1. Coefficient of Determination
Section 3.3

2. Coefficient of determination
r = correlation
r2 = coefficient of determination
What this means?
Tells us the proportion (or percent) of variability in y that is explained by the LSRL and variation in x.

3. Coefficient of determinationx 3 6 y 10 2
Scatterplot for data

4. Coefficient of determinationx y
𝒚− 𝒚 2 16 3 10 36 6 2 4
Squares of deviations about y-bar (4)

5. Coefficient of determinationx y
𝒚− 𝒚 𝟐 16 9 3 10 36 6 2 4
Squares of deviations about y-hat (0-3), (10-4), (2-5)

6. Coefficient of determinationx y
𝒚− 𝒚 𝟐 16 9 3 10 36 6 2 4 56 54
SST
SSE
SST = sum of squares of the deviations
SSE = sum of squares for error

7. Coefficient of determinationx y
𝒚− 𝒚 𝟐 16 9 3 10 36 6 2 4 56 54
SST
SSE
𝑟 2 = 𝑆𝑆𝑇−𝑆𝑆𝐸 𝑆𝑆𝑇
𝑟 2 = 𝑆𝑆𝑇−𝑆𝑆𝐸 𝑆𝑆𝑇 = 56−54 56 =0.0357
SST – SSE measures the amount of variation of y that can be explained by the regression line of y on x.

8. Coefficient of determination
𝑟 2 = 𝑆𝑆𝑇−𝑆𝑆𝐸 𝑆𝑆𝑇 = 56−54 56 =0.0357
3.57% of the variation in y is explained by least-squares regression of y on x.

9. Coefficient of determinationx 5 10 y 7 8
Scatterplot for data

10. Coefficient of determinationx y
𝒚− 𝒚 2 25 5 7 4 10 8 9

11. Coefficient of determinationx y
𝒚− 𝒚 𝟐 25 1 5 7 4 10 8 9

12. Coefficient of determinationx y
𝒚− 𝒚 𝟐 25 1 5 7 4 10 8 9 38 6
SST
SSE
𝑟 2 = 𝑆𝑆𝑇−𝑆𝑆𝐸 𝑆𝑆𝑇
= 38−6 38 =0.842
84.2% of the variation in y is explained by least-squares regression of y on x.

13. Coefficient of determination
Please note:

14. Comparing r2 Values, Ex. 3.12, p. 162
Distinction between explanatory and response variables is essential.
Scatterplot of data that played a central role in the discovery that the universe is expanding. Distances from earth of 24 spiral galaxies and the speed at which these galaxies are moving away from us. There is a positive linear relationship, r = , so that more distant galaxies are moving away more rapidly. Astronomers believe that there is a perfect linear relationship and that the scatter is caused by imperfect measurements.
The two lines on the plot are the two least-squares regression lines. Regression line of velocity on distance is solid, distance on velocity is dashed. Their correlation will be the same, but the regression lines will be different.

15. Please remember
𝑟= 𝑟 2
If you are given the percent of variation, you can find the correlation. Same for correlation  variation.

16. Practice Problems
Exercises 3.42, 3.43