"’Analysis of variance’ is in some ways a misleading name for a collection of statistical methods and models that deal with differences in the means of a variable across groups of observations. While ‘analysis of means’ may be a better name, the methods all employ ratios of variances in order to establish whether the means differe, and the name analysis of variance is here to stay. The name is often abbreviated to ANOVA, whch a student of ours for a long time thought was the name of an Italian statistician" (Iversen & Norpoth, 1987, p. 7).
"Twoway ANOVA also introduces a concept not known in a oneway analysis: the concept of interaction. This refers to the way in which a category of one explanatory variable combines with a category of the other explanatory variable to produce an effect on the dependent variable that goes beyond the sum of the separate effects" (Iverson & Norpoth, 1987, p. 91).
This hypothetical study investigated the effect of alcohol (no alcohol versus strong alcohol) on the ability of 20 introductory psychology students to correctly recall either a list of nondirty words or a list of highly dirty words. All participants were randomly assigned to conditions. This design was 2 X 2 and completely randomized. The dependent variable was the number of repetitions required for perfect recall. Here are the results of the study:
A1B1 Strong Alcohol/ NonDirty Words 21 22 19 18 21 
A2B1 No Alcohol/ NonDirty Words 19 20 18 17 21 
A1B2 Strong Alcohol/ Highly Dirty Words 2 5 6 3 4 
A2B2 No Alcohol/ Highly Dirty Words 7 7 5 6 8 
a. Graph these results, using a line graph
b. Interpret your graph
c. What means could you calculate to explore your interpretation?
Calculating the cell and column means would help you get a clearer sense of whether there is a main effect or effects, and wteher there is an interaction:
Alcohol (column) 

Words (row) 
Strong 
No 

Nondirty words 
20.2 
19 
19.6 
Highly dirty words 
4 
6.6 
5.3 
12.1 
12.8 
12.45 
A complete statistical analysis requires additional calculations, some of which are shown below:
Alcohol 

Words 
Strong 
No 

Nondirty words 
21 22 19 18 21 =20.2 
19 20 18 17 21 =19 
SX = 196 SX^{2 }= 3992 n = 10 = 19.6 
Highly dirty words 
2 5 6 3 4 =4 
7 7 5 6 8 =6.6 
SX = 53 SX^{2 }= 313 n = 10 = 5.3 
SX = 121 SX^{2 }= 2212 n = 10 = 12.1 
SX = 128 SX^{2 }= 2093 n = 10 = 12.8 
SX = 249 SX^{2 }= 4305 N = 20 
The following is a typical summary table, showing the results of a complete analysis of variance on this data. Note that there is a significant main effect for the Words factor. There is also a significant interaction effect.
Source  SS  df  MS  F  p 
Alcohol  2.45  1  2.45  1.0089  .312 
Words  1022.45  1  1022.45  454.42  0.000 
Words x Alcohol  18.05  1  18.05  8.022  0.012 
Error  36.00  16  2.250  
Total  1078.95  19 