Given I have multiple values in the range of [0,1] and I want to condense them into a single value (again, in the range [0,1]). What's the semantic difference of the following two approaches?
-
Arithmetic mean:
- Sum up the values
- Divide the sum by the number of values
Example: $\frac{0.1 + 0.7 + 0.4}{3} = 0.4$
-
Normalized vector length:
- Interpret the values as being vector coordinates in a n-dimensional space
- Calculate the vector length
- Normalize the length by dividing it by the maximum possible vector length, i.e., a vector with all coordinates being 1.0
Example: $\frac{\sqrt{0.1^2+0.7^2+0.4^2}}{\sqrt{1.0^2+1.0^2+1.0^2}}=0.469…$
I understand the differences in calculation, but can't make sense of why the results differ. Which approach should I use to get a sensible summary of the values?
Best Answer
I realized that my way of calculating the "Normalized vector length" average is algebraically equivalent to the Quadratic Mean (https://en.wikipedia.org/wiki/Average#Summary_of_types), so I now have a term to look for further information on the subject.
In short, the Quadratic Mean emphasizes higher values, which the arithmetic mean doesn't.