(A) The
distances are on average zero.
(B) The distances are independent
of one another.
(C) The distances have the same variance at each
X-value.
(D) The distances are approximately normally
distributed.
The assumptions are all concerned with the distance from the points to the true regression line. Unfortunately we don't know what the true regression line is, but we can estimate these distances by looking at the distance from the points to the fitted regression line. These distances are called the residuals, and we check the assumptions by looking at plots of the residuals, that is diagnostic residual plots. It turns out that we can't actual check assumption (A) so we will forget it for now. If assumptions (B) is violated in a time series context then it is termed "autocorrelation" and if assumption (C) is violated we call it "heteroscedasticity". Assumption (D) is not too important so long as you have a reasonable amount of data and the histogram of the residuals looks roughly symmetric about zero.
The second issue concerns the effects that outlying points can have on a regression. The critical thing to realize is that a single data point has the potential to drive an entire regression analysis so that we we need ways of classifying and identifying these important points. There are two basic concepts. The first is leverage. A point is called leveraged if it is an outlier in the X-direction. The second concept is the size of the residual in the Y-direction. A point can have BIG or LITTLE leverage and either a BIG or LITTLE residual. There are clearly four possibilities, but the one to watch out for is when a point has a BIG leverage and a BIG residual. This is the sort of point that can be single handedly driving a whole regression and we call it a point of high influence
In this class we learn about problems in a regression by looking at residual plots and in later classes we will learn how to fix some of them up.