A mean in which values are weighted by themselves 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?

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: $f_i=x_i.$
