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Given the data set {10,10,14,14,15,15,17,17,27,27,30,30}

is the mode {10,14,15,17,27,30}

or is there no mode?

Nick Cox
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Mohamed Helmy
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    Have you tried looking up the definition of "mode"? https://en.wikipedia.org/wiki/Mode_(statistics) – jbowman May 02 '18 at 04:33

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As you're realising, the naive definition of a mode as the most common value breaks down when there are ties for mode, especially in the extreme case when every value is distinct, which is very common with measured data with fractional parts and a small dataset. Perhaps every person would have a distinct height if you measured to the nearest nanometre, but that shouldn't stop you thinking about the mode of a height distribution. So conventions about measurement enter too.

More crucially, there are other ways to get at modes. One is to apply a kernel or other density estimate and look for the position of a peak of the estimated density. Another, similar in spirit but not in detail, is to look recursively for the midpoint of an interval where values are densest. There might be reservations about how far either idea carries over to discrete variables.

The half-sample mode procedure discussed in much more detail within How to find the mode of a probability density function? gives 14 as the answer in your case.

Drawing a graph of your distribution to see any estimate of mode in context and thinking about how you might want to use the mode are basic but both important steps.

Nick Cox
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There is no single mode in a multimodal data set. Your particular data set is completely uniform, so it has no mode at all. Since all your observations are integers, it seems likely that the variable is not continuous, but it isn't clear what they are, the spacing between them is odd (i.e. they aren't dice rolls or something like that).

Peter Flom
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    "Completely uniform" seems to contradict "the spacing between them is odd." – whuber May 02 '18 at 12:58
  • I'm not sure that it does. That would be so if this were a discretization of a continuous variable, but I don't think it is if it's an inherently disecrete variable. – Peter Flom May 02 '18 at 13:18
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    Implicitly, then, it looks like you are assuming these values are the entire support of the distribution from which they are drawn: there is zero probability, say, of observing 11, 12, or 13. That seems to be the sense in which you mean "completely uniform," but that would be inconsistent with the remark about spacing. That's why it looks like there's a contradiction. In trying to explain this it also occurs to me that it would help to distinguish remarks about the data (as in "particular data set") from remarks about a model for them (as in "the variable is not continuous"). – whuber May 02 '18 at 14:30