# Measure of dispersion around the mode

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?

• The width of the smallest interval containing half the data (the shortest half, or shorth) is sometimes used. See stats.stackexchange.com/questions/76848/… – Glen_b -Reinstate Monica Nov 20 '18 at 13:00
• @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
• @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 -Reinstate Monica Nov 20 '18 at 13:13
• 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
• 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