Regression line (best fit) passes through $(\bar{x},\bar{y})$. $e_1,e_2,\ldots,e_6$ represent th

Question

Regression line (best fit) passes through $(\bar{x},\bar{y})$.
$e_1,e_2,\ldots,e_6$ represent the $y$ errors.
Aim to minimise the values of the errors,
$\sum e_k^2=e_1^2+e_2^2+\cdots+e_6^2$
Regression line of $y$ on $x$ obtained.

Regression line (best fit) passes through $(\bar{x},\bar{y})$.
$e_1,e_2,\ldots,e_6$ represent the $x$ errors.
Aim to minimise the values of the errors,
$\sum e_k^2=e_1^2+e_2^2+\cdots+e_6^2$
Regression line of $x$ on $y$ obtained.

Important notes:

With nonzero standard deviations and $0<|r|<1$, the $x$-on-$y$ line is steeper in absolute slope when both are plotted with $x$ horizontal and $y$ vertical. They coincide when $|r|=1$. When $r=0$, the lines are $x=\bar{x}$ and $y=\bar{y}$.
The point$\left( \bar{x},\bar{y} \right)$ lies on both the regression lines of “$y$ on $x$” and “$x$ on $y$”.

Accepted Answer

The $y$-on-$x$ regression minimises squared $y$-residuals; the $x$-on-$y$ regression minimises squared $x$-residuals. Both pass through $(\bar{x},\bar{y})$. Choose according to the variable being predicted. Interpolation stays within the observed range; strong correlation alone does not guarantee reliable extrapolation.

The Method of Least Squares

The Method of Least Squares

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