must be computed. ∑Xm is the sum of scores for each participant; (∑Xm)2 is the square of the sum of the scores for each participant; S is the number of participants; and K is the numbers of trials or tests.

Formula 11 - 5. Computational formula for the total sum of squares

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This equation is identical to the numerator of sample variance, which we said in Chapter 6 was sometimes called the sum of squares or SS.

Formula 11-6. Computational formula for the within-group sum of squares

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This is just the sum of squares equation computed for each group and then summed across groups.
For three groups, the computational formula for SSw becomes

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Formula 11-7. Computational formula for the between-groups sum of squares

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For three groups, the computational formula for SSb becomes

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Formulas 11-8, 11-9, and 11-10. Equations for between-groups degrees of freedom, within-groups degrees of freedom, and total degrees of freedom, respectively

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Formula 11-11.Equation for the between-groups mean square

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Formula 11-12.Equation for the within-groups mean square

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Formula 11-13.Equation for F ratio in one-way between-subjects ANOVA

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Formula 11-14.Least significant difference (LSD) between pairs of means

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Formula 11-15.Honestly significant difference (HSD) between pairs of means

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Formula 11-18.Computational formula for within-subjects sum of squares in one-way repeated measures ANOVA

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For three subjects, the computational formula for SSsubj becomes

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Formula 11-19.Computational formula for error sum of squares in one-way repeated measures ANOVA

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Formula 11-20.Computational formula for error degrees of freedom

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Formula 11-21.Computational formula for mean square error in one-way repeated measures ANOVA

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Formula 11-22.Computational formula for F ratio in one-way repeated measures ANOVA

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The degrees of freedom for the F ratio are the df associated with the numerator (dfb = K – 1) and df associated with the denominator [dferror = (K – 1)(S – 1)].