1. Coefficient of Determination
AP Statistics

2. What is a 'Coefficient of Determination'
The coefficient of determination is a measure used in statistical analysis that assesses how well a model explains and predicts future outcomes. It is indicative of the level of explained variability in the data set. The coefficient of determination, also commonly known as "R-squared," is used as a guideline to measure the accuracy of the model.

3. BREAKING DOWN 'Coefficient of Determination'
The coefficient of determination is used to explain how much variability of one factor can be caused by its relationship to another factor. It is relied on heavily in trend analysis and is represented as a value between zero and one. The closer the value is to one, the better the fit, or relationship, between the two factors. The coefficient of determination is the square of the correlation coefficient, also known as "R," which allows it to display the degree of linear correlation between two variables.

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The correlation is known as the "goodness of fit." A value of one indicates a perfect fit, and therefore it is a very reliable model for future forecasts. A value of zero, on the other hand, would indicate that the model fails to accurately model the data.

5. Analyzing the Coefficient of Determination
The coefficient of determination is the square of the correlation between the predicted scores in a data set versus the actual set of scores. It can also be expressed as the square of the correlation between X and Y scores, with the X being the independent variable and the Y being the dependent variable.

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Regardless of representation, an R- squared equal to zero means that the dependent variable cannot be predicted using the independent variable. Conversely, if it equals one, it means that the dependent of variable is always predicted by the independent variable.

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A coefficient of determination that falls within this range measures the extent that the dependent variable is predicted by the independent variable. An R-squared of 0.20, for example, means that 20% of the dependent variable is predicted by the independent variable.

8. Video

9. There is a (Strength, Direction, Linear) relationship between (explanatory variable) and (response variable).
Example 1
What is the correlation coefficient of this data set?
(1, 0) (2, 3) (3, 2) (0, 9) (8, 3) (2, 5) (6, 1)

10. The Coefficient of determination, r², is:
Calculator steps to copy R value from table
(TI-Nspire) Click on r value from table. Ctrl C
Press Scratchpad button
Ctrl V
Enter
(You may have to delete the “=“ preceding the number)
Square this value
You should get….
A linear association between (explanatory variable) and (response variable) predicts (r² percent value) % of variability in (response variable).
Scratchpad
Square

11. Now find the Coefficient of Determination.
A linear association between X and Y predicts % of the variability in Y.
Example 2
What is the correlation coefficient of this data set?
(5, 4) (2, 3) (3, 2) (1, 2) (5, 3) (2, 1)
Now find the Coefficient of Determination.

12. What is the correlation coefficient of this data set?
Now Calculate and explain the Coefficient of Determination.
Example 3
What is the correlation coefficient of this data set?
(5, 8) (2, 6) (-1, 2) (1, 2) (6, 6) (-5, 7) (-2, 3)
Lets say that the explanatory variable (x) is degrees in temperature in February and the response variable (y) is number of flowers bloomed at the location come spring.
R is explained as: There is a weak, positive, linear relationship between temperature in February and the number of flowers that bloom at the location come spring.
R² is explained as: A linear association between Temperature in February and Number of flowers at the location come spring predicts 9.5% of variability in Number of flowers that bloomed at the location come spring.