I am a conceptual person, so it helps me to understand if I know why we are dong things. In a standard deviation we square the mean and then add it all up. My question is, why do we square the difference scores? Why not simply take the sum of the absolute values of the differences between each score and the mean?
Well we don't square the mean, we square the difference between each observation and the mean.
The standard deviation describes the average variation around the mean. Conceptually its a measure of how tightly values are centred around the mean.
Two reasons we square each difference is to make them all positive and then to make the bigger differences stand out. This helps bound the smaller differences.
So far as I know, the primary reason why squares were chosen instead of absolute values is that squares are easier to work with analytically (because they are differentiable). But you can make a good argument for absolute values.
A paper for the "Department of Educational Studies", and the first sentence indicates the author can safely be ignored:Originally Posted by tom@ballooncalculus
"This paper discusses the reliance of numerical analysis on the concept of the standard deviation, and its close relative the variance."
Then we have the statement:
"The main reason that the standard deviation (SD) was created like this was because the squaring eliminates all negative deviations, making the result easier to work with algebraically" - rather than the variance is a smooth function so allowing the methods of calculs to be deployed. In fact there is not a mention of calculus in this piece but a lot about algebra, makes me think the author is not that familiar with calculus and how smoothness helps.
I could go on ...
CB
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