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Why must the R-Squared value of a regression be less than 1?

1 Answer

Oct 27, 2015

$S S R e g \leq S S T$ $SSReg le SST$

Explanation:

Note that $R^{2} = \frac{SSReg}{S S T}$ $R^2= ("SSReg")/(SST)$ where SST= SSReg +SSE and we know that sum of squares are always $\geq 0$ $ge 0$ .

So $S S E \geq 0$ $SSE ge 0$
$\Rightarrow S S R e g + S S E \geq S S R e g$ $implies SSReg +SSE ge SSReg$
$\Rightarrow S S T \geq S S R e g$ $implies SST ge SSReg$
$\Rightarrow \frac{S S R e g}{S S T} \leq 1$ $implies (SSReg)/(SST) le 1$
$\Rightarrow R^{2} \leq 1$ $implies R^2 le 1$

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