On the Comparative Anatomy of Transformations

The attention of statisticians has usually been focussed on single transformations, rather than on families of transformations. With a growing appreciation of the advantages of examining the behavior of data or approximations over whole families of transformations (Moore and Tukey 2, Anscombe and Tukey 1), there arises a need for rationally planned charts for representing families of transformations. The contributions which (i) the topology of the family and (ii) a definition of the strength of a transformation can make to charting are studied in general and applied to the charting of the simple family of transformations. This family is defined to include all transformations of the form y is replaced by z = (y + c) ᵖ and all their limits. It thus includes z = (y + c), z = e^my and the special case equation*z = cases0, (2) Where y + c is always safely >0, and the range of y is through not many powers of 10.