# Statistic not sensitive to coefficients

I'm looking for some kind of function I can compute on a bunch of numbers. The idea is that if I increase all numbers by some coefficient (say 2) the value of this should not change.

e.g.

f(2,4,6,9)= f(4,8,12,18)

Edit: I know I can just divide all the numbers by the largest and then compute any other statistic. I'm more looking for some function that doesn't care about the actual numbers but the relative differences betwen them.

The problem with just dividing everything by the largest is that it would make this extremely sensitive to outliers or bad data

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This question is too vague to answer as it stands. Is there something you do want the statistic to be able to detect? In addition, in your example, you are not increasing all numbers by a constant, but multiplying them by a constant. This makes what you are looking for hard to figure out. – gung Jul 26 '12 at 17:36
Sorry for the confusion. That's why i said "constant coefficient". – Dez Udezue Jul 26 '12 at 17:49
Fair enough. You want this statistic to not be sensitive to multiplying by a constant coefficient; what do you want it to be sensitive to? – gung Jul 26 '12 at 17:56
Hi @DezUdäzue, I think it would help a lot to know what exactly you plan to do with this statistic. Right now I'm still a little confused about what exactly you want and why. It seems like correlation might fit the description you're talking about, since it is invariant to multiplicative (and additive) changes, but I'm not sure. Also, note that standard deviation is sensitive to multiplicative changes, by the way. – Macro Jul 26 '12 at 18:04
This question is asking for a homogeneous function of degree 1. These are easy to come by: pick any probability density $d\mu$ on the positive real line and any function $f$ for which $t\to f(t x_1, \ldots, t x_n)/t$ is integrable with respect to $\rho$, and define $f^{*}(\mathbf{x}) = \int_0^\infty f(t\mathbf{x})d\mu(t)/t$. This gives us extraordinary freedom to focus on what matters: namely, what is the purpose of $f$? – whuber Jul 26 '12 at 19:58
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## 1 Answer

Like the commenters, I do wonder what you are trying to do. One thing you could do is take the log of all the numbers, then take the standard deviation, or the range, or other things. E.g

f(2,4,6,9)= f(4,8,12,18)

becomes f(0.69, 1.39, 1.79, 2.2) = f(1.39, 2.07, 2.48, 2.89)

and the range of each is 1.5 and the sd of each is 0.64.

That fulfills your request, but it's hard to know if it is what you want.

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I'm not sure whether to upvote this. I was thinking of exactly the same thing, but I can't tell if it's an answer to the question b/c I really can't understand what the goal is here (not that that's your fault). – gung Jul 26 '12 at 19:37