# Best formula to normalize non linear scores to scale of 1-100

I have lists of scores, which can be very non linear. For example:

20332
18000
599
39
19
2


I need to normalize these scores to a 1-100 scale so I can do some comparisons with other score sets which may be in different scale, such as:

294
39
8
5
3
1


The thing is that, due to the non linearity, I want to give the high outliers less relative weight. So the scores above will normalize to something closer to a rank. For example:

100
60
18
12
7
1


What is the best way to do normalization in such a case?

• A simple method is to just compute the empirical CDF & rescale it. Here's an example: stats.stackexchange.com/a/380284/22311
– Sycorax
Jun 15, 2022 at 18:17
• This sounds mighty arbitrary. If you would like it to have some meaning, please explain what these scores might be measuring and what the comparison is supposed to reflect.
– whuber
Jun 15, 2022 at 18:31
• I’m with whuber that we will need a lot more detail to be able to address this in a way that gives meaning to your normalized scores.
– Dave
Jun 15, 2022 at 18:39

Firstly, are you certain you want to transform to 1-100, as opposed to 0-100? (The latter scale would be much more sensible in most contexts.) Assuming you actually want to transform to the latter scale, if your data is $$\mathbf{x}=(x_1,...,x_n)$$ one option would be to define the empirical distribution function:
$$F_\mathbf{x}(x) = \frac{1}{n} \sum_{i=1}^n \mathbb{I}(x \leqslant x_i),$$
$$y_k = 100 \times\frac{n F(x_k) - \alpha}{n+1-2\alpha},$$
where $$0 \leqslant \alpha \leqslant 1$$ is a shifting parameter used for the transformation (a good value here is $$\alpha = \tfrac{1}{2}$$). The values $$y_k$$ are essentially just scaled sample quantiles of the corresponding values in $$\mathbf{x}$$. If you were to use a transformation of this kind then comparisons of points across different data sets would essentially be comparing the sample quantiles of corresponding points.