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/
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