- Describe how the slope, shape of the data, and the coefficient of determination are connected.
- Find [latex]R^2[/latex] and describe how [latex]R^2[/latex] describes the relationship in a data set.
Now that you have developed some intuition about [latex]R^2[/latex] and the shape of a plot, let’s explore how the spread of a plot affects the value of [latex]R^2[/latex].
Step 1: From the drop-down menu, select Linear Relationship.
Step 2: Select the boxes that will display [latex]r[/latex] and [latex]r^2[/latex].
Step 3: Toggle back and forth between the different options for spread (small, medium, or large) and click the Refresh button, if needed. Make a note for yourself about how [latex]r^2[/latex] changes as you change the spread from large to medium to small. As you do this, note that squaring [latex]r[/latex] does, in fact, yield [latex]r^2[/latex].
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Possible values of [latex]R^2[/latex]
Depending on the tools you use, [latex]R^{2}[/latex] may be expressed as a decimal or as a percentage. Even though the tool expresses [latex]R^{2}[/latex] using a percentage, it is important to be able to read values of [latex]R^{2}[/latex] in both decimal and percentage form.
Recall that [latex]R^{2}[/latex] is the proportion of the variation in the response variable that can be explained by its linear relationship with the explanatory variable. It doesn’t make sense for this proportion to be negative. Can more than [latex]100\%[/latex] of the variation in the response variable be explained by its linear relationship with the explanatory variable? No, it can’t. For these reasons, the value of [latex]R^{2}[/latex] will always be between [latex]0[/latex] and [latex]1[/latex] or, in percentage form, [latex]0\%[/latex] and [latex]100\%[/latex].