A mean in which values are weighted by themselves

Question

I'm looking for the name of the following expression:

$$ \frac{\sum_{i} x_i^2}{\sum_i x_i} $$

This can be understood as a type of average in which each value is weighted by itself – in other words, a type of average in which near-zero values are ignored. Here are some examples:

• $[0, 0, 100, 0, 100] \rightarrow 100$
• $[10, 10, 10, 10, 10] \rightarrow 10$
• $[1000, 100, 0, 0, 0] \rightarrow 918.18$

Does this type of mean have a name? What are some typical use cases?

Answer 1

Weighted aM is given as $$\bar x=\frac{ \sum_{i=1}^n f_i x_i}{\sum_{i=1}^n f_i}$$ In your case it a particular weighted AM, where the frequency of the variables $x_i$ is $x_i$ itself: $f_i=x_i.$

Answer 2

According to this article, you could call it a "self-weighted mean", and for a set of positive values it should actually lie between the arithmetic mean and the maximum.

Answer 3

We can think of the expression

$$ \frac{x^n}{\sum_i x_i^n} $$

as a polynomial approximation of the softmax weighting

$$ \frac{e^{x/T}}{\sum_i e^{x_i/T}}. $$

One difference is that $e^0 = 1$ while $0^n = 0$. The other is that the exponential grows much faster, especially when $T$ is small; similarly, larger $n$ will increase the bias towards the maximum and away from the average.

We can therefore frame the expression given in the question as a polynomial approximation of a softmax-weighted mean with $n=1$:

$$ \sum_j \frac{x_j^1}{\sum_i x_i^1} x_j $$