I heard an interesting fact a while ago about how people draw a line through a cloud of points on a scatter plot. Usually, when calculating lines of best fit, we use the minimal the sum of squares of residuals. But if you draw it (to the best of your ability) by eye, you likely won't draw a squared error fit. I believe the exponent is slightly lower than two. This means we try to draw the line close to many points, and don't worry about a few extreme outliers.
Does anyone know what sort of error term we use intuitively? I realise the true objective function we minimise subconsciously probably has a more complex form, but this is just to get an indication for the form of the error penalties relative to least squares.
I'm writing a chapter for my thesis about regression analysis and I want to mention the effect of using different exponents on the error term. I thought would be nice to throw this bit of information in, too!
Edit 1: I've just posted this photo for my Facebook friends to do some line of best fit by eye. I have asked them to either send me a picture of the graph with line, or just the extreme y-values of the line. It would be great if we could all have a go! Link: http://www.freeimagehosting.net/ujx9h
Sorry for the annoying hosting site. Suggestions for good hoster welcome.
Edit 2: (Following discussion in comments.) For simplicity's sake, I am restricting the investigation to error functions of the form of sum of simple exponents of the residual.