Two Displays for a Small Set of Values
Suppose a group of students records how many hours each spent studying yesterday. A graph should show more than the list of numbers: it should make it easy to see where observations gather, how widely they spread, and whether any values sit apart from the rest. For a small data set, a dotplot and a stemplot do this while keeping the individual observations visible.
Both displays rely on the same basic rule: represent every observation, not just every distinct value. If three students each report the same number of hours, the graph must show that value three times. A display that loses repeated observations can give a false impression of how common each value is.
Worked Example: One Data Set, Two Displays
Worked Example: One Data Set, Two Displays
Twelve students report studying for the following whole numbers of hours yesterday: \(1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 6, 10\). Construct a dotplot and a stemplot. Then describe the concentration, range, and any value that stands apart.
Dotplot. Use an axis with one-hour steps from 1 through 10. Place one dot above a value for every student who reported that many hours. For example, the three observations of 3 hours require three dots. The completed display is:
| Study hours | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| Dots | • | • • | • • • | • • | • • | • | • |
The columns with no dots represent values between the observed values 6 and 10. They are included so the one-hour scale remains clear; they are not observations.
Stemplot. Use the tens digit as the stem and the ones digit as the leaf. Sort the leaves from smallest to largest within each stem:
| Stem | Leaves |
|---|---|
| 0 | 1 2 2 3 3 3 4 4 5 5 6 |
| 1 | 0 |
Key: \(0\mid 1 = 1\) hour; \(1\mid 0 = 10\) hours. The stem and leaf together reproduce the original value.
In the dotplot, the observations gather between 2 and 5 hours, with the tallest stack at 3 hours. The smallest observation is 1 hour and the largest is 10 hours, so the range, defined as maximum minus minimum, is \(10-1=9\) hours. The 10-hour observation sits apart from the main group in this display. That is a visual description, not a claim that the observation is wrong.
How to Build and Check the Displays
A dotplot is easiest to read when the axis is numerical, evenly spaced, and labelled with units. Choose a scale that includes every observed value. Place one dot for each observation directly above its value, stacking dots when values repeat. Do not spread repeated values sideways across the axis: their shared position is what shows that the recorded values are equal.
A stemplot requires a consistent place-value split. The stem contains the leading digit or digits, and the leaf contains the final digit. For whole-hour values from 1 to 10, stems 0 and 1 with one-digit leaves preserve the values. The key is essential: without it, a reader might not know whether \(1\mid 0\) means 10 hours, 1.0 hours, or something else.
Use these checks after drawing either display:
- Count the marks. A dotplot should have one dot per observation. A stemplot should have one leaf per observation. The total here is 12.
- Check the endpoints. The lowest and highest plotted values should match the smallest and largest observations in the list.
- Check repeated values. A value that occurs three times should appear as three dots or as three matching leaves.
- Check the key and scale. The key must decode leaves into the recorded units, and the dotplot axis must show the actual numerical spacing.
These checks give a quick way to catch omissions and transcription errors. In the first example, the stemplot contains 11 leaves on stem 0 and 1 leaf on stem 1, for 12 observations altogether. Its lowest and highest decoded values are 1 and 10, matching the data list and the dotplot.
Why the Two Displays Preserve the Data
The displays arrange observations differently, but neither should discard or alter them. In a dotplot, an observation at value \(x\) contributes one dot at \(x\). In a stemplot, the chosen place-value rule assigns that observation one stem-and-leaf pair that reconstructs \(x\). Thus each observation contributes exactly one mark in either graph.
This also explains a useful check. If the data set contains \(n\) observations, the dotplot must have \(n\) dots and the stemplot must have \(n\) leaves. In either display, the smallest and largest decoded observations are still present, so the range can be found by subtracting the smallest from the largest.
The rule does not depend on whether the data include repeats: repeated observations each contribute a separate mark, while the endpoints remain the least and greatest observed values.
Worked Example: Values on a Half-Hour Scale
Worked Example: Values on a Half-Hour Scale
A different hypothetical group reports study times of \(0.5, 1.0, 1.5, 1.5, 2.0, 2.5, 3.0, 3.0, 3.5,\) and \(4.0\) hours. Make both displays and identify where the observations are concentrated.
Dotplot. Since all values fall on half-hour increments, use equally spaced positions from 0.5 to 4.0 hours. There are two observations at 1.5 hours and two at 3.0 hours, so those positions each need two dots.
| Study hours | 0.5 | 1.0 | 1.5 | 2.0 | 2.5 | 3.0 | 3.5 | 4.0 |
|---|---|---|---|---|---|---|---|---|
| Dots | • | • | • • | • | • | • • | • | • |
Stemplot. Let each stem give the whole-hour part and each leaf give the tenths digit. The leaves are ordered within each stem:
| Stem | Leaves |
|---|---|
| 0 | 5 |
| 1 | 0 5 5 |
| 2 | 0 5 |
| 3 | 0 0 5 |
| 4 | 0 |
Key: \(1\mid 5 = 1.5\) hours. For example, the two leaves 5 on stem 1 preserve the two reports of 1.5 hours.
The dotplot shows repeated observations at 1.5 and 3.0 hours; the stemplot shows the same repeats as duplicate leaves. The values extend from 0.5 to 4.0 hours, giving a range of \(4.0-0.5=3.5\) hours. The observations are spread across that interval, with no single value occurring more than twice.
Worked Example: A Value That Sits Apart
Worked Example: A Value That Sits Apart
Eleven students report studying \(1, 2, 2, 3, 3, 4, 4, 5, 5, 6,\) and \(12\) hours. Make displays that help assess whether any value stands apart, and calculate the range.
Dotplot. Use one-hour steps from 1 through 12 so the gap between the main group and 12 is visible.
| Study hours | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Dots | • | • • | • • | • • | • • | • | • |
Stemplot. With the tens digit as the stem and the ones digit as the leaf:
| Stem | Leaves |
|---|---|
| 0 | 1 2 2 3 3 4 4 5 5 6 |
| 1 | 2 |
Key: \(1\mid 2 = 12\) hours. There are 10 leaves on stem 0 and 1 on stem 1, giving 11 observations. The range is \(12-1=11\) hours. The value of 12 hours is separated from the other observations by the empty positions from 7 through 11, so it stands apart visually. The displays alone do not establish why that report differs or whether it is inaccurate; those questions require checking the measurement and its context.
What Concentration, Range, and Unusualness Tell You
Concentration describes where observations gather. A tall stack in a dotplot or many leaves in a nearby part of a stemplot signals that several observations fall in that region. Describe the values and units: for example, “most reports are between 2 and 5 hours,” rather than simply “the data are concentrated.”
Range describes the distance from the minimum to the maximum observation in the variable’s units. It uses only those two endpoint values, so it tells how far the data extend but not how observations are distributed between the endpoints. In the first example, a range of 9 hours does not mean that students typically differ by 9 hours; it is the difference between the largest and smallest reports.
A value may look unusual when it is separated from the rest or occurs in a region with few nearby observations. “Unusual” in this graphical description means that the value stands out relative to the displayed data. It does not by itself identify a mistake, explain the value, or establish a formal cutoff. Check the original record and relevant context before deciding what to do with a value that appears unusual.
Common Mistakes
- Counting distinct values instead of observations. If 3 hours appears three times, the dotplot needs three dots and the stemplot needs three leaves. One mark would conceal the frequency of that report.
- Leaving out the key. A stemplot is incomplete unless the reader can tell what a stem and leaf represent. State the key with units, such as \(1\mid 5 = 1.5\) hours.
- Using unequal spacing on a dotplot axis. Numerical distances should correspond to equal visual distances. Otherwise, the graph can make gaps or clusters look larger or smaller than they are.
- Calling a separated value an error. A graph can show that a value stands apart, but it cannot show that the value was recorded incorrectly. Verify the observation before making that judgment.
Check Your Understanding
Use small, hypothetical study-hour data sets to answer these questions.
- For the observations \(1, 1, 2, 4\), how many dots should appear above 1, and how many leaves should represent 1 in a stemplot?
- Write a key for a stemplot in which stem 2 and leaf 5 represent 2.5 hours.
- For \(2, 3, 3, 5, 8\) hours, find the range and state which two observations determine it.
- Why should a dotplot axis include evenly spaced positions between observed values when showing a gap?
- A stemplot shows one leaf well separated from the others. What can you conclude from the display, and what can you not conclude without checking the context?