if r = 0.6, then how much part of total variation of the dependent variable can be explained by regression model ?
Answers
Step-by-step explanation:
R² = the proportion (fraction) of the total variance that is "explained" by the regression model.
The variance of the data is the average squared distance of the values to their mean.
Applying a regression model means that the "mean" (=the model prediction) will depend on some predictors(s). The distances to these "means" are called residuals, and the average suqared residuals is the residual variance.
The residual variance (against the model predictions) will always be smaller than the variance (against a constant mean value). This reduction in the variance is said to be "explained" by the model, and R² is this reduction relative to (=divided by) the variance.
Example:
x = 1,2,3,4
y = 2,5,6,9
variance = 8.333*
linear regression model: y = 2.2*x + 0.0
predicted values = 2.2, 4.4, 6.6, 8.8
residuals = -0.2, 0.6, -0.6, 0.2
residual variance = 0.267
reduction in variance due to the model = 8.333 - 0.267 = 8.067
R² = 8.067 / 8.333 = 0.986
* the variance shown is the "sample variance" (what is usually reported as "variance" by most software), that is the sum of the squared diveiations divided by n-1 (here: 4-1=3), and not by n. This denominator is the calculated in the same way for the residual variance. Therefore this denominator will cancel out in the calculation of R² anyway and it does not matter what kind of variance (sample or population variance) is used. One could thus say that R² is the reduction in the "sum of squared deviation" explained by the model relative to the "sum of squared deviations" of the data.