- 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.
[latex]R^{2}[/latex] and Scatterplot Shape
The coefficient of determination, [latex]R^2[/latex], is a measure of the proportion of the variation of a response variable in linearly related bivariate data that can be explained by its relationship with the explanatory variable. You should understand that:
- [latex]R^2[/latex] is equivalent to the square of the correlation coefficient [latex]r[/latex] and will always be a positive number between [latex]0\%[/latex] and [latex]100\%[/latex].
- [latex]R^2[/latex] should be interpreted and written as a percentage.
Consider what you already understand about the shape and spread of a scatterplot.
- The strongest linear relationships appear in plots as data that is roughly linear in shape with data points that lie very close to some line.
- Weaker relationships may be very roughly linear in shape and more spread out, with data points that lie further from some line.
- Non-linear relationships have data points that either form other shapes or are randomly scattered across the plot.