The short answer to your least squares question is
'because Matheron was a least squares person' ;-)
There are three basic schools of statistics:
(1) least squares (sometimes known as frequentist)
which probably includes the majority of
non-statisticians doing statistics. The least squares
approach might be paraphrased as "closest to the real
answer on average". It is based on the concept of
approximately Normal 'errors'
(2) maximum likelihood. This could be paraphrased as
'that solution from which the samples are most likely
to have come'. This yields much more general but not
always unbiassed answers. For example, the maximum
likelihood estimator of the variance is divided by 'n'
not 'n-1' as in least squares. Maximum likelihood
demands a pretty fair knowledge of the underlying
distribution of the samples, not the errors as such.
(3) bayesian estimation. Based somewhat on a maximum
likelihood method, you can build in your own prior
knowledge about the situation to affect the final
answers. (1) and (2) above rely solely on hard data.
This is, of course, a massive over-simplification but
serves, I think, to emphasise why there are many
different answers to the same problem and why you have
to define what you mean by "best" before you get the
"best answer". I try to cover these concepts in
non-rigorous terms in all my teaching.
It is also true that the mathematics is much simpler
if you use variances than if you use any other
function of differences.
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