The Coefficient of Determination (College Board AP® Statistics)

Coefficient of determination

What is the coefficient of determination?

• On a scatterplot, if all data points lie exactly on the least-squares regression line then there is a perfect linear relationship

  • Any variations (differences) seen in the -coordinates of the data points (the response variable) would be due to corresponding variations in the -coordinates of the data points (the explanatory variable)

    • Variations in the response variable are said to be fully explained by the linear relationship with the explanatory variable

• In reality, data points do not lie exactly on the least-squares regression line

  • Variations in the response variable consist of:

    • variations that are explained by the linear relationship

    • and other variations that are not explained by the linear relationship

• The coefficient of determination is the proportion of the total variation in the response variable () that is explained by the linear relationship with the explanatory variable

How do I calculate the coefficient of determination?

• The coefficient of determination for a least-squares regression line is the square of the correlation coefficient,

  • Coefficient of determination =

    • where

• Values of close to 1 indicate that the least-squares regression line is a very good model for the data

How do I interpret the coefficient of determination in context?

• You can interpret the coefficient of determination, , in context in the following way:

  • First change the decimal into a percentage

  • Then use the sentence:

    • "The coefficient of determination indicates that [percentage] of the total variation in the [-variable in context] is explained by the linear relationship with the [-variable in context]"

• Note that the remaining percentage is the percentage that is unexplained by the linear relationship

How do I find the correlation coefficient from the coefficient of determination?

• To find the correlation coefficient, , from the coefficient of determination,

  • take the square root of

  • then check the least-squares regression line to see if the slope is positive or negative

    • If positive, the correlation coefficient is the positive square root

    • If negative, the correlation coefficient is the negative square root