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.

An alternative is to not use normalization but rather, for ex., standardization, which uses the mean and SD of the variable to scale. Then it becomes a question of which statistic is more reliable/efficient, the min/max or the mean/SD? Yes this is not the same as normalization since the output is no longer in [0,1] but it also has nice interpretation. If the variables in question do allow extreme values (someone buying 100 houses) then some robust normalization might be even better.
