When I’m done grading an exam, I like to see how well the class did as a whole. Most of the time there’s not a calculator handy, so I don’t start by calculating the mean or median
. Instead it’s helpful to see how the scores are distributed. Do they resemble a bell curve? Are the scores bimodal? One type of graph that displays these features of the data is called a stem and leaf plot, or stemplot. Despite the name, there is no flora or foliage involved. Instead the idea is that stems form one part of a number, and leaves are the rest of that number.
To make my stemplot, each score is broken into two pieces, the stem and leaf. In this particular case the tens digits are stems, and the ones digits form the leaves. The resulting stemplot produces a distribution of the data similar to a histogram, but all of the data values are retained in a compact form. Features of the students’ performance can be easily seen from the shape of the stem and leaf plot.
Suppose that my class had the following test scores: 84, 65, 78, 75, 89, 90, 88, 83, 72, 91, 90 and we wanted to see at a glance what features were present in the data. We rewrite the list of scores in order and then utilize a stem and leaf plot. The stems are 6, 7, 8, 9, corresponding to the tens place of the data. This is listed in a vertical column. The ones digit of each score is written in a horizontal row to the right of each stem:
9| 0 0 1
8| 3 4 8 9
7| 2 5 8
6| 2
The data can be easily read from the stemplot. For example, the top row contains the values 90, 90 and 91.
What’s the Stem and What’s the Leaf?
With test scores, as well as other data, that range between 0 and 100 points the above strategy works for choosing stems and leaves. But how do we know what to pick for a stem and a leaf with other sets of data? For data with more than two digits there are some options. It all comes down to how the data values are distributed.
If we wanted to make a stem and leaf plot for the data 100, 105, 110, 120, 124, 126, 130, 131, 132 what would we choose for a stem? If we say the highest place value, that is the hundreds digit is the stem, our resulting stemplot is not very helpful because none of the values are separated from any of the others:
1|00 05 10 20 24 26 30 31 32
Instead we could try to make the stem the first two digits of the data. The resulting stem and leaf plot does a better job at depicting the data:
13| 0 1 2
12| 0 4 6
11| 0
10| 0 5
Expanding and Condensing
The two stemplots in the previous section show how versatile stem and leaf plots are. They can be expanded or condensed by changing the form of the stem. One strategy for expanding a stemplot is to evenly split a stem into equally sized pieces. Consider the stemplot:
9| 0 0 1
8| 3 4 8 9
7| 2 5 8
6| 2
We expand this stem and leaf plot by splitting each stem into two. This results in two stems for each tens digit. The data with 0-4 in the ones place value are separated from those with digits 5-9 in the ones place:
9| 0 0 1
8| 8 9
8| 3 4
7| 5 8
7| 2
6|
6| 2
Here the 6 with no numbers to the right shows that there are no data values from 65 to 69.
