More Than One Mode?

Date: 10/07/2002 at 17:02:08
From: Robin Redsull
Subject: Statistical mode

Dr Math - 

I teach one section of statistics to advanced math students and we 
came upon an answer we did not agree with. We would like your help in 
determining if the answer is a misprint. And if it is not a misprint, 
we would like an explanation for the answer!

By definition, a statistical mode is the value(s) that occur most 
frequently in the data set. What is the mode of the data set:
13, 13, 14, 14, 15, 15? 

We say 'none' because no number appears more frequently than any other 
number. The answer key said 13, 14 and 15. What do you say??

Thank you - Robin Redsull

Date: 10/07/2002 at 17:59:54
From: Doctor Achilles
Subject: Re: Statistical mode

Hi Robin,

Thanks for writing to Dr. Math.

The other day, I was running a race against two other people. The 
other two people finished ahead of me, but at EXACTLY the same time.  
So they BOTH got first place and I got third place (there was no 
second place).

By definition, the winner of a race is (are) the runner(s) who finish 
with the fastest time. If two runners tie for first, they both win.  
If three, four, or more runners all get the EXACT same time, then they 
all win.

So the way to decide the winner of the race is to give each runner a 
time, like this:

  Runner A: 1min23sec
  Runner B: 1min25sec
  Runner C: 1min23sec

The winner(s) is (are) the runner(s) who has (have) the fastest time.  
To figure that out, we find the fastest time:

  1min23sec

So EVERY runner with a time of 1min23sec is a winner.

In the data set:

  1, 1, 1, 2, 2, 2, 3

1 and 2 both have the most occurrences, so they are both modes. Since 
they tie for first place, they both get it. 3 is not a mode because it 
occurs less frequently.

So the way to find a mode is to give each number a score based on how 
many times it occurs.

  1: occurs three times
  2: occurs three times
  3: occurs one time

The mode(s) is (are) the value(s) that occur(s) the most. The most any 
value appears is:

  three times

So every number that occurs three times is a mode.

Hope this helps. If you have other questions about this or you're 
still stuck, please write back.

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

Date: 10/08/2002 at 12:22:50
From: Robin Redsull
Subject: Statistical mode

Dr Math - Thanks for the quick reply to my question. I totally agree 
with your comments and completely understand them. But you didn't 
really address my question. If your data set is: 13, 13, 14, 14, 15, 
15,  is there a mode, and what is it?

Robin Redsull

Date: 10/08/2002 at 14:16:35
From: Doctor Achilles
Subject: Re: Statistical mode

Hi Robin,

Thanks for writing back to Dr. Math.

In that data set, the modes are: 13, 14, and 15.

In the data set:

  1, 13, 13, 14, 14, 15, 15

The modes are also: 13, 14, and 15.

You can have multiple modes in a data set.

Hope this helps. If you'd like to talk about this some more, please 
write back.

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

Date: 10/08/2002 at 12:30:08
From: Doctor Sarah
Subject: Re: Statistical mode

Hi Robin - thanks for writing to Dr. Math.

When no number occurs more than once in a data set, there is no mode.  
If each of two numbers occurs twice, we say the set is bimodal. Your 
set is trimodal.

See:

   Statistics - edHelper.com
   http://www.edhelper.com/statistics.htm 

   Mean, Median, and Mode Discussion - Shodor Education Foundation
   http://www.shodor.org/interactivate/discussions/sd1.html 

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

Date: 10/08/2002 at 12:35:51
From: Robin Redsull
Subject: Thank you (Statistical mode)

Thank you for your answer! What a neat resource to have access to.  
Now that I have found it - I'm sure my students will take advantage of 
it and try to come up with interesting questions to try to stump you 
all.