 Statistics for the Behavioral Sciences, 4/e Michael Thorne,
Mississippi State University  Mississippi State Martin Giesen,
Mississippi State University  Mississippi State
OneWay Analysis of Variance With Post Hoc Comparisons
Symbols and FormulasSYMBOLS Symbol  Stands For 
 (0.0K)  total mean or grand mean (GM)  (0.0K)  score within a group  (0.0K)  mean of a group  SS_{tot}  total sum of squares  SS_{w}  withingroups sum of squares  SS_{b}  betweengroups 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  (0.0K)  sum over or across groups  MS_{b}  mean square between groups  MS_{w}  mean square within groups  df_{b}  betweengroups degrees of freedom  K  number of groups or number of trials in a repeated measures ANOVA  S  number of participants (subjects)  df_{w}  withingroups 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}, ∑X^{2}_{g}, N_{g}, ∑X, ∑X^{2}, and N. ∑X_{g} is the sum of the scores within each group; ∑X^{2}_{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; ∑X^{2} is the sum of all the squared scores; and N is the total number of observations. In addition, for
oneway 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 (1.0K)
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 116. Computational formula for the withingroup sum of squares (3.0K)
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 (3.0K) Formula 117. Computational formula for the betweengroups sum of squares (2.0K)
For three groups, the computational formula for SS_{b} becomes (2.0K) Formulas 118, 119, and 1110. Equations for betweengroups degrees of freedom, withingroups degrees
of freedom, and total degrees of freedom, respectively (1.0K) Formula 1111.Equation for the betweengroups mean square (1.0K) Formula 1112.Equation for the withingroups mean square (1.0K) Formula 1113.Equation for F ratio in oneway betweensubjects ANOVA (1.0K) Formula 1114.Least significant difference (LSD) between pairs of means (1.0K) Formula 1115.Honestly significant difference (HSD) between pairs of means (1.0K) Formula 1118.Computational formula for withinsubjects sum of squares in oneway repeated measures ANOVA (2.0K)
For three subjects, the computational formula for SS_{subj} becomes (2.0K) Formula 1119.Computational formula for error sum of squares in oneway repeated measures ANOVA (1.0K) Formula 1120.Computational formula for error degrees of freedom (1.0K) Formula 1121.Computational formula for mean square error in oneway repeated measures ANOVA (1.0K) Formula 1122.Computational formula for F ratio in oneway repeated measures ANOVA (1.0K)
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)]. 
