The fit of graphical displays to patterns of expectations

Graphical displays of multivariate data often clearly exhibit features of the expectations even though the data themselves are poorly fitted by the displays. Thus, it often occurs that ordinations and biplots that poorly fit the sample data still reveal salient characteristics such as clusters of similar individuals and patterns of correlation. This paper provides an explanation of this seemingly paradoxical phenomenon and shows that when many variables are analyzed, the common measure of goodness of fit of a lower rank approximation often seriously underestimates the closeness of the fit to underlying patterns. The paper also provides some guidelines on better estimates of the latter goodness of fit.