# How to scale one dataset in relation to another?

I have two sets of numbers, that represent grades from two different groups of students, A and B, these grades go from 0 up to 1000. These sets have very different distributions, while set A has a lot of lower values (e.g. 220,270,300,...) set B has a lot of higher values (e.g. 620, 700, 800). Thus, a grade of 650 from a student of group A would be considered extremely high since not many students have a grade so high as that, however, that same grade in set B would be considered a low grade.

Therefore, I want to find a way to put these datasets on the same scale (or rescale one of them) so that I can make comparisons between them, comparisons that take into account the distribution of the grades.

This, way a score of 650 in A, which is considered extremely high, would be equivalent in the set B to 950 which is also an extremely high grade in B, for example.

I tried to normalize and use the z-score, but nothing seems to make sense.

PS: the numeric values are just illustrative.

• Have you tried quantiles?
– Dave
Commented Dec 17, 2020 at 2:39
• What was wrong with z-scoring? That would have been my first instinct.
– Izzy
Commented Dec 17, 2020 at 2:39
• @Dave what about quantiles exactly? I indeed want to match the quartiles, but I don't know how to do so. Thanks! Commented Dec 17, 2020 at 23:54
• @Izzy z-scoring does not take the frequencies into account, so even after scaling the data using z-score a low grade of 650 (which is considered high) from group B would still be much lower than a grade that is also considered high from group A, and therefore, the grades could not be properly compared. Commented Dec 18, 2020 at 0:04
• Z-scoring takes into account both the mean and spread of the data. If the scores in group A are low, then the mean score will also be low, so if 650 is much higher than the mean, it will result in a large z-score. If the mean score is high in group B, and 650 is much lower than that mean, then that will result in a small z-score (i.e. a very negative value). So, this should be a way to get the kind of comparison you want, unless your data is skewed in some way that makes the z-scores unreliable. Another idea you could look into (maybe this is what Dave means) is quantile normalization.
– Izzy
Commented Dec 18, 2020 at 3:54