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How do you show that if you only have two data points $(x_1, y_1)$ and $(x_2,y_2)$ then the best fit line given by the method of least squares is the line through $(x_1,y_1)$ and $(x_2,y_2)$

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  • $\begingroup$ Simple: the residuals are zero for the best fit line, and zero is the smallest sum you can get with the residuals. $\endgroup$ – Parcly Taxel Mar 13 '18 at 6:08
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If $x_1 \ne x_2$, then we can construct a line linking them and since the points lie on the line, there is no error.

However, precaution is the case where $x_1 = x_2$. In that case we want to minimize $$(\hat{y}-y_1)^2+(\hat{y}-y_2)^2$$

and the minimal is attain as long as $\hat{y}= \frac{y_1+y_2}2$, that is the best fit line has to passes through $(x_1, \frac{y_1+y_2}2)$.

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