Arithmetic mean vs. normalized vector length in multidimensional space

average

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:

    1. Sum up the values
    2. Divide the sum by the number of values

    Example: $\frac{0.1 + 0.7 + 0.4}{3} = 0.4$

  • Normalized vector length:

    1. Interpret the values as being vector coordinates in a n-dimensional space
    2. Calculate the vector length
    3. 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.

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