​In this example, we are testing 2 Factors for their effect on the y Variable, Cleanliness. 

• Factor A is the Detergent type.  There are 2 "levels": Detergent #1 and Detergent #2. 
• Factor B is the Water Temperature. There are 3 levels: Cold, Warm, and Hot.

 
We see from the graph that -- for all three levels of Factor B, Detergent #1 cleans better than Detergent #2. The lines are substantially separated, indicating that the difference is Statistically Significant. (The ANOVA numbers will tell us for sure.) 
 
If Factor B did have an effect, the lines would be slanted. Again, separated lines tell us that Factor A has an effect. 

​If the lines were not separated, as below, then Factor A does not have an effect. 

The purpose of Regression analysis is to develop a cause and effect "Model" in the form of an equation. To keep things simple, let's talk about Simple Linear Regression, in which the equation is

                                                                           y = bx + a
 
The Regression analysis comes up with the values for b and a.

​Residuals represent the Error in the Regression Model. They represent the Variation in the y variable which is not explainedby the Regression Model. 
 
So, Residuals must be Random.  If not -- if Residuals form a pattern -- that is evidence that one or more additional factors (x's) influence y.
 
A Scatterplot of Residuals against y-values should illustrate Randomness: 

​Being Random means that the Residuals 

• are Normally distributed
• have constant Variance
• show no patterns when graphed 
• have no unexplained Outliers

 
Here are some patterns which indicate the Regression Model is incomplete.