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Exploring the Goodness of Fit in Linear Models - Introducing Noise or Scatter: The Scatter Tab

Scott A. Sinex

If you have not already opened the scatter.xls spreadsheet, please do so now. Then click on the Scatter tab. The worksheet should look something like Figure 6.

Figure 6. The Scatter worksheet
(Click on image for larger view.)

This worksheet allows students to vary the scatter (or noise) level, by adjusting the scroll bar or by clicking on the arrows, to see how the slope and intercept of line respond to the addition of scatter to the data, while monitoring the value of r2. Here the r2 value is treated as a mystery variable. Getting science students to use the language of mathematics is also a hurdle to overcome here, but it should not be avoided. How does r2 vary with the scatter of the data about the regression line? Students should quickly discover that as scatter increases the value of r2 decreases. (Because the scatter cannot be set to zero in this worksheet, they will only see r2 get close to 1, not actually reach it.)

The use of r2 as a measure of the goodness of fit is realistic because it describes the fraction (or if multiplied by 100, the percentage) of the y-variable that is explained by the variation of the x-variable. You can also discuss statistical significance at this point (For 10 data points, an r2 > 0.795 is needed at the 95% confidence level to have significance.)


Journal of Online Mathematics and its Applications