When you finish grading an exam, you might want to determine how your class performed on the test. If you do not have a calculator handy, you can calculate the mean or median of the test scores. Alternately, it is 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 stem forms one part of a number, and the leaves make up the rest of that number.

### Constructing a Stemplot

In a stemplot, each score is broken into two pieces: the stem and leaf. In this example, the tens digits are stems, and the one 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. You can easily see features of the students’ performance from the shape of the stem-and-leaf plot.

### Stem and Leaf Plot Example

Suppose that your class had the following test scores: 84, 65, 78, 75, 89, 90, 88, 83, 72, 91, and 90 and you wanted to see at a glance what features were present in the data. You would rewrite the list of scores in order and then use a stem-and-leaf plot. The stems are 6, 7, 8, and 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, as follows:

9| 0 0 1

8| 3 4 8 9

7| 2 5 8

6| 2

You can easily read the data from this stemplot. For example, the top row contains the values of 90, 90, and 91. It shows that only three students earned a score in the 90th percentile with scores of 90, 90, and 91. By contrast, four students earned scores in the 80th percentile, with marks of 83, 84, 88, and 89.

### Breaking Down the Stem and Leaf

With test scores as well as other data that range between zero and 100 points, the above strategy works for choosing stems and leaves. But for data with more than two digits, you'll need to use other strategies.

For example, if you want to make a stem-and-leaf plot for the data set of 100, 105, 110, 120, 124, 126, 130, 131, and 132, you can use the highest place value to create the stem. In this case, the hundreds digit would be the stem, which is not very helpful because none of the values is separated from any of the others:

1|00 05 10 20 24 26 30 31 32

Instead, to obtain a better distribution, make the stem the first two digits of the data. The resulting stem-and-leaf plot does a better job of 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 the versatility of stem-and-leaf plots. 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:

9| 0 0 1

8| 3 4 8 9

7| 2 5 8

6| 2

You would expand this stem-and-leaf plot by splitting each stem into two. This results in two stems for each tens digit. The data with zero to four in the ones place value are separated from those with digits five to nine:

9| 0 0 1

8| 8 9

8| 3 4

7| 5 8

7| 2

6|

6| 2

The six with no numbers to the right shows that there are no data values from 65 to 69.