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I wonder what are the ways for measuring the degree of differences between two numbers on a given scale. For example, if two persons are asked to each choose a number between $0$ to $10$, how can we measure the degree of the difference between their choices (e.g. $3$ and $9$)?

A plausible way is to normalize the difference by the scale, so for the example above it would be $\frac{\left|3-9\right|}{10}$. Thus the maximum degree of difference here is $1$, whereas the minimum is $0$. Conversely, if we want to measure the level of similarity, we can do $1-\frac{\left|difference\right|}{scale}$ here.

Is there any other alternative/better way? Thanks.

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What are you trying to accomplish with your measurement? – Austin Mohr Sep 10 '11 at 0:04
@Austin Mohr: like in the example above, one possible scenario is to compare how different/similar two persons' choices are given the same scale. – skyork Sep 10 '11 at 1:07
Just using absolute value measures this just as well. For your application, what is the purpose of dividing by the length of the scale? – Austin Mohr Sep 10 '11 at 1:10
dividing by the length of the scale gives the degree of difference/similarity. for example, a difference of 3 on a scale of 10 may be more pronounced than the same difference on a scale of 100. – skyork Sep 10 '11 at 3:10

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