This post is directed at content developers, elementary math. I would like you to develop a Voxitatis tutorial on this topic to help parents explain elementary school mathematics to their children. The tutorial, when developed, may also help students who are struggling with the content described.
The topic of the tutorial should be stem-and-leaf plots: what they show about data, how to make a stem-and-leaf plot, the importance of the key, etc.
When should you use a stem-and-leaf plot?
Teachers have sort of “invented” the idea of a stem-and-leaf plot as an alternative way to represent data. I don’t want to imply that they are useless, but they are “coined” representations of data that teach elementary students another way to represent data. That said, they are fairly useful if you want to get an overall view of how often values in certain ranges occur in the data, compared to the frequency of values in other ranges.
A stem-and-leaf plot represents integers, which as students will know by the time they get to stem-and-leaf plots, are whole numbers. Integers do not include fractions or decimal points. A set of integers is like this: { … , -4, -3, -2, -1, 0, 1, 2, 3, 4, … }.
Then a stem-and-leaf plot “groups” integers by tens. For example, all the numbers between 20 and 29 would be in one category, numbers between 30 and 39 in the next category, 40 to 49 in the next, and so on.
A typical problem students would be asked to solve is to be presented with a large set of data, all of which are integers, and put them in a stem-and-leaf plot. The first step in that task is to group the integers by tens. In one science class, for example, students received the following scores on their chapter test:
| Student | Score |
| Sarah | 83 |
| Bill | 77 |
| John | 88 |
| Keith | 82 |
| Jessica | 75 |
| James | 89 |
| Kenneth | 55 |
| Lisa | 79 |
| Jennifer | 95 |
| Ashli | 90 |
| Megan | 84 |
We want to make a stem-and-leaf plot of these scores, and the first step is to group the scores by tens. How would you group these scores by tens?
In order to group these scores by tens, we have to know a little about place value. The last digit in a whole number (the one on the right, like the 7 in 467) is said to be in the one’s place. The digit to the left of that (like 6 in 467) is said to be in the ten’s place.
In order to group integers by tens, ignore the digit in the one’s place (for now). That is to say, 467 would be in the same group as 462, 464, and 469, because only the one’s digit is different among these four number, and we don’t care about the one’s digit when we’re grouping by tens. The important point is that all the digits are the same, except the one’s digit possibly, when we group integers by tens.
Let’s get back to our science test scores. Since the lowest score in the class is 55 and the highest is 95, we only have to go from the 50’s to the 90’s when we group by tens. We would come up with the following grouping:
From here, you can easily make a stem-and-leaf plot…
What does a stem-and-leaf plot look like?
As you can see from the table in the previous comment, more students scored in the 80’s than in any other grouping, Students may be able to analyze this data further, or people may be interested in certain trends in the data. For example, we see that no student got a score in the 60’s. It’s easier to see that in a table like this than in the original list of data.
But a table like the one above also has lots of extra information that you really don’t need to see the trend. For example, since the ten’s digit is the same in all the scores in each group, we don’t need to write it for every one.
That’s the designing principle of stem-and-leaf plots. We write just enough so we know what we mean. This results in a table that will look strange to all those who went through third grade before about 1990, but it’s simply an efficient way to write the table from the previous comment.
Instead of writing 70’s in one column and then 77, 75, 79 in another for that row in the table, we simply write the tens digit once in the left column (this column is called the stem in our stem-and-leaf plot), and then we don’t need to repeat it for each of the scores. Then, we write only the one’s digit in the right column (called the leaves), like this:
Since this stem-and-leaf plot just “represents” the data, we need to provide a key or legend as to what these representations mean. This is usually done to the side of the stem-and-leaf plot, or just below it, with a small legend that picks one or two values from the stem-and-leaf plot and uses it as an example, like this:
You will also notice I made a few changes in going from the table to the stem-and-leaf plot:
First, I sorted the scores in each group of tens. This helps to spot trends in the data, but since stem-and-leaf plots are sort of a “made-up” concept in the first place, there are no real “rules” about whether or not the data should be sorted in each row. Some teachers insist that it be sorted, while others don’t really care. Check with your teacher.
Second, I took out the commas. They are not necessary, since each data point is represented by exactly one digit in the stem-and-leaf plot anyway. Some teachers like to use commas, while others insist that they not be used. I can’t say with any great certainty whether or not they should be used. It’s up to the teacher. All I know is they are not necessary.
A nice lesson for stem-and-leaf plots, which uses some animated flash videos, is online at Wisc-Online (link).
A more difficult stem-and-leaf plot
Let’s say some of the scores on the science test we used in this example were like 127, 116, and 120. How would we add those to the stem-and-leaf plot?
Well, since our stem-and-leaf plot only goes up to a maximum of 99, we would need to add more “stems” to the plot. We do this by adding numbers below the 9 in the stem so that we go up to 12 “tens” in the plot, like this:
This gives us all the stems we need. Now, to add the leaves to those stems. Even though two digits are written in the stem, we still only write one digit for each leaf in the right-hand column. Thus, we write 6 next to the “11” to represent 116, and we write 0 7 next to the “12” to represent 120 and 127, respectively.
Question: Even though there are no scores in the 60s, do you still have to write the 6 in the stem?
Yes. The stem has to be written vertically like that, in order, including every number for the tens column, even if there are no numbers to write for the leaf on that line.
Question: Can the stem-and-leaf plot be written sideways?
Yes. In fact, if you rotate the stem-and-leaf plot used in this example 90 degrees counter-clockwise, it will resemble a histogram. That is another way to represent data graphically, where bars go higher in groups where there are more data points.
Question: How can I come up with practice problems?
If you don’t have a random number generator handy, you can put all kinds of real-world data in a stem-and-leaf plot. If you would like to try using a random number generator on the Web, you can find one here.
On the other hand, if you’re not into the randomness of the whole thing, and you like baseball, how about putting the number of games won last season by the teams (which you can find in any newspaper) into a stem-and-leaf plot?
If Rock ‘n’ Roll is more your speed, get a few CDs and put the number of seconds each song takes into a stem-and-leaf plot. For example, if three of the songs on one CD are 4:21, 3:40, and 3:26 in length, you could plot 21, 40, and 26.
And if football is your game, try plotting the quarterback rating for all the quarterbacks in the NFL, rounded to the nearest integer. These numbers must be available on some fantasy football site somewhere.