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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?

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    $\begingroup$ The width of the smallest interval containing half the data (the shortest half, or shorth) is sometimes used. See stats.stackexchange.com/questions/76848/… $\endgroup$ – Glen_b -Reinstate Monica Nov 20 '18 at 13:00
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    $\begingroup$ @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.) $\endgroup$ – Nick Cox Nov 20 '18 at 13:05
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    $\begingroup$ @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. $\endgroup$ – Glen_b -Reinstate Monica Nov 20 '18 at 13:13
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    $\begingroup$ 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. $\endgroup$ – Nick Cox Nov 20 '18 at 13:35
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    $\begingroup$ 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. $\endgroup$ – Determinant Nov 20 '18 at 13:55
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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.]

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