Start With the Key
In Building a Stemplot With Split Stems, you learned how a quantitative data set can be organized by placing each observation into a stem and a leaf. Reading a stemplot reverses that task: use its key to combine each stem and leaf and recover the original observations. Once the values are clear, the arrangement also helps you describe the distribution’s shape and notice clusters or gaps.
The key is the guide to the scale. Do not assume that a stem and leaf always represent a whole number, or that a leaf is automatically a decimal digit. For example, the key \(4|2=42\) minutes means stem 4 and leaf 2 represent 42 minutes. A different key, \(4|2=4.2\) centimeters, means the same pair of digits represents 4.2 centimeters. The key also identifies the variable’s units.
For a split-stem plot, the split changes which leaves appear on each row, not how the key is read. If stem 7 is split into a 0–4 row and a 5–9 row, the leaf 3 belongs on the first row and the leaf 8 on the second. With a key of \(7|3=73\) points, they represent 73 and 78 points. Read the pair as a value, not as two separate counts.
Read Values, Repeats, and Intervals
To decode a plot, first read its key. Then combine the stem and each leaf on a row to list the values. In a typical whole-number plot, stem 2 with leaves 1, 4, and 4 represents 21, 24, and 24. There are three observations on that row, and the repeated leaf means two observations have the value 24.
The ordered arrangement makes some questions quick to answer. The smallest value is represented by the first leaf in the earliest nonempty row; the largest is the last leaf in the latest nonempty row. The number of leaves on a row gives the count of observations in that row’s range. In a split-stem plot, be sure to distinguish the two ranges for the same stem: the lower row covers leaves 0–4 and the upper row covers leaves 5–9.
A blank row represents an interval with no observed values. For instance, a blank stem 3 in a whole-number plot with tens as stems means there are no observations from 30 through 39. A row with a leaf 0 is different: it represents an actual value ending in zero, such as \(3|0=30\). These distinctions matter when you describe gaps.
Use the Pattern to Describe Shape and Gaps
A stemplot displays the distribution of a quantitative variable. Describe what the pattern shows, not just what the individual values are. Look for where leaves are concentrated, whether the distribution has one or more apparent clusters, whether it is roughly balanced or has a tail extending farther on one side, and whether any values stand apart from the rest.
A cluster is a region where observations are relatively concentrated. A gap is a stretch of values or intervals with no observations in the displayed data. One missing leaf, such as no leaf 6 between leaves 5 and 7, is a small break in the observed values; it does not necessarily create a substantial gap in the distribution. Several empty rows in succession make a broader gap easier to see.
Use context and units when interpreting a pattern. A careful description might say that most recorded travel times cluster in the teens and twenties, with a gap before a smaller group in the forties. Avoid claiming that no travel time in the gap is possible: the plot shows the observations collected, not every value that could occur. Also avoid calling one unusually large observation an error unless there is evidence that it was recorded incorrectly.
Worked Example: Read a Stemplot of Practice Times
Worked Example: Read a Stemplot of Practice Times
A fictional group of students recorded how many minutes they spent on a short practice activity. The stemplot is shown below. Read the values and describe the distribution.
| Stem | Leaves |
|---|---|
| 1 | 4 7 9 |
| 2 | 1 2 2 4 6 8 |
| 3 | 1 3 5 |
Read the key. The key is \(2|4=24\) minutes. Therefore, stem 2 and leaf 4 represent 24 minutes, not 2.4 minutes. The complete list of recorded times is 14, 17, 19, 21, 22, 22, 24, 26, 28, 31, 33, and 35 minutes.
Read counts and extremes. There are three observations in the teens, six in the twenties, and three in the thirties, for a total of \(3+6+3=12\) observations. The smallest recorded time is 14 minutes and the largest is 35 minutes. The repeated leaf 2 on stem 2 represents two students who each recorded 22 minutes.
Describe the pattern. The observations are most concentrated in the twenties, with three in the teens and three in the thirties. The counts by decade rise and then fall in a roughly balanced pattern, with the twenties forming the center of the visible concentration. There is no broad empty interval between the observed values: all three decades from 10 through 39 contain observations. Some individual values are absent, such as 23 minutes, but that alone is not a broad gap. This description refers to the 12 recorded times, not to every possible practice time.
Worked Example: Find Empty Intervals in a Split-Stem Plot
Worked Example: Find Empty Intervals in a Split-Stem Plot
A fictional delivery team recorded the number of minutes required for 12 short trips. Here is a split-stem stemplot. Read the values and identify its clusters and gaps.
| Stem | Leaf range | Leaves |
|---|---|---|
| 1 | 0–4 | 2 4 |
| 1 | 5–9 | 7 8 |
| 2 | 0–4 | 2 4 |
| 2 | 5–9 | 5 7 |
| 3 | 0–4 | |
| 3 | 5–9 | |
| 4 | 0–4 | 1 3 |
| 4 | 5–9 | 6 |
| 5 | 0–4 | |
| 5 | 5–9 | |
| 6 | 0–4 | |
| 6 | 5–9 | 8 |
Decode using the key. The key is \(4|1=41\) minutes. The values are 12, 14, 17, 18, 22, 24, 25, 27, 41, 43, 46, and 68 minutes. There are 12 leaves, matching the 12 trips described.
Locate clusters and gaps. Eight trips took between 12 and 27 minutes, while three took between 41 and 46 minutes. No trip took a time in the 30s, so the two blank rows for stem 3 show an empty interval from 30 through 39 minutes. The rows for stem 5 and the lower row for stem 6 are also empty. After the value 46, the next recorded time is 68, leaving observed values from 47 through 67 absent.
Describe the overall pattern. The values form a larger cluster in the teens and twenties and a smaller cluster in the low-to-mid forties, with a broad gap in the thirties between them. The value 68 minutes stands apart from the other recorded times. It is reasonable to call it unusually high relative to this group, but the stemplot alone does not establish that it is a mistake or an outlier for a larger population. The display describes these trips; it does not prove that trips taking 30–39 or 47–67 minutes are impossible.
Worked Example: Read a Key With Decimal Values
Worked Example: Read a Key With Decimal Values
A fictional set of seedlings was measured for height in centimeters. The stemplot records measurements to the nearest tenth. Read the heights and describe where they are concentrated.
| Stem | Leaves |
|---|---|
| 2 | 1 3 4 5 5 7 8 |
| 3 | 0 1 1 2 4 |
Use the stated scale. The key is \(2|5=2.5\) centimeters. Here the stem gives the whole-number part and the leaf gives tenths. The measurements are 2.1, 2.3, 2.4, 2.5, 2.5, 2.7, 2.8, 3.0, 3.1, 3.1, 3.2, and 3.4 centimeters. The leaf 0 on stem 3 represents 3.0 centimeters; it is an observation, not an empty row.
Read repeats and spread. The leaf 5 appears twice on stem 2, so two seedlings measured 2.5 centimeters. The smallest height is 2.1 centimeters and the largest is 3.4 centimeters. The difference between these recorded extremes is \(3.4-2.1=1.3\) centimeters.
Describe the pattern in context. The heights are concentrated from about 2.3 to 3.2 centimeters, and both stems contain observations. There is no empty stem between 2 and 3 and no broad gap in the displayed measurements. The missing tenths, such as 2.6 and 2.9, are unobserved values within an otherwise fairly continuous range; they do not split the data into separate clusters. This is a useful reminder that a few missing leaves do not automatically indicate a meaningful gap.
Common Mistakes and Full-Credit Communication
- Ignoring or misreading the key. A pair such as \(2|5\) could represent 25, 2.5, or another scale. State the value using the actual key and include the units.
- Treating leaves as counts. Each leaf is one observation. Two leaves of 2 on stem 2 mean two observations of 22 under a whole-number key, not a count of two attached to one value.
- Forgetting that repeats matter. Repeated leaves are repeated data values. Include them when listing values and counting observations.
- Confusing empty rows and zero leaves. A blank row means no observations in that interval. A leaf 0 represents an actual value, such as 30 under a key of \(3|0=30\).
- Calling every missing value a gap. A missing leaf can simply be an unobserved value between nearby observations. Describe a gap in terms of the range and the pattern shown, especially when multiple rows or intervals are empty.
- Overstating what the plot proves. A gap means the displayed data contain no values in that interval. It does not prove that such values cannot occur outside the observed group.
- Describing shape without context. A complete description names the variable and uses its units. For example, say “trip times cluster in the teens and twenties” rather than “there is a big group on the left.”
For a full-credit response, show that the key has been used correctly, support claims about clusters or gaps with specific values or intervals, and describe the pattern in context. When asked about shape, do more than list the smallest and largest observations: explain where values are concentrated and whether the distribution appears balanced, clustered, or separated by gaps.
Check Your Understanding
Use the key and the displayed leaves carefully. For questions about shape or gaps, describe only what the observations show.
- A plot has key \(5|3=53\) points and stem 5 has leaves 1, 3, 3, 8. What four original values are represented, and how many observations equal 53?
- A split-stem plot uses a 0–4 row and a 5–9 row for each stem. Where does leaf 6 belong, and what interval does that row represent for stem 4?
- A whole-number stemplot has key \(2|0=20\) minutes, no leaves on stem 3, and observations on stems 2 and 4. What interval has no recorded observations?
- A decimal stemplot has key \(3|0=3.0\) centimeters. What does leaf 0 on stem 3 represent, and how is it different from an empty row?
- Why is it not justified to conclude from an empty interval in a stemplot that no value in that interval could ever occur?