The proportion of the variation in the response variable that is explained by the regression model. Show
If there is a perfect linear relationship between the explanatory variable and the response variable there will be some variation in the values of the response variable because of the variation that exists in the values of the explanatory variable. In any real data there will be more variation in the values of the response variable than the variation that would be explained by a perfect linear relationship. The total variation in the values of the response variable can be regarded as being made up of variation explained by the linear regression model and unexplained variation. The coefficient of determination is the proportion of the explained variation relative to the total variation. If the points are close to a straight line then the unexplained variation will be a small proportion of the total variation in the values of the response variable. This means that the closer the coefficient of determination is to 1 the stronger the linear relationship. The coefficient of determination is also used in more advanced forms of regression, and is usually represented by R2. In linear regression, the coefficient of determination, R2, is equal to the square of the correlation coefficient, i.e., R2 = r2. Example The actual weights and self-perceived ideal weights of a random sample of 40 female students enrolled in an introductory Statistics course at the University of Auckland are displayed on the
scatter plot below. A regression line has been drawn. The equation of the regression line is  The coefficient of determination, R2 = 0.822 This means that 82.2% of the variation in the ideal weights is explained by the regression model (i.e., by the equation of the regression line). Curriculum achievement objectives reference $\begingroup$ I know that the r^2 value for the data is 0.9832. Is there a way to use that value to find the percent variation in Y is explained by X? Or do I need to use the data given to me? asked Nov 15, 2015 at 5:00
$\endgroup$ 3 $\begingroup$ $r^2*100$ is the percentage of variance explained by $X$. When you regress $Y$ on $X$ you get $\hat{Y}=a+r\frac{s_y}{s_x}X$ And $Var(\hat{Y})=r^2Var(Y)$ from the above equation. So $\frac{Var(\hat{Y})}{Var(Y)}*100=r^2*100$ is the percentage of variance explained by x. answered Nov 15, 2015 at 5:07
$\endgroup$ What percentage of the variation in Y can be explained by the corresponding variation in X and the least squares line what percentage is unexplained?Answer and Explanation: Therefore, the percentage of the variation in y that can be explained by the corresponding variation in x and the least-squares line is 94.64% . The percentage which is unexplained is 5.36% i.e. (100%−94.64%) .
What percentage of the variation in Y has been explained by the regression?In linear regression, the coefficient of determination, R2, is equal to the square of the correlation coefficient, i.e., R2 = r2. This means that 82.2% of the variation in the ideal weights is explained by the regression model (i.e., by the equation of the regression line).
What percentage of the variance in one variable is explained by the other?Essentially, an R-Squared value of 0.9 would indicate that 90% of the variance of the dependent variable being studied is explained by the variance of the independent variable.
Is the percent of variation in the dependent variable that is explained by the regression equation?The coefficient of determination is denoted as r2 , which can be obtained by squaring the correlation coefficient. By definition, the coefficient of determination is the percentage of the variance in the dependent variable that can be described by the independent variables (or regression equation).
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