Solution 11748: Explanation of r^2 and R^2 on the TI-83 Family or TI-84 Family Graphing Calculator.
Why is R^2 computed for some regressions on a TI-83 family or TI-84 family graphing calculator?
The following information details when r2 and R2 are computed using a TI-83 family or TI-84 family graphing calculator.
When regression models are executed, a TI-83 family or TI-84 family graphing calculator computes and stores diagnostics values for r (correlation coefficient) and r2 (coefficient of determination) and R2 (coefficient of multiple determination).
r2 expresses the proportion of the total variation in the values of the variable Y that can be accounted for or explained by a linear relationship with the values of the random variable X. Thus, a correlation of 0.6 means that 0.36 or 36% or the total variation of the values of Y in our sample is accounted for by a linear relationship with the values of X. R2 is used when independent variables are created from the regression. This quantity merely indicates what proportion of the total variation in the response Y is explained by the fitted model. (Probability and Statistics for Engineers and Scientists, 4th edition, Walpole and Myers, pp. 393 and 423)
The reason that R2 is used instead of r2 for quadratic (QuadReg), cubic (CubicReg), and quartic (QuartReg) regressions is because, for each of these, we are artificially forming more than one independent variable. For example, for quadratic regression, the independent variables are x and x2, for cubic, x, x2, x3. In other words, we are really doing multiple linear regressions for these three "nonlinear" regressions.
On the other hand, r2 is used for linear (LinReg), logarithmic (LnReg), power (PwrReg), and exponential (ExpReg) regressions because each of these have only one independent variable.