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Statistics for the Behavioral Sciences, 4/e

Michael Thorne, Mississippi State University -- Mississippi State
Martin Giesen, Mississippi State University -- Mississippi State

One-Way Analysis of Variance With Post Hoc Comparisons

Symbols and Formulas

SYMBOLS

Symbol	Stands For

	total mean or grand mean (GM)
	score within a group
	mean of a group
SS_tot	total sum of squares
SS_w	within-groups sum of squares
SS_b	between-groups sum of squares
SS_subj	subjects sum of squares
SS_error	error sum of squares
N_g	number of subjects within a group
N	total number of subjects or total number of scores in a repeated measures ANOVA
	sum over or across groups
MS_b	mean square between groups
MS_w	mean square within groups
df_b	between-groups degrees of freedom
K	number of groups or number of trials in a repeated measures ANOVA
S	number of participants (subjects)
df_w	within-groups degrees of freedom
df_tot	total degrees of freedom
df_subj	subjects degrees of freedom
df_error	error degrees of freedom
F	F ratio, ANOVA test
F_comp	your computed F ratio
F_crit	the critical value of F from Table C
LSD	least significant difference
HSD	honestly significant difference
q	studentized range statistic
LSD_a, HSD_a	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: ∑X_g, ∑X²_g, N_g, ∑X, ∑X², and N. ∑X_g is the sum of the scores within each group; ∑X²_g is the sum of the squared scores within each group; ∑N_g 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, ∑X_m, (∑X_m)², S, and K must be computed. ∑X_m is the sum of scores for each participant; (∑X_m)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

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

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

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

For three groups, the computational formula for SS_b becomes

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

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

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

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

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

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

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

For three subjects, the computational formula for SS_subj becomes

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

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

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

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

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

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