Comparing distances between points in different dimensions

Question

I have pairs of points, each pair is in the same dimension, and I need to measure the distance between each pair:

For example:

Points:    dim:     distance:
x1,x2      2        0.8
x3,x4      8        0.2
x5,x6      12       0.5

I need to compare the distances (sort, find the smallest etc). I think they are not comparable due to the different dimensions. Is there a way to normelize the distances so I compare them?

Thanks !

Answer 1

Distances are measured in linear units no matter what the dimension. I can talk about two points in the plane being 1 unit apart and two points in space being 1 unit apart using the same unit. You can compare them without a problem. Whether it is useful to do so is another issue, but that depends on what your data looks like and what you are trying to do with it. Even in the same dimension, you might have very different scales in different directions. Think about if your two dimensions were age and annual income-the scales are very different.