mode of no-repeats set

[Mike Oliver]

Lindsay A. B. Moniz wrote:
> Chris Aitken wrote:
> 
>> Chris Aitken wrote:
>>
>>> This is off topic but as it is my first, I hope you'll indulge me. 
>>> You are the smartest people I know.
>>> ;-)
>>>
>>> How can you calcualte the mode of a set of numbers when all numbers 
>>> are unique (no repeats)? I've checked a math text and some websites 
>>> and they are all clear: mode is the most frequently occuring number.
>>>
>>> What if the set is 221, 204, 254, 194, 165 and 176?
>
> There is no mode to that set of numbers.


Well, *literally* there's not, no.  And practically there's not,
either, with such a small sample size.

But lots of times you *can* meaningfully speak of a mode, even
when no two samples are exactly equal.  Basically you're thinking
of the samples as being drawn from an infinite population with
a probability distribution, and the "mode" is where the probability
distribution has a maximum.

You can get a good idea where that is from a finite set of data,
by grouping them into equal-sized bins and making a histogram.  For
well-behaved data, the peak of the histogram won't depend much on
either the bin size or the endpoints of the bins, until you make
the bins so small that there aren't enough samples in each bin
to make them well behaved.
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