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I've created a composite indicator which is now in a normalized form. I want to rescale this variable so that the mean in the baseline year is equal to 100. In this way, all observations above 100 will be better than the mean and the ones above 100 will be worse than the mean.

I don't want to rescale the variable with the min-max method, as this variable refers to evolution across time, and because it will need to be updated year after year.

Can anyone help me? Thanks!

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  • $\begingroup$ You mention only changing the mean, what you mean by scaling in here? $\endgroup$
    – Tim
    Commented May 11, 2021 at 19:06
  • $\begingroup$ @Tim, I want to keep the same distance to the mean for each point, but rescaling the "relation" to one where the mean is equal to 100, but I want the distance to this mean to be proportionally equivalent to the original one. I believe this is some sort of re-scaling not centering. But I might be mistaken. $\endgroup$
    – Gigi39
    Commented May 12, 2021 at 10:28
  • $\begingroup$ What do you mean by "distance" in here? If mean was 5 and one of the samples was 7, than it is 2 "units" from the mean, and you move all your data by +95, than the mean is 100 and the sample is 102, so it is still 2 "units" from the mean. $\endgroup$
    – Tim
    Commented May 12, 2021 at 10:31

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What you describe doesn’t sound like scaling but centering. If you subtract the mean from all the samples, their mean would become zero. To convince yourself, try it yourself

$$ 0 = \frac{1}{n} \sum_{i=1}^n \Big[x_i - \Big(\frac{1}{n} \sum_{i=1}^n x_i \Big)\Big] $$

If next, you add 100 to them, the mean would be equal to 100, the same logic applies as above.

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  • $\begingroup$ The question asks how to rescale the variable, not recenter it. The most general solution is of the form $x\to 100+\lambda(x-\bar x)$ where $x$ is the vector of data and $\lambda$ is any real number. Recentering is the specific case $\lambda=1,$ which is no rescaling at all. $\endgroup$
    – whuber ♦
    Commented May 11, 2021 at 18:29
  • $\begingroup$ @whuber the description mentions only that the mean needs to change, I’ve read it as using the word “re-scaling” incorrectly while OP seems to mean centering. $\endgroup$
    – Tim
    Commented May 11, 2021 at 18:49
  • $\begingroup$ Tim, when you choose to make an unusual interpretation of a term, consider asking the OP to clarify their intention first. The reference to the "min-max method" suggests to me that some amount of scaling is part of the intended operation. $\endgroup$
    – whuber ♦
    Commented May 11, 2021 at 18:53

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