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.
 A: 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).
A: 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.
A: If you just want to have a scale in $[0,1]$ to allow for a straight interpretation that, say, 55% of the maximum was achieved, the absolute min and max should be used rather than the observed. This may be helpful for data presentation. This would not be achieved by using min and max in the data as then future data could lead to values outside the $[0,1]$ range.
This however does not mean that different scales constructed in different ways that are normalised in this way can be compared, as one may have a maximum that corresponds to an ideal state that in reality can be hardly ever achieved, and existing observations may therefore end up with a max of 0.8 or so, whereas another scale may be constructued so that real existing democracies end up with a value of 1. If you have different indexes you can plot them against each other (after having transformed them both to $[0,1]$) to see this. In such a case I don't think straight comparability can be achieved by any kind of mathematical normalisation, as different scales by construction measure at least slightly different things. One can study and try to explain the differences, but this requires subject matter knowledge and not only numerical comparison (although it could be of interest if there is a very strong connection between two indexes; they may turn out to be approximately equal, or may show a regression slope other than 1 between them, or even nonlinearity).
Standardisation/normalisation is also often used when aggregating information from different scales, like constructing a single new scale out of several old ones. In that case it can be argued that data dependent normalisation could be more appropriate, because if you want to define a formal rule without a tedious discussion of the specific indexes (that may not imply how to normalise them anyway), the variation in the observed data is informative. In this case the min/max observed in the data may be used, arguing for example that if the realistically achievable maximum of one index is 0.8 (even though it theoretically goes up to 1) whereas the other one in reality reaches values up to 1, it may be appropriate to reflect the variation of the data better if 0.8 of the first index is transformed to 1 (correspondingly if the index range is 2-12 or whatever). In that case however one may want to use more information about the distribution than just the minimum and the maximum, and standardisation to mean 0 standard deviation 1 (or robust alternatives) may be better, see the answer of @user2974951 (whether it actually is better is hard to say and depends on a number of considerations that partly again refer to the subject matter).
Also keep in mind that many statistical methods are invariant or equivariant against linear transformations, meaning that the results, if interpreted correctly, do not depend on whether and how you transform your variables. Invariance means that results do not change, equivariance means that they change in mathematically appropriate ways and can be transferred from one transformation to another without actually repeating the analysis. This holds for example for correlations and PCA computed on correlations (invariant against linear transformations of the individual variables), and linear regression (equivariant) if you know how (and how not) to interpret the coefficients. Certain other techniques (such as $k$-means clustering) are not equivariant and require comparable scales and therefore normalisation/standardisation at least as long as the different amounts of variation in different variables do not indicate their importance for the task of interest.
