Mode of a Uniform Distribution

Date: 09/06/2000 at 23:11:02
From: David Kennedy
Subject: Modes

I am the father of a 6th-grade student who uses the text _Everyday 
Mathematics_ from the University of Chicago School of Mathematics 
Project. My question is: what is the mode of a set of numbers, if none 
of the numbers repeat themselves? Here is the data set:

   338, 324, 270, 229, 209, 193, 170, 168, 154, 140

What about a set like:

   1, 1, 2, 2, 3, 3

How do I explain this so that it makes sense?

Thanks.

Date: 09/14/2000 at 11:12:04
From: Doctor TWE
Subject: Re: Modes

Hi David - thanks for writing to Dr. Math.

Your question is a good one. After consulting with several colleagues 
and college professors (including my wife, who teaches graduate-level 
statistics courses), I find that the consensus is that there is no 
consensus. The problem with the definition of mode is that it doesn't 
explicitly say what to do in the case of a uniform discrete 
distribution.

The definition says that the mode is "the most frequently occurring 
value in a sequence of numbers." By a strict interpretation, this 
means that in a uniform distribution (a sequence in which all values 
occur with equal frequency) such as your series, all values are modes, 
since there is no value that occurs more often. However, some 
references say that if all elements in a data set have the same 
frequency, then the data is said to be of no mode.

My wife says that she would accept either "all values are modes" or 
"there is no mode" as an answer for that problem. The software package 
she uses in her classes, "Adventures in Statistics," only accepts the 
answer "there is no mode."

Personally, I am a stickler for exactness in definitions and 
definition interpretation, so I would say any uniform discrete 
distribution is multimodal with all values being a mode. (But that's 
just my opinion.)

My wife pointed out another interesting situation. Consider the set 
produced by taking the absolute value of all of the integers. In this 
set, the values 1, 2, 3, ... occur twice each, but the value 0 occurs 
only once.

Therefore, it is not a uniform distribution. Does this make it 
multimodal with all values except 0 being modes? If so, does removal 
of the 0 cause it to have no mode?

As to explaining it to a sixth-grader, I would stick with a simple 
explanation. Ask either "what value occurs more than the others?" (the 
answer would be "none"), or "which value or values occur the most 
often?" (the answer would be "all of them"). You might then ask "which 
answer - none or all - is more useful?" This is an opportunity to get 
him or her thinking more deeply about math as a tool for understanding 
other things.

I hope this helps. If you have any more questions, write back.

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