Mode, Mean, and Median in Stemplots

Date: 02/21/2002 at 20:00:39
From: Linda Davidson
Subject: Stem and leaf graphs/plots

I'm trying to help my 6th grader do homework. How do I find a "mode,"
"mean," and "median" using a stem/leaf plot?

Problem:
           stem  leaf
           1     889
           2     035579
           3     138
           4     235

Date: 02/22/2002 at 00:35:55
From: Doctor Twe
Subject: Re: Stem and leaf graphs/plots

Hi Linda - thanks for writing to Dr. Math.

Each stem-and-leaf combination represents a data point in our set. So 
to find the mode, mean, and median of the set, we have to figure out 
how to interpret their definitions for this type of representation.

The mode is defined as the data value that occurs most often. So we 
are looking for the leaf (number) that occurs the most often on one 
stem of the diagram. In your example, there are two 8 leafs on the 
1 stem (i.e. two data points of value 18), and two 5 leafs on the 
2 stem (i.e. two data points of value 25). So the data set is 
"bi-modal" with modes of 18 and 25.

Note that I did not count the 5 leaf on the 4 stem because it 
represents a different value (45) - it just happens to have the same 
last digit as my mode of 25. I similarly did not count the 8 leaf on 
the 3 stem, nor the three different 3 leaves.

The mean is the conventional "average," and perhaps the best way to 
find this is to do it the conventional way - add the values and divide 
by the number of numbers. With the stem-and-leaf plot, that means that 
we'll have to "read" each stem-and-leaf as a conventional number. For 
your example we'll get:

  (18+18+19+20+23+25+25+27+29+31+33+38+42+43+45) / 15 = 436/15 = 29.1

(Do you see how I got the numbers I added?)

The median is the middle value in the set. This is relatively simple. 
Start crossing off pairs of high and low leaves. Start with the 
leftmost leaf on the bottom stem and the rightmost leaf on the top 
stem. When you only have one (or two) leaves left that have not been 
crossed out, that value (or the average of the two values) is the 
median. In your example (I'm using matching symbols to show which two 
were crossed out as a pair):

     stem  leaf
     1     X*#
     2     -+=@7@
     3     =+-
     4     #*X

The one I'm left with is the 7 leaf on the 2 stem, so the median is 
27.

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

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