## The Relationship Between the Possible Regression Lines Relating Two Quantities

When quantities
$a$
and
$b$
are related, we may try and find a straight line relationship between them. We have two possibilities.
We can find the 'regression line of
$a$
on
$b$
. This takes the form
$a=m_1 (b- \bar{b})+ \bar{a}$
where
$m_1 = \frac{s_{ab}}{s_{bb}}=\frac{\sum_i a_ib_i -\frac{\sum_i a_i \sum_i b_i}{n}}{\sum_i b^2_i -\frac{(\sum_i b_i)^2}{n}}$
and
$\bar{a}=\frac{\sum_i a_i}{n}, \: \bar{b}=\frac{\sum_i b_i}{n}$
.
Alternatively we can find the regression line of
$b$
on
$a$
. This takes the form
$b=m_2 (a- \bar{a})+ \bar{b}$
where
$m_2 = \frac{s_{ab}}{s_{bb}}=\frac{\sum_i a_ib_i -\frac{\sum_i a_i \sum_i b_i}{n}}{\sum_i a^2_i -\frac{(\sum_i a_i)^2}{n}}$
and
$\bar{a}=\frac{\sum_i a_i}{n}, \: \bar{b}=\frac{\sum_i b_i}{n}$
.
These regression line relates the same quantities, so we might expect the lines to be the same. THEY ARE NOT! The lines are related however. Both lines pass through the point
$(\bar{a}, \bar{b})$
$m_1, \: m_2$
$\frac{m_1}{m_2}=\frac{s_{aa}}{s_{bb}}$