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This is almost definitely a case for "not knowing what to search for", but am interested in how to refer to the following:

For a set of $n$ reals (say non-negative), I'm interested in characterizing how "concentrated" the values are in the top few (a la Pareto $80/20$). This could be for instance for thinking about percent of total variance of top few principal components.

Is there a way to refer to this function? I'd want a single value, similar to AUROC in statistics.

Context: I'm interested in optimizing over this function as a tradeoff, similar to penalties on norms like $L^1$ or $L^0$.

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    $\begingroup$ This may be relevant: en.wikipedia.org/wiki/Gini_coefficient#Definition $\endgroup$ Jul 2 at 0:14
  • $\begingroup$ What is the percent of total variance? And what are principal components of a set? There are like eight other phrases I don't understand but maybe they are economics terms. $\endgroup$
    – pancini
    Jul 2 at 0:17
  • $\begingroup$ Thanks @TravisWillse , I think Gini coefficient works! $\endgroup$
    – dashnick
    Jul 2 at 0:19
  • $\begingroup$ Cheers. In that case I'll write a short answer. $\endgroup$ Jul 2 at 0:58

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One such metric is the Gini coefficient $G$, which quantifies how much a distribution of values deviates from perfect equality: Concisely, it is half of the relative mean absolute difference of a list of values, i.e., if the values are $x_1, \ldots, x_n$, we define $$\boxed{G = \frac{1}{n^2 \bar x}\sum_{1 \leq i < j \leq n} |x_i - x_j|} ,$$ where $\bar x$ is the mean of $x_1, \ldots, x_n$. Provided that all $x_i$ are nonnegative, the Gini quotient takes values between $0$ (perfect equality) and $1$ (perfect inequality). It is often used to quantify inequality in income or wealth.

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