One Scale, Two Distributions
In Reading a Stemplot and Its Key, you learned to recover observations from a stemplot and use its pattern to describe a quantitative distribution. A back-to-back stemplot extends that display to two groups: the groups’ leaves appear on opposite sides of a shared column of stems. This makes it possible to compare two distributions while still seeing the individual values.
Suppose two classes take the same quiz. A back-to-back stemplot can show how their scores are distributed across the same score scale. The shared stems make corresponding score ranges easy to compare. Each leaf still represents one student’s score, and the key still determines what each stem–leaf pair means.
The main construction detail is the direction of the leaves. On the right, list leaves in increasing order as you move away from the stem. On the left, list leaves in decreasing order as you move away from the stem. That way, the smallest left-side leaf sits closest to the stem, just as the smallest right-side leaf does. Read each pair from the stem outward to recover the value.
For example, with a key of \(6|2=62\) points, the left-side leaves \(8\ 5\ 2\) beside stem 6 represent 68, 65, and 62 points. Read from the stem outward, the leaves are 2, 5, 8, in increasing order. On the right, leaves \(2\ 5\ 8\) represent the same three scores. The two sides are arranged in opposite directions so each group’s leaves increase as you read away from the stem.
A Careful Construction Routine
Start by identifying the variable, units, and groups. Here, the variable is quiz score, measured in points, and the groups are the two classes. Then choose stems that cover the scores in both groups. The stems should use the same place-value convention and the same rows for both groups; do not give one class a different scale to make its display look fuller.
For whole-number scores, the tens digit can be the stem and the ones digit the leaf. Include every stem needed to cover the lowest and highest scores in either group.
For each score, record its ones digit beside the matching tens stem. Put the first group on the left and the second group on the right.
On the left, arrange leaves in decreasing order from left to right, so they increase when read outward from the stem. On the right, arrange leaves in increasing order from left to right.
Identify both groups, name the variable and units, and provide a key. Count the leaves on each side and confirm that each group’s leaf count equals its number of scores.
Keep empty rows when they help show the scale or make the groups’ patterns easier to compare. An empty row means that group has no observations in that stem’s range. A row may contain leaves on one side and none on the other; that is a valid feature of the display, not a reason to omit the stem.
Worked Example: Build a Plot for Two Quiz Classes
Worked Example: Build a Plot for Two Quiz Classes
The following scores are invented for this example. Class A earned 62, 65, 68, 71, 72, 74, 75, 78, 81, 84, 87, and 89 points. Class B earned 58, 63, 66, 69, 70, 73, 76, 77, 79, 82, 83, and 86 points. Construct a back-to-back stemplot and make a brief comparison.
Choose the common stems and key. The lowest score is 58 and the highest is 89, so stems 5, 6, 7, and 8 cover both classes. Use the key \(6|2=62\) points. Both classes have 12 scores, so each side of the completed plot should contain 12 leaves.
Place and order Class A’s leaves on the left. Class A’s scores in the 60s have leaves 2, 5, and 8. Written on the left, they appear as \(8\ 5\ 2\). Its 70s have leaves 1, 2, 4, 5, and 8, so the left-side order is \(8\ 5\ 4\ 2\ 1\). Its 80s have leaves 1, 4, 7, and 9, written \(9\ 7\ 4\ 1\). Class A has no score in the 50s.
Place and order Class B’s leaves on the right. Its score in the 50s gives leaf 8. In the 60s, the leaves are 3, 6, and 9; in the 70s, they are 0, 3, 6, 7, and 9; in the 80s, they are 2, 3, and 6. The right-side leaves are written in increasing order.
| Class A leaves | Stem | Class B leaves |
|---|---|---|
| 5 | 8 | |
| 8 5 2 | 6 | 3 6 9 |
| 8 5 4 2 1 | 7 | 0 3 6 7 9 |
| 9 7 4 1 | 8 | 2 3 6 |
Check the plot and compare. Class A has \(3+5+4=12\) leaves, and Class B has \(1+3+5+3=12\), matching the two lists. Both distributions have their greatest concentration in the 70s. Class A’s scores range from 62 to 89 points, while Class B’s range from 58 to 86 points. The scores overlap substantially, so the plot does not show one class scoring uniformly above the other. It does show that the observed extremes differ: Class B has the lower minimum, and Class A has the higher maximum. These descriptions apply to the scores displayed, not to every score either class might earn.
Compare the Distributions, Not Just the Leaves
A back-to-back stemplot is useful because it preserves individual values and makes broad patterns visible side by side. Compare the groups using features of their distributions: where scores are concentrated, how spread out the scores appear, whether the patterns have similar shapes, and whether either group has a visible gap or value that stands apart. As in earlier tutorials on reading stemplots, use the key to check the actual score ranges before describing a pattern.
Keep each comparison tied to its group and the quiz context. Instead of writing “the left side is higher,” say, “Class A’s scores extend to 89 points, while Class B’s extend to 86 points.” If one class has more leaves in a score range, distinguish a larger count from a greater proportion: the count comparison is most direct when the groups have equal sizes. When group sizes differ, first note that there are different numbers of students; the plot still allows you to compare the locations and patterns of their recorded scores, but raw leaf counts alone do not represent equal-sized groups.
A back-to-back stemplot does not automatically establish that one group performed better overall. A few high values, a lower minimum, or a denser stem does not tell the whole story. Describe the overall distributions and support your comparison with specific stems, leaves, or score ranges. The display is a summary of the observations collected; an empty row is not evidence that scores in that range are impossible.
Worked Example: Read and Compare Two Class Distributions
Worked Example: Read and Compare Two Class Distributions
A second fictional pair of classes completed the same quiz. The plot below uses a key of \(7|3=73\) points. Describe similarities and differences in their observed score distributions.
| Class C leaves | Stem | Class D leaves |
|---|---|---|
| 8 4 | 5 | 7 |
| 8 6 4 1 | 6 | 2 3 7 |
| 9 5 2 | 7 | 1 3 4 6 |
| 7 3 | 8 | 2 5 8 |
| 1 | 9 |
Read the scale and count observations. The key means stem 7 with leaf 3 represents 73 points. Class C has \(2+4+3+2+1=12\) scores. Class D has \(1+3+4+3=11\) scores. The empty Class D row for stem 9 indicates that it has no score in the 90s.
Compare location and range. Both classes have several scores in the 70s, so that range is a visible area of concentration for each. Class C’s observed scores range from 54 to 91 points, and Class D’s range from 57 to 88 points. Thus, Class C has both the lower minimum and the higher maximum in these samples. Their ranges overlap, and Class D’s scores are not all below Class C’s: for instance, the plot includes Class D scores in the 80s as well as scores in the 70s.
State the comparison carefully. The plot shows that both classes have observations in the 50s, 60s, 70s, and 80s, while only Class C has an observation in the 90s. Class C’s scores extend farther in both directions than Class D’s observed scores. Since the classes have different sample sizes, the numbers of leaves in a range should not be treated as a direct comparison of percentages. These are comparisons of the displayed quiz scores, not proof of a difference in all students’ performance.
Worked Example: Construct a Plot With an Empty Row
Worked Example: Construct a Plot With an Empty Row
Two more fictional classes have these quiz scores. Class E: 61, 64, 68, 72, 75, 78, 83, 86. Class F: 57, 63, 67, 71, 74, 79, 82, 88. Construct the plot and identify the empty range for each group.
Set up the stems. Scores range from 57 through 88, so use stems 5, 6, 7, and 8. The key is \(6|1=61\) points. Class E has no score in the 50s, while Class F has the score 57. Both classes have scores in the 60s, 70s, and 80s.
Sort each group’s leaves. For Class E, stem 6 has leaves 1, 4, and 8; stem 7 has 2, 5, and 8; stem 8 has 3 and 6. Place each left-side group in reverse order. For Class F, the 50s leaf is 7; the 60s leaves are 3 and 7; the 70s leaves are 1, 4, and 9; and the 80s leaves are 2 and 8. Place right-side leaves in increasing order.
| Class E leaves | Stem | Class F leaves |
|---|---|---|
| 5 | 7 | |
| 8 4 1 | 6 | 3 7 |
| 8 5 2 | 7 | 1 4 9 |
| 6 3 | 8 | 2 8 |
Check and interpret. Each side has eight leaves, as required by the eight scores in each list. The Class E side is empty for stem 5, meaning it has no recorded score from 50 through 59 points. Neither group has an empty stem from 6 through 8. The plot shows both groups represented in the 60s, 70s, and 80s, with Class F’s scores extending into the 50s. An empty stem on one side is meaningful to describe, but it does not mean that scores in that interval could not occur.
Common Mistakes and Full-Credit Communication
- Putting both groups on the same side. A back-to-back stemplot has one group on each side of the shared stems. Label which group is on the left and which is on the right.
- Sorting both sides in the same direction. The left leaves should decrease from left to right; the right leaves should increase from left to right. On both sides, the leaves increase when read outward from the stem.
- Using different stems or keys. That makes the groups’ values difficult to compare and can misrepresent their scales. Use one set of stems and one key for both.
- Leaving out observations or repeats. Each leaf represents one score, including a score that appears more than once. Count both sides and check against the group sizes.
- Comparing counts without considering group sizes. If one class has more students, it may have more leaves in a range simply because it is larger. Name the group sizes and avoid treating counts as percentages.
- Making a conclusion from one feature alone. A maximum, minimum, or dense stem does not by itself describe an entire distribution. Compare the overall patterns and give evidence from the plot.
- Forgetting context and units. A complete comparison names the classes, the quiz scores, and points. For example, say “Class E has no recorded score in the 50s” rather than “that side has a gap.”
For full-credit communication, identify the key, check the number of leaves for each group, and compare specific features in context. Say what the displayed scores show; do not claim that the plot proves a cause or describes scores beyond the students represented.
Check Your Understanding
Use the shared key and remember that left-side leaves are written in the opposite direction from right-side leaves.
- With a key of \(7|4=74\) points, what score does the left-side entry \(9\ 4\ 1\) beside stem 7 represent?
- Why are left-side leaves usually written in decreasing order from left to right?
- Two classes have different sample sizes. Why should you be cautious when comparing the number of leaves each class has in the 70s?
- One class has an empty row for stem 8 and the other has several leaves there. What does the empty row show, and what does it not prove?
- What checks can confirm that a completed back-to-back stemplot includes all the scores exactly once?