The fourth worksheet is similar to the scatter worksheet but now permits the slope to be changed by entering a value in the yellow box. (See Figure 11, and click on the image to see a full-size picture.)

Figure 11. The Scatter, Curvature, and Residuals worksheet

(shown with slope = 2, no scatter, and no curvature)

A residuals plot is also included in this worksheet (Figure 12).

Figure 12. A residuals plot with small scatter and no curvature

Both scatter and curvature of the data are introduced in the generated data, the latter by an *x*^{2} term in the *y* column. Explore this worksheet one variable at a time. How does the residuals plot behave? If a small amount of curvature occurs, would the value of *r*^{2} alert you to it?

Using only *r*^{2}, you will not discover the presence of curvature. It is seen only in the residuals plot, as in Figure 13, which shows a nice trend.

Figure 13. A residuals plot with small scatter and small curvature

Ideally, residual plots should show a random distribution of points, i.e., no trends, and the sum of the squared residuals should be kept to a minimum. Now, with a contribution of curvature, as seen in the residuals plot (or in Figure 13), increase the noise/scatter. What happens? You find that a non-linear model can be very difficult to uncover if scatter is apparent in the data. So, some system models can start out linear and, as measurement techniques for these systems improve, may evolve into non-linear models.

Journal of Online Mathematics and its Applications