Which is the Best Description?

Date: 03/25/2001 at 17:59:57
From: Sandra Notesetine
Subject: Mode, mean, range, median; which best describes?

I am asked to answer the question; "which best describes the number of 
times "Popular Song" was played on the radio per day?" given the 
following information:

     Sunday:     9
     Monday:    13
     Tuesday:   13
     Wednesday: 12
     Thrusday:  14
     Friday:     3
     Saturday:  10


I calculated that the range is 11, the median is 12, the mode is 13, 
and the mean is 10.5.

How should this question be answered? Is there an answer? I would 
guess the answer to be mode, since 13 occurs twice in the list. But I 
don't know how to answer the question. 

Please help.

Date: 03/26/2001 at 15:35:06
From: Doctor TWE
Subject: Re: Mode, mean, range, median; which best describes?

Hi Sandra - thanks for writing to Dr. Math.

Statistics are used to describe a data set. Two of the most common 
characteristics we want to describe about a set of data are its 
"central tendency" and its "spread."

There are three common measures of central tendency; the mean, median 
and mode. You are apparently already aware of the definitions of these 
and how to calculate them. (By the way, the mean is 10.571... which 
would round off to 10.6, not 10.5 - but that's a minor point.) The 
question is; which *best* describes the data set? The answer to that 
depends on what exactly you want to describe. Let me give you another 
example.

Let's say we were comparing two basketball players, Anne and Rich. In 
five games they've made the following points:

     G#   Anne   Rich
     --   ----   ----
      1    12     10
      2     0     13
      3    13      9
      4    25      8
      5     0     10

When comparing means, Anne and Rich seem to be equivalent, since each 
averages 10 points per game. If we look at their median scores, Anne 
seems to be the better player; her median is 12 as compared to Rich's 
median of 10. If we consider their modes, the "most likely outcome" is 
that Anne will score 0 points, but Rich will score 10. Rich's mode is 
better than Anne's.

Let's look at these numbers all together:

             Anne   Rich
             ----   ----
     Mean     10     10
     Median   12     10
     Mode      0     10

Who's better? It depends on what you're looking for.

If, for example, you were to have an "office pool" on how many points 
Anne or Rich will score in the next game (and you had to get the score 
exactly to win the pool), you'd want to use their modes. If you had a 
pool and the winner was the one who came closest to the player's 
score, you'd want to use their means. If you wanted to know what score 
you'd have to get to have a better-than-even chance of outscoring Anne 
or Rich, you'd want to use the median (or more precisely, the median 
+ 1). The numbers don't indicate a clear "winner," but they do paint a 
good picture of the types of players Anne and Rich are.

You are correct in eliminating the range as an option - range (along 
with variance and standard deviation) is a measure of "spread," or 
distribution of the data, not central tendency.

My assumption is that the problem was not asked with a right or wrong 
answer in mind, but rather to see if the student can justify his or 
her choice. Does it ask the student why he or she gave their answer? 
This type of question can be used to test the student's understanding 
of the underlying concepts rather than just their ability to "number 
crunch."

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

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

Date: 03/26/2001 at 23:29:09
From: Sandy
Subject: Re: Mode, mean, range, median; which best describes?

Thank you for your response.

No explanations are requested. The question is multiple choice.  

   Which one best describes the number of times "popular song" was 
   played on the radio per day?

There is no comparison to anything else.

   1. Mean
   2. Median
   3. Mode
   4. Range

Thanks. I am so glad you are here on the Web.

Date: 03/27/2001 at 11:57:20
From: Doctor TWE
Subject: Re: Mode, Mean, Range, median which best describes?

Hi Sandy - thanks for writing back.

This is why I dislike closed form (i.e. multiple choice, true/false, 
etc.) interpretation questions. There are valid arguments that can be 
made for any of the first three choices, depending on why the 
information is being sought. But in lieu of more information, I would 
say that the mode *probably* best describes the number of times the 
song was played.

As a teacher, I avoid using multiple choice except for "factual" 
recall questions. (E.g. "which of the following is the median of the 
data set?") When it comes to interpreting data and understanding 
concepts, I want to see what the student is thinking, and the only way 
to do that is with open-ended questions. (E.g. "which measure of 
central tendency best describes the number of times the song was 
played on the radio per day? Defend your choice.")

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

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