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Suppose we are looking at a right-skewed size distribution such as the distributions of income, casualty losses, or flood sizes, and we want to look at some alternative measures to jointly represent the central tendency of that distribution. The median -- the ratio of, e.g., income per person where half the people have incomes above that ratio and half below -- is one appealing measure. Is there any commonly used, named statistic that represents the level of income where half of the dollars of income (loss value, gallons of water) are received by people (accidents, floods) above that level and half below?

If so, does it have a name? A literature? In right-skewed populations, will these two statistics bracket the mean?

kjetil b halvorsen
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andrewH
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  • If I understand correctly, the statistic you describe is actually the median of some related distribution. If the original pdf is $f(x)$ then it sounds like you're looking for the median of $\frac{1}{z}xf(x)$ where $z$ is a normalizing constant. (I don't know if this has a name though.) – tddevlin Mar 07 '19 at 06:52
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    this post is related, and proposes a name: tantile (or medial). This is related to partial moments, search this site. – kjetil b halvorsen Mar 07 '19 at 13:06
  • Even without a single name, there’s a standard sentence structure for this: “Most US income goes to the top 18% of earners”; “Most Americans are represented by Senators for the top 18% of most populous states”; “in the desert, most of the year’s rain may come on the three wettest days”. – Matt F. Mar 19 '24 at 02:14

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