[dschiptsov]

To make it positive and to amplify it (as a side-effect).

BTW, "error" is a misleading term - it communicates some fault, at least in the common sense. Distance would be much better term.

So, "squared distance" makes much more sense, because negative distance is nonsense.

[tonyedgecombe]

Well it will only amplify values > 1.

[esrauch]

That's not correct. Even though the magnitudes of the value in isolation shrinks, the relative magnitudes are still amplified which is what matters.

Consider values 1/2 and 1/4: in the original space it's double but in the squared space it becomes 1/4 and 1/16 so the difference is 4x. Also relevantly if you compare eg 0.9 and 1, the gap between them is amplified after squaring.

[tonyedgecombe]

Those values aren't compared individually, they are summed to calculate the deviation, the result of that sum will be reduced if the values are < 1.

[klodolph]

That's a compelling case for why we should not use "distance", because distance cannot be negative, but the error term can.

Just look at bog-standard linear regressions, say Y_i = m X_i + b + ε_i. It makes no sense to call the ε_i terms "distance".