The Proportion of Variability in Y Accounted for
by the Linear Relationship Between X and Y
Bruce Weaver
28-Mar-2003
Coefficient of Determination
• In simple linear regression, r2 is often called the
coefficient of determination, because it is equal to
the proportion of variability in Y (the outcome
variable) that is explained by (or determined by)
the linear relationship between X and Y. Students
often struggle to understand exactly what that
means.
• The following series of figures may help in that
regard.
Variance of Y = 900 in all plots
• Each of the figures shows a scatterplot in which
the SD of the Y-scores is equal to 30 (so the
variance of the Y scores is equal to 900).
• But notice that the correlation between X and Y
varies from one figure to the next.
• Notice also that as the correlation changes, so does
the proportion of the variance of Y that is
explained by the linear relationship.
Figure 1. Pearson r = 0. The least squares regression line is a horizontal
line through the mean of Y. The proportion of Y variability accounted for
by the linear relationship = r2 = 0. The proportion of Y variability left
unexplained = 1, or 100%.
Figure 2. Pearson r = 0.2. The proportion of Y variance explained by the
linear relationship between X and Y = r2 = 0.04, or 4%. The proportion of
Y variability left unexplained = 0.96, or 96%.
Figure 3. Pearson r = 0.4. The proportion of Y variance explained by the
linear relationship between X and Y = r2 = 0.16, or 16%. The proportion
of Y variability left unexplained = 0.84, or 84%.
Figure 4. Pearson r = 0.6. The proportion of Y variance explained by the
linear relationship between X and Y = r2 = 0.36, or 36%. The proportion
of Y variability left unexplained = 0.64, or 64%.
Figure 5. Pearson r = 0.8. The proportion of Y variance explained by the
linear relationship between X and Y = r2 = 0.64, or 64%. The proportion
of Y variability left unexplained = 0.36, or 36%.
Figure 6. Pearson r = 0.9. The proportion of Y variance explained by the
linear relationship between X and Y = r2 = 0.81, or 81%. The proportion
of Y variability left unexplained = 0.09, or 9%.
Figure 7. Pearson r = 1. The proportion of Y variance explained by the
linear relationship between X and Y = r2 = 1, or 100%. The proportion of
Y variability left unexplained = 0.