# 3.21 FAQ-247 Why is R-Square greatly different when intercept is fixed in linear fit?

Last Update: 12/26/2018

Why is R-Square greatly different when intercept is fixed in linear fit?

It is because the R-Square is calculated with equation below:

$R^2=\frac{SSR}{TSS}=1-\frac{RSS}{TSS} \,\!$

where SSR is the sum of squares due to regression, TSS is the total sum of squares, and RSS is the residual sum of squares. For TSS:

• $TSS=\sum_{i=1}^n(y_i-\overline{y})^2 \,\!$, when intercept is not fixed
• $TSS=\sum_{i=1}^n y_i^2 \,\!$, when intercept is fixed

Please look at the detailed explanation below why we use uncorrected sum of squares for TSS calculation when intercept is fixed.

##### When intercept is fixed

When intercept is fixed in linear fit, it follows the relation below:

$\sum_{i=1}^n y_i^2 = \sum_{i=1}^n (y_i-f(x_i))^2 + \sum_{i=1}^n (f(x_i))^2$

Then TSS and SSR need be redefined, and RSS is unchanged.

$TSS = \sum_{i=1}^n y_i^2$
$SSR = \sum_{i=1}^n (f(x_i))^2$

And the coefficient of determination (R-Square) is redefined as follows:

$R^2=\frac{SSR}{TSS}=1-\frac{RSS}{TSS}=1-\frac{\displaystyle \sum_{i=1}^n (y_i-f(x_i))^2}{\displaystyle \sum_{i=1}^n y_i^2}$