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.
- The coefficient of determination ranges from 0 to 1.
- An R2 of 0 means that the dependent variable cannot be predicted from the independent variable.
- An R2 of 1 means the dependent variable can be predicted without error from the independent variable.
- An R2 between 0 and 1 indicates the extent to which the dependent variable is predictable. An R2 of 0.10 means that 10 percent of the variance in Y is predictable from X; an R2 of 0.20 means that 20 percent is predictable; and so on.
The coefficient of determination (R2) for a linear regression model with one independent variable is:
\[R^{2} =\left ( \frac{1}{N} \frac{ \sum \left ( \left ( x_{i}-\overline{x }\right ) \left ( y_{i} -\overline{y}\right )\right )}{\sigma _{x}\sigma _{y}} \right )^{2} \]
where N is the number of observations used to fit the model, Σ is the summation symbol, xi is the x value for observation i, x is the mean x value, yi is the y value for observation i, y is the mean y value, σx is the standard deviation of x, and σy is the standard deviation of y.
For calculation
Let value of xi = X and yi = Y \[\sigma _{x} \] = a , \[\sigma _{y} \] = b , \[\overline{x} \] = x , \[\overline{y} \]= y
