1) squared error: (actual-estimate)^2 --> best estimate is the mean

2) absolute error: |actual-estimate| --> best estimate is the median

I have two questions.

1) Why do people use the squared error method? The absolute error method makes much more intuitive sense. You get the difference between the actual and the estimate. Plain and simple. If you square the difference, then won't you get "warped" values depending on the size of the difference?

2) This also got me thinking about what is "expected value." Expected value is defined as the mean. However, the best estimate under the absolute error loss function is the median. So is the "expected value" the median? I would very much appreciate it if someone can help me clarify my thinking. Thanks.