Coefficient of Determination: Apply It 1

  • 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.

Thinking About Education

Let’s approach this activity from the perspective of the secretary of education in your state. You noticed that many public school students in your state are not showing good results on their high school math exams. So, you would like to introduce a policy change that will lead to better results. Your first step should be to collect data about high school students to see what factors best predict their math performance.

Students and a teacher standing and smiling in front of a chalkboard that reads "Back to School"
Figure 1. Before introducing new education policies, it’s important to understand the students they’ll affect—collecting data from students can help identify what influences math success.
By comparing [latex]R^{2}[/latex], we can make some decisions regarding the data set because [latex]R^{2}[/latex] represents the percent of variation in the dependent (predicted) variable [latex]y[/latex] that can be explained by variation in the independent (explanatory) variable [latex]x[/latex] using the regression (best-fit) line.

“Correlation does not imply causation.”

A common mistake people make when describing the relationship between two quantitative variables is that they confuse association and causation. This confusion often occurs when there is a strong relationship between the two quantitative variables.

In the case of a linear relationship, people mistakenly interpret an [latex]r[/latex]-value that is close to [latex]1[/latex] or [latex]-1[/latex] or an [latex]R^{2}[/latex] that is close to [latex]1[/latex] as evidence that the explanatory variable causes changes in the response variable. In this case, the correct interpretation is that there is a statistical relationship between the variables, not a causal link. In other words, the explanatory variable and the response variable vary together in a predictable way. There is an association between the variables. But this should not be interpreted as a cause-and-effect relationship.