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To my understanding arithmetic mean income is influenced by people on high incomes more than medium and low incomes, Geometric mean income is equally influenced by people on high, medium, and low incomes, and Harmonic mean income is more influenced by people on low incomes than on medium or high incomes. Harmonic mean income is therefore the best of the three types of mean to assess poverty.

Are these types of mean a part of a sequence of means? If so, what would be the next mean in the sequence? If used to summarise mean income, would the next mean in the sequence be even less influenced by high and medium incomes than the Harmonic Mean while being even more influenced by low incomes in the dataset?

The reason I want to know is because I am interested in producing a summary statistic to indicate levels of absolute poverty.

novice
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    this might answer your question: https://en.wikipedia.org/wiki/Mean#Generalized_means, – Albert Nov 01 '22 at 15:27
  • In particular, from the link in the comment above, you can see that the most extreme mean is simply "take the minimum", in which case you ignore all data except the very lowest income. Presumably you're looking for something between that and the harmonic mean. – Théophile Nov 01 '22 at 15:38
  • @Theophile Exactly. And as part of my paper I would like to present the type of mean (=measure of poverty) which correlates most closely with variable Y (while being clear about what I've done). It's a shame the next means in the sequence seem not to have any official names, since having to present the actual mathematical expressions for them in the paper might look a bit overwhelming to the kind of person who would be reading it. – novice Nov 01 '22 at 15:52
  • @novice But by the same token, if the audience would be overwhelmed by such an expression, then they wouldn't necessarily be better served by a technical name like quadratic harmonic mean (which incidentally does appear here and there in the literature for the choice of $m=-2$). – Théophile Nov 01 '22 at 18:54
  • Note that there's nothing stopping you from using a non-integer value, which is what you'll end up with if you're looking for the closest fit. Given that, the best "official name" is really just going to be a description like "a generalized mean with $m=-2.93$". – Théophile Nov 01 '22 at 19:01
  • Usually instead of moving to more esoteric means, if you're analysing skewed distributions like income it's common to look at other statistics like quantiles, which then lead to inequality measures like the Pareto index. – ConMan Nov 02 '22 at 00:45
  • Consider also asking for more guidance on https://stats.stackexchange.com/. – Théophile Nov 02 '22 at 19:01

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