In this part we consider how to measure the quality of our fit by the model function to the given data. You have already overlaid a semilog or log-log plot (whichever is appropriate) of your model function on a semilog or loglog plot of the given data. If you have constructed your function $f\left(t\right)$ correctly, its graph should appear as a straight line, and the line should come reasonably close to most of the data points. If this is not the case, go back now to the previous page, and rethink your construction of the function.

You have also overlaid an Cartesian plot of your model function on the initial plot of the data points. Again, your model function should come reasonably close to all of the data points, but it will probably miss some of them.

The visual information from the preceding page can help you decide whether you followed the right modeling strategy and whether you have reasonable values for the parameters \(a\) and \(b\) in the model function. However, these graphs don't really measure the accuracy of our fit to the data. We can get a better sense of that by calculating residuals, that is, the numerical differences between the numbers of cases and the numbers predicted by the model at corresponding times. Specifically, if the \(j\)-th time is \(Months_j\), the number of cases that month is \(Cases_j\), and your model function is \(f(t)\), then the \(j\)-th residual is the difference ${\mathrm{Cases}}_j-f\left({\mathrm{Months}}_j\right)$ .

1. Use the Residual Plot tool to plot the residuals as a function of time. If you have a good fit, the residuals should be relatively small – are they?

2. Even with a very good fit, you will probably see that the residuals tend to grow in magnitude as time goes on, some positive and some negative. There is a trick word in the preceding question: “relatively.” If the large residuals are differences in very large numbers, they may still be relatively small. We can “scale” the residuals by dividing each one by the number of Cases at the corresponding time – this gives us relative residuals. Select the Relative Residuals plot, and then comment (again) on whether your residuals are relatively small.