What's the right way to rescaling (min-max normalization)? Say I have a scale 2-12 and want to transform it into the range between 0-1. I would use the following formula:
$$ y={\frac {x-{\text{min}}(x)}{{\text{max}}(x)-{\text{min}}(x)}}$$
Sometimes I just saw this formula and other times they said explicitly to transform the observed data. But if my observed data doesn't include the min and/or max value, wouldn't it lead to a false transformation espcially when comparing it with other data (i.e. measuring the same on another scale) that has the min and max values observed or is already scaled between 0-1?
Further information as suggested in the comments:
In my special case I want to compare two measurements of democracy scores: Freedom House and Varities of Democracy. There are studies from Högström and Vaccaro. Högström compares Freedom House with Polity and Vaccano Freedom House and Varities of Democraacy inter alia.
Both are transforming there scales to 0-100.
Freedom House is measured between 1-7 (1 is best, 7 is worst) and is ordinal scaled. Min and max values are observed in the dataset.
Varities of Democracy is measured between 0-1 (0 is worst, 1 is best) and is interval scaled. Min and max values are not observed. In the papers from VDem and methodology part I haven't found if the min and max are only theoretical or just not observed yet.
Vaccaro uses the formula above but with only the observed values. The worst score therefore is 0 transformend and the best 100 but on the original scales the min and max is not achieved while in the case of Freedom House it is the case that min and max are achieved.
Also in the studies even if they mention it they just don't really care about scales, using OLS inter alia to compare them.

If there are in principle absolute minimum and maximum limits then it may make sense to use those instead of the observed minimum and maximum, particularly if you wish to compare two or more datasets with similar flavour.
There are several more or less obvious reservations here. If you don't know any such, you can't apply them. For example, in household data, the minimum number of children is clearly zero, but who knows what the maximum number is? One simple strategy is not to normalize at all in this circumstance.
Advice to normalize data in this way seems immensely more common in machine learning circles than in mainstream statistics. It's not that people in the latter group are unaware of the difficulties of mixing variables with quite different magnitudes and units of measurement. Rather, the art lies in recognizing and tackling that otherwise. For example, in regression, coefficients will have different units and magnitudes if predictors do, but those coefficients typically can or should bear substantive interpretations. In principal component analysis, a mixture of variables implies using the correlation matrix as the basis for calculation (not that solves all the problems of comparing results from similar but different datasets).
