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I usually associate the standard deviation with the mean and the IQR with the median. Is there a measure of dispersion typically associated with the mode?

Determinant
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    The width of the smallest interval containing half the data (the shortest half, or *shorth*) is sometimes used. See https://stats.stackexchange.com/questions/76848/retrieving-minimum-width-that-contains-specified-fraction-of-all-values – Glen_b Nov 20 '18 at 13:00
  • I don't think there is anything typical as a measure of dispersion. If "the" mode -- however determined; that's a different story -- is quite different from the mean or median, then the main implication is that you need to report skewness too. – Nick Cox Nov 20 '18 at 13:01
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    @Glen_b's helpful comment implies that the shortest half will contain the mode. In fact, that's not guaranteed, although it would be very common. (Counter-example: 0 has 40% of the values and is the mode. 9 and 10 have 30% of the values each.) – Nick Cox Nov 20 '18 at 13:05
  • I’m looking for a generalized answer if possible, but right now, in particular, I’m looking for dispersion around the mode of a beta PDF that has shape parameters such that there is only one mode. – Determinant Nov 20 '18 at 13:11
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    @Nick I definitely didn't intend to imply anything more than that the shortest half is sometimes used as a measure of spread along with the mode; there's certainly no guarantee that the interval would include the mode. Indeed, there may be several such shortest intervals and it may be that none of them include the mode. I agree there's no typical measure. – Glen_b Nov 20 '18 at 13:13
  • @Glen_b We agree completely. – Nick Cox Nov 20 '18 at 13:17
  • If you're focusing on the beta distribution, then are you assuming that you know the parameters or are you trying to estimate them? – Nick Cox Nov 20 '18 at 13:19
  • In the end I will be determining the parameters given a known mode and a known value for the dispersion metric, but constrained by the fact that PDF have a single peak. I’ve seen this done with the mean and variances on the beta PDF but for my purposes the mode is more intuitive and useful. – Determinant Nov 20 '18 at 13:27
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    With a distribution thought to be (close to) beta, I would just use maximum likelihood (ML). This sounds like a homegrown method bound to be very sensitive to how you calculate the mode in the first place. With ML the mode is just estimated as a side-effect provided your parameter estimates are consistent with unimodality. On a different level, this thread is morphing into something quite different, so I suggest asking a new question. – Nick Cox Nov 20 '18 at 13:35
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    I think the answer to this thread might be, “No, but you can try the shorth,” which is what I needed. But regarding the ML comment, I wouldn’t be calculating the mode. I would be supplying the location of the mode and the dispersion metric. For example, “What are the shape parameters of a unimodal Beta PDF given a mode at 0.8 and a [dispersion metric] of [number].” I could turn this into a new question. Let me think about it. – Determinant Nov 20 '18 at 13:55
  • To be clear, I’m not fitting the PDF to data, I’m hypothesizing that the data be exactly a PDF with shape parameters I provide. Not sure I just made sense. – Determinant Nov 20 '18 at 13:58
  • There are various ways to parameterise the beta distribution. I don't recall one that matches your view and it's hard to see that one could exist, as other parameters determine the mode when it's well-defined, but not _vice versa_. – Nick Cox Nov 20 '18 at 14:17
  • I believe it can be done with the mode and some dispersion metric (right?). Perhaps the parameters can’t be solved for explicitly but it’s essentially two equations (mode and dispersion metric) and two unknowns (alpha and beta). It has definitely been done with mean and variance. – Determinant Nov 20 '18 at 14:27
  • @Glen_b if you want to post your comment about the shorth with the caveat mentioned by Nick the I would accept that answer. Serious thanks to both of you! – Determinant Nov 20 '18 at 14:32

1 Answers1

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There isn't really a typical measure of spread associated with the mode.

Of course one could calculate a root-mean-square-deviation-around-the-mode (a standard-deviation measure with the mode taken as the center) or a mean-absolute-deviation-around-the mode, but neither of those are common measures.

A measure of spread that is sometimes used with the mode is the width of the smallest interval containing half the data (the shortest half, or shorth). This measure is discussed by Nick Cox here, but it isn't "about the mode" -- the shortest interval containing half the data needn't include the mode (indeed there may be multiple such intervals, and any or all of them may fail to include the mode).

However with a continuous unimodal distribution, if mode and the shortest half are unique, then the shortest half will contain the mode (otherwise, some shortest half will contain a mode).

[Incidentally that linked article on unimodality does mention a result relating to the root mean square deviation around the mode, indicating that the first measure I mentioned does crop up in some situations.]

Glen_b
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