From Individual Scores to a Stemplot
In Building a Dotplot Step by Step, you learned to display each quantitative observation while stacking equal values. A stemplot also preserves the individual observations, but it organizes them by place value. That makes a list of exam scores easier to scan while keeping the original scores visible.
To make a stemplot, divide each score into a stem and a leaf. For whole-number exam scores, a natural choice is to use the tens digit as the stem and the ones digit as the leaf. For example, a score of 76 is represented by stem 7 and leaf 6. The stem and leaf together recover the original score; neither part is a count or a rounded value.
A regular stem row can become crowded when many scores have the same tens digit. Splitting each stem into two rows gives more room and can make the distribution easier to see. In the first row for a stem, place scores whose ones digits are 0, 1, 2, 3, or 4. In the second row, place scores whose ones digits are 5, 6, 7, 8, or 9. Thus, a score ending in 5 belongs on the second row, not the first.
The leaves on each row must be written in increasing order from left to right. Stems are arranged from least to greatest down the display, and each stem appears twice in succession. A repeated score produces a repeated leaf: if two students scored 74, the leaf 4 appears twice on the lower row for stem 7. Do not erase a repeat just because the two scores are equal; each leaf represents one observation.
A key is essential. The notation \(7|4=74\) points says that stem 7 and leaf 4 represent an exam score of 74 points. Without a key, a reader might not know whether the display represents scores such as 74 or values such as 7.4. Include the unit when the variable has one, and make clear which digit is the leaf.
Worked Example: Make a Split-Stem Plot From Scores
Worked Example: Make a Split-Stem Plot From Scores
A fictional class has these 16 exam scores, in points: 62, 64, 67, 68, 69, 71, 72, 74, 75, 76, 78, 81, 83, 84, 87, and 92. Construct a split-stem stemplot.
Choose the stems and leaves. Every score is a whole number with a tens digit and a ones digit. Use the tens digit as the stem and the ones digit as the leaf. For instance, 62 becomes stem 6 and leaf 2, while 92 becomes stem 9 and leaf 2.
Place and order the leaves. For stem 6, leaves 2 and 4 go on the 0–4 row, while leaves 7, 8, and 9 go on the 5–9 row. Apply the same rule to each stem. The resulting display is:
| Stem | Leaf range | Leaves, in order |
|---|---|---|
| 6 | 0–4 | 2 4 |
| 6 | 5–9 | 7 8 9 |
| 7 | 0–4 | 1 2 4 |
| 7 | 5–9 | 5 6 8 |
| 8 | 0–4 | 1 3 4 |
| 8 | 5–9 | 7 |
| 9 | 0–4 | 2 |
| 9 | 5–9 |
Add the key. The key is \(7|4=74\) points. It shows how to read every stem–leaf pair and identifies the units. The final row has no leaves because this data set contains no score from 95 to 99 points. A blank row means there are no observations in that interval; it does not represent a score.
Check the construction. Count the leaves: \(2+3+3+3+3+1+1+0=16\). This matches the 16 scores in the original list. Also check that, within each row, leaves are in order and belong in that row’s range. For example, 75 is on the 5–9 row for stem 7, and 74 is on the 0–4 row.
Why Split Stems and Ordered Leaves Matter
Splitting a stem changes the layout, not the underlying data. The two rows for stem 7 still represent scores in the 70s. The first covers 70–74, and the second covers 75–79. Together, they show every score with a tens digit of 7. The split can spread observations across more rows, making concentrations within a decade easier to notice than they would be in one crowded row.
Ordering the leaves makes the display readable and gives each row a consistent direction. If the leaves for the 70–74 row are 1, 2, and 4, write them as \(1\ 2\ 4\), not \(4\ 1\ 2\). You do not need to rearrange the data set into a new list for any mathematical reason, but you do need to sort the leaves so readers can follow the display and find values efficiently.
Stems should cover the range of the data in order, with both rows shown for each stem used. When a half-row has no values, leave its leaves cell empty rather than inventing a zero leaf. A leaf 0 is a real score ending in zero: for example, \(8|0\) represents 80 points. An empty row and a row containing leaf 0 therefore mean different things.
Worked Example: Handle Repeated Scores and an Unsorted List
Worked Example: Handle Repeated Scores and an Unsorted List
A second fictional class has these 20 exam scores, listed in the order recorded: 74, 63, 91, 55, 86, 74, 68, 97, 72, 82, 63, 89, 58, 77, 94, 70, 85, 79, 66, and 72. Make a split-stem stemplot.
Sort by stem and split range. Use the tens digit as the stem and the ones digit as the leaf. For example, the two scores of 63 both give a leaf 3 on stem 6’s 0–4 row. The scores 74 and 74 give two leaves 4 on stem 7’s 0–4 row. The order in which scores were recorded does not determine the order of leaves.
| Stem | Leaf range | Leaves, in order |
|---|---|---|
| 5 | 0–4 | |
| 5 | 5–9 | 5 8 |
| 6 | 0–4 | 3 3 |
| 6 | 5–9 | 6 8 |
| 7 | 0–4 | 0 2 2 4 4 |
| 7 | 5–9 | 7 9 |
| 8 | 0–4 | 2 |
| 8 | 5–9 | 5 6 9 |
| 9 | 0–4 | 1 4 |
| 9 | 5–9 | 7 |
Write a key. Use \(6|3=63\) points. This makes the two repeated leaves on stem 6’s lower row understandable as two students who each scored 63 points.
Check the leaves against the list. The row counts are \(0+2+2+2+5+2+1+3+2+1=20\), matching the 20 listed scores. The lower 70s row includes 70, 72, 72, 74, and 74; the upper 70s row includes 77 and 79. This confirms that the split boundary is being applied correctly and that repeated scores have not been dropped.
Worked Example: Build a Clear Display for a Clustered Set
Worked Example: Build a Clear Display for a Clustered Set
A third fictional class has 12 exam scores: 76, 73, 82, 75, 77, 74, 81, 78, 87, 75, 80, and 84. Construct a split-stem stemplot and explain why the split is useful here.
Separate each score. The scores in the 70s have stem 7, and those in the 80s have stem 8. Place leaves 0–4 on the lower row and leaves 5–9 on the upper row. In particular, 74 goes on stem 7’s lower row, while 75 and 78 go on the upper row.
| Stem | Leaf range | Leaves, in order |
|---|---|---|
| 7 | 0–4 | 3 4 |
| 7 | 5–9 | 5 5 6 7 8 |
| 8 | 0–4 | 0 1 2 4 |
| 8 | 5–9 | 7 |
Include the key. Here the key is \(8|2=82\) points. The split is useful because it displays the lower and upper parts of the 70s and 80s separately. For example, five scores fall from 75 through 79, while two fall from 70 through 74. Those counts are visible from the leaf counts on the corresponding rows.
Verify the display. The four rows have \(2+5+4+1=12\) leaves, exactly the number of original scores. The repeated leaf 5 appears twice for the two scores of 75. All leaves are ordered, and each is in the correct range. The two 80s rows together represent 80, 81, 82, 84, and 87.
Common Mistakes and Full-Credit Communication
- Leaving out the key. A reader needs to know how the digits combine and what the values’ units are. A full-credit stemplot includes a key such as \(8|2=82\) points.
- Putting a leaf in the wrong half. Leaves 0–4 belong on the first row and leaves 5–9 on the second. In particular, 5 is in the upper half, not the lower half.
- Writing leaves in the order they appear in the data. The original list may be unsorted. Sort leaves from least to greatest within each split row.
- Removing repeated leaves. Each leaf represents one observation. If three students earned the same score, show that leaf three times.
- Confusing an empty row with a zero leaf. An empty row means no values belong to that range. A leaf 0 represents an actual score ending in zero.
- Using a split without showing both rows. A split stem has a lower row and an upper row. Showing both makes the row ranges clear, including when one half contains no scores.
- Failing to check the number of observations. Count all leaves, including repeats, and compare the total with the number of scores in the data set.
A complete response presents the stemplot with stems in order, both rows for each split stem, ordered leaves, and a key with units. It also checks that every score appears once and that the number of leaves matches the data set size. These checks catch common construction errors before anyone uses the display to describe the scores.
Check Your Understanding
For each question, use the stated exam scores and show how the stem and leaves are arranged.
- Make a split-stem stemplot for scores 61, 64, 65, 68, 72, 74, and 79. Include a key.
- In a split-stem plot, where does a score of 85 belong: the 0–4 row or the 5–9 row for stem 8?
- A class has scores 73, 73, 75, and 78. What leaves should appear on each of stem 7’s two rows, and why does one leaf repeat?
- What does an empty 0–4 row for stem 9 mean? How is that different from showing leaf 0 on that row?
- A data set contains 18 scores, but a proposed stemplot has 17 leaves. Name one important check to make before accepting the plot.