Is There a Mode when All Data Points Occur Equally?

Date: 05/02/2007 at 13:56:34
From: TJ
Subject: Is there a "mode" when every number is different?

If you have a set of data such as 2, 67, 39, 20, 15, and 56, what 
would the "mode" be? 

Date: 05/02/2007 at 21:06:16
From: Doctor Rick
Subject: Re: Is there a 

Hi, TJ.

Take a look at this answer in the Dr. Math Archive:

  More Than One Mode?
    http://mathforum.org/library/drmath/view/61375.html 

At the end you'll see a reference stating that a set in which no 
element occurs more than once has no mode.  You could also argue that, 
in any set in which all the elements occur the same number of times 
(including once), every element is a mode.  It really doesn't matter.

In real life, we wouldn't bother calculating statistical measures for 
such a small set; those measures are meant to distill a *large* set of 
numbers into a small set of numbers (mean, median, mode(s), standard 
deviation, ...) that characterize the distribution.  We study small 
sets like your example in order to see up close how statistical 
measures work, but we shouldn't focus on these toy examples too much, 
or on the issues that arise in dealing with them.

The larger the set, the less likely it is that every number will 
appear exactly the same number of times.  If the number of occurrences 
is *nearly* the same for all numbers, then the mode (or modes) is 
meaningless; what is significant is that the distribution of the data 
is nearly flat.

If each number appears only once, it suggests that you'd get more 
interesting results by binning the numbers to increase the number of 
occurrences in each bin.  (For instance, total the counts for numbers 
1 to 10 in one bin, the counts for 11 to 20 in the next bin, etc.) 
You may well find that a mode (or modes) will become evident when 
you do this.  Thus there is no point in worrying about what to call 
the mode when each number appears only once; in practice you won't 
continue to work with the data in that form.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/