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### Mode of a Uniform Distribution

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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

1, 1, 2, 2, 3, 3

How do I explain this so that it makes sense?

Thanks.
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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

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/
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Associated Topics:
High School Statistics
Middle School Statistics

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