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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!

Thanks.

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.

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closed as off topic by Gerry Myerson, tomasz, Noah Snyder, userNaN, Jason DeVito Oct 8 '12 at 15:55

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What do you mean by "I believe the exponent is slightly lower than two."? –  Patrick Li Sep 29 '12 at 19:31
    
Given a clean linear distribution of points with one random bump a human would probably draw a line to fit the linearly distributed points perfectly, which is not least-anything. In general case, you could ask a bunch of people to draw intuitive lines and deduce that term. I doubt you'll get a better answer. –  Karolis Juodelė Sep 29 '12 at 20:04
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The only question asked here is, "Does anyone know what sort of error term we use intuitively?" This is, in my opinion, not a Mathematics question. Rather, it's a question to be answered by searching the psychology literature (or by posting it somewhere where people familiar with the psychology literature hang out). –  Gerry Myerson Sep 30 '12 at 10:12
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Mosteller, F., Siegel, A. F., Trapido, E. and Youtz, C. (2006) Fitting Straight Lines by Eye, in Exploring Data Tables, Trends, and Shapes (eds D. C. Hoaglin, F. Mosteller and J. W. Tukey), John Wiley & Sons, Inc., Hoboken, NJ, USA. doi: 10.1002/9781118150702.ch6 –  Gerry Myerson Sep 30 '12 at 10:16
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@GerryMyerson Thanks for the link! That pretty much is what I was looking for. And you're right. It's more a question for mathematical modelling and cognitive science. If you post that as an answer I'll accept. –  dynamo Sep 30 '12 at 12:12

1 Answer 1

up vote 2 down vote accepted

Mosteller, F., Siegel, A. F., Trapido, E. and Youtz, C. (2006) Fitting Straight Lines by Eye, in Exploring Data Tables, Trends, and Shapes (eds D. C. Hoaglin, F. Mosteller and J. W. Tukey), John Wiley & Sons, Inc., Hoboken, NJ, USA. doi: 10.1002/9781118150702.ch6

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Figure 6-3 shows a comparison of least squares, least absolute and fit-by-eye for a set of 15 points, where the fit-by-eye is between the two others. –  dynamo Oct 1 '12 at 8:37

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