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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 *r*^{2}. Here the *r*^{2} 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 *r*^{2} vary with the scatter of the data about the regression line? Students should quickly discover that as scatter increases the value of *r*^{2} decreases. (Because the scatter cannot be set to zero in this worksheet, they will only see *r*^{2} get close to 1, not actually reach it.)

The use of *r*^{2} 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 *r*^{2} > 0.795 is needed at the 95% confidence level to have significance.)

Scott A. Sinex, "Exploring the Goodness of Fit in Linear Models - Introducing Noise or Scatter: The Scatter Tab," *Convergence* (March 2005)

Journal of Online Mathematics and its Applications