R-Squared vs Correlation

1.R-Squared:

The coefficient of determination, denoted R² or r² and pronounced “R squared”, is the proportion of the variance in the dependent variable that is predictable from the independent variable(s).

It is a stastical used in the context of statistical models whose main purpose is either the prediction of future outcomes or the testing of hypothetes, on the basis of other related information. It provides a measure of how well observed outcomes are replicated by the model, based on the proportion of total variation of outcomes explained by the model.

There are several definitions of R² that are only sometimes equivalent. One class of such cases includes that of simple linear regression where r² is used instead of R². When an intercept is included, then r² is simply the square of the sample correlation coefficient (i.e., r) between the observed outcomes and the observed predictor values. If additional regressors are included, R² is the square of the coefficient of multiple correlation. In both such cases, the coefficient of determination normally ranges from 0 to 1.

The most general definition of the coefficient of determination is: SST = SSR+SSE

{\displaystyle R^{2}\equiv 1-{SS_{\rm {res}} \over SS_{\rm {tot}}}\,}

2.Correlation

Pearson’s correlation coefficient is the covariance of the two variables divided by the product of their standard deviations. The form of the definition involves a “product moment”, that is, the mean (the first moment about the origin) of the product of the mean-adjusted random variables; hence the modifier product-moment in the name.

Pearson’s correlation coefficient when applied to a population is commonly represented by the Greek letter ρ (rho) and may be referred to as the population correlation coefficient or the population Pearson correlation coefficient. Given a pair of random variables {\displaystyle (X,Y)}, the formula for ρ^[7] is: