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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

According to your formula (d = |difference | and error = d/ 10), the possible values of error must lie between 0 and 1. Thus, we can use sinθ to represent the situation, i. e. sinθ = d/10.

In fact, you can build a “d vs θ” reference table [like when d = 1, the deviation is arcsin(0.1) = 5.74 degrees etc.]. Then, your “degree of the difference between their choices” is simulated by “degree (of an angle)”.

Another good thing about this system is that the guess need not be restricted to integers lying between 0 and 10.

As for the degree of similarity, you can use cosθ to represent it. But the formula need to be changed to $\sqrt (1 – d^2/(10^2))$ because $cosθ = sqrt(1 – sin^2θ)$.

Furthermore, you can redefine your formula as:-

D = first number chosen – second number chosen;

True Error = D/10; and

Sinθ = D/10

When θ lies between 0 and 90 (sinθ is positive), then you know that the first is greater than the second. Otherwise, the negative result is represented by an angle lying between 180 and 270 (or between 270 and 360).

As for the similarity, the above formula still applies except that there is no negative value and hence the result can only be represented by an angle lying between 0 and 90.

One can argue that the distinction between positive and negative similarities is ignored.

If still not happy, we can ‘borrow’ the negative sign from above and use the formula $cosθ = – sqrt(1 – sin^2θ)$

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