# Extrapolation error of linear regression lines for a two-cluster data set

I am studying on some linear regression problems using the least-squares method and stumbled upon a problem regarding the error when extrapolating far-away datapoints.

About the problem: For a point $$y$$ given by best fitting line, described by

$$y = ax + b$$,

where $$a$$ and $$b$$ are the coefficients achieved by applying the least-squares-method to a given data set, we know that generally, the error or the variance for $$y$$, will increase, the further we move away from our actual data, while the variance will be minimal at the mean position of $$x$$ of our given data set. This is so far totally clear for me.

But now I wondered:

Say, I have a given data set that consists of two clusters, with many datapoints around a very small negative position $$j$$ and another cluster at a very big positive position $$k$$.

Wouldn't the overall error of the best fitting line be decreased now? Will our line, for example for interpolating points in between, be a better fit with this kind of measurement, than a line that comes from only one data cluster int the 'middle' of the two-cluster version?

I hope I explained my problem sufficiently and I am excited for your suggestions!