curves. Smooth curves would by their smoothness imply a degree of accuracy in the data much greater than would ordinarily be justifiable. By using the straight lines instead of the smooth curves to connect points, the reader is warned that the chart represents facts as found, rather than facts which are assumed to be in accordance with any definite laws. Where smooth curves are proper, as in Fig. 163, it is feasible to have a mathematical formula to represent the shape of the smooth curve. As the determination of mathematical formulas to suit the shape of any curve is a whole study in itself, there will be no attempt to cover that subject in this book.
Biologists have constant use for correlation curves. Fig. 164 shows a curve used in the solution of a typical problem such as biologists are constantly attacking. It was desired here to find out what laws govern the physical characteristics of the offspring when certain characteristics are found in the mother. In all, 1,370 cases were measured, both mother and daughter, to get the data from which Fig. 164 was plotted. All the cases were classed according to the span of the mother, with the class limits made 1 inch apart. A single point was plotted on the chart as an average for each class. A curve line was then drawn so as to represent most closely the data shown on the chart, and it is seen that the curve is a straight line. There were naturally fewer observations at either end of the chart, for the very small spans and for the very large spans, than for the intermediate spans. It was probably because of the small number of observations at either end of the curve that the dots there are so far away from the curve line. If more numerous observations were taken it is probable that all the averages would fall more closely on the curve line than seen in Fig. 164.
The ordinary course of procedure in making a correlation chart is to plot all the observations by the method seen in Fig. 163. Sometimes, however, the observations are so many as to make an extremely confusing chart if the observations are shown in the form of separate dots. In Fig. 164 the observations in any 1-inch class have been averaged and all the observations in one class are represented by a single dot. In Fig. 165 the dots would be very numerous, so numerous as to make it not feasible to show them on a small size chart. For Fig. 165 it was desired to show the maximum income and the minimum income, as well as the average income. Instead of showing all the dots on the chart, cross-hatching was used to represent the area in
