Why Use Dotplots to Compare Small Groups?
A graph can summarize a group while hiding the observations that produced it. When a group is small, it is often useful to see every measured value: the exact locations of the observations, which values repeat, and where there are gaps. A dotplot does this by representing each observation with one dot.
To compare groups, place their dotplots on the same numerical scale and align the rows. Then a position on the horizontal axis represents the same value for every group. This makes it possible to compare patterns without losing sight of the individual data values. Dotplots are most useful when the groups are small enough that plotting every observation remains readable.
The horizontal axis shows the variable and its units. Each group row has its own dots, and each dot represents one individual observation—not a group average or a range of values. The height of a stack shows how many observations in that group have that exact value; it does not indicate a larger value on the horizontal scale.
Reading an Aligned Dotplot
Consider hypothetical quiz scores for two small groups. The score is measured in points, and both groups have eight students. Each column below corresponds to the same score for both groups. Dots on separate lines in a cell are stacked at that score.
| Group | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|
| A | ● | ||||||||
| ● | ● | ||||||||
| ● | |||||||||
| ● | |||||||||
| B | ● | ||||||||
| ● | |||||||||
| ● | ● | ||||||||
| ● | |||||||||
| ● | ● | ● |
The plot shows Group A's scores as 2, 3, 3, 5, 6, 6, 6, and 8. Group B's scores are 1, 4, 4, 5, 7, 7, 8, and 9. The blank cells are positions with no observations in that group. For example, Group A has no score of 4 or 7, while Group B has no score of 2, 3, or 6.
The stack of two dots at 3 in Group A means two students received exactly 3 points. The three dots at 6 mean three students received exactly 6 points. In Group B, two dots at 4 and two at 7 likewise show repeated values. A single dot in each group at 5 means each group has one observation at 5; it does not mean either group has a repeated value at 5.
Worked Example: Describing Clusters and Gaps
Use the aligned quiz-score plot to describe where observations cluster and where there are gaps. Check that the displayed dots account for all observations in each group.
Group A. The values are 2, 3, 3, 5, 6, 6, 6, and 8. The dots are concentrated at 3 and 6, with a particularly tall stack at 6. There are no observations at 4 or 7, so those positions are gaps in this small set of observed scores.
Group B. The values are 1, 4, 4, 5, 7, 7, 8, and 9. The repeated values are 4 and 7. There are no observations at 2, 3, or 6. The dots from 7 through 9 form a short run of occupied positions, while the group also has observations at 1 and 4.
Dot-count check. If \(c_g(v)\) is the number of dots for group \(g\) at value \(v\), then the group size is the sum of those counts. For Group A, the nonzero counts are 1, 2, 1, 3, and 1, so \(1+2+1+3+1=8\). For Group B, they are 1, 2, 1, 2, 1, and 1, so \(1+2+1+2+1+1=8\). Both plots therefore display the stated eight observations.
Constructing a Dotplot Correctly
To make a comparison useful, the two groups must use the same horizontal scale, with equal distances representing equal changes in the measurement. Start the scale at a value that includes the data from every group, and label the variable and units. For each group, place one dot at the appropriate position for every observation. When a value occurs more than once in that group, stack the corresponding dots vertically.
A compact mathematical description helps explain why a dotplot preserves the data. For a group \(g\), let \(c_g(v)\) be the number of observations equal to \(v\). Every observation has exactly one value, so the observations can be partitioned into sets according to their values. Those sets do not overlap, and together they contain all \(n_g\) observations. Therefore:
The sum only needs to include values that occur; all other counts are zero. Since each observation contributes one dot to exactly one stack, the total number of dots must equal the group size. This is also a useful check when drawing by hand.
Worked Example: Building Two Rows From Raw Data
Suppose a hypothetical exercise records the number of minutes needed to complete a short task. Group C has observations 4, 4, 5, 7, 7, and 8 minutes. Group D has observations 3, 5, 5, 6, 8, and 8 minutes. Construct an aligned dotplot using integer minute positions from 3 through 8.
Step 1: Count each value. For Group C, the counts at 3, 4, 5, 6, 7, and 8 are 0, 2, 1, 0, 2, and 1. For Group D, they are 1, 0, 2, 1, 0, and 2.
Step 2: Place one dot per observation. The plot below uses one row per group and a shared minutes scale. Dots are vertically stacked when the count is greater than one.
| Group | 3 min | 4 min | 5 min | 6 min | 7 min | 8 min |
|---|---|---|---|---|---|---|
| C | ● ● | ● | ● ● | ● | ||
| D | ● | ● ● | ● | ● ● |
Step 3: Verify the totals. Group C has \(0+2+1+0+2+1=6\) dots, matching its six recorded times. Group D has \(1+0+2+1+0+2=6\) dots, also matching its six observations. This check catches missing observations and accidental extra dots.
The repeated times are clear: two observations in Group C took 4 minutes and two took 7 minutes; two in Group D took 5 minutes and two took 8 minutes. The plot retains the exact values rather than combining nearby values into intervals.
What Comparisons Can the Plot Support?
An aligned dotplot supports direct visual descriptions of the observed groups. You can note whether one group's dots tend to lie at higher or lower values, whether one group occupies a wider span of the displayed scale, and whether the groups have different concentrations or gaps. Use the variable's units and name the groups in your description. A plot alone does not explain why the patterns differ.
For example, in the quiz-score plot, Group A has its largest stack at 6 points, while Group B has repeated scores at 4 and 7 points. Group A's observed scores run from 2 to 8 points, and Group B's from 1 to 9 points. Those statements describe the plotted observations; they do not establish that membership in a group caused a score difference.
Do not assume that every visual feature is stable or important. A small group can show an apparent gap or a tall stack simply because it contains few observations. Report what the graph displays, and avoid claiming that a pattern must hold in a wider population unless the study design and an appropriate analysis support that conclusion.
Worked Example: Distinguishing Repetition From Alignment
Suppose a hypothetical clinic records waiting times, in minutes, for four patients in each of two service areas. Area E has times 10, 12, 12, and 15. Area F has times 8, 12, 14, and 15. Interpret the dots at 12 and 15 on an aligned dotplot.
At 12 minutes, Area E has two dots stacked in its row, while Area F has one dot in its row. Thus, 12 minutes is repeated within Area E, but appears once in Area F. At 15 minutes, there is one dot in each row. The two dots line up horizontally because both areas have an observation at 15 minutes; they are two separate patients, not a repeated value within either area.
The total-count check is \(1+2+0+1=4\) observations for Area E, corresponding to 10, 12, 12, and 15 minutes. For Area F it is \(1+1+1+1=4\), corresponding to 8, 12, 14, and 15 minutes. The aligned position at 15 minutes represents two observations across the two groups combined, but the plot keeps them in their separate group rows.
Common Mistakes
- Drawing one dot for each distinct value instead of each observation. That loses repetitions. If a group has three observations at 6, draw three vertically stacked dots at 6, not one.
- Treating dots in different group rows as a within-group repeat. Two dots aligned at the same horizontal value but in different rows belong to different groups. Count repeats within each row.
- Using separate horizontal scales for separate groups. The same horizontal position could then stand for different values, making visual comparison misleading. Use one common scale with the same units and spacing.
- Leaving out the units or unclear tick marks. Without a labeled, interpretable scale, a reader cannot reliably tell which measurements the dots represent.
- Reading a gap as proof that a value is impossible. A gap means only that the value was not observed in that group of data. It does not establish that no individual could have that value.
Check Your Understanding
Answer these questions using the definitions and methods in this tutorial.
- A group has measurements 2, 2, 4, and 6. How many dots should appear at 2, and how many dots should the full row contain?
- In an aligned plot, one dot in each of two group rows appears above the same horizontal value. Is that a repeated observation within either group? Explain.
- Why should aligned dotplots use a common horizontal scale with the same spacing and units?
- A row has stacks of 1, 3, and 2 dots at its occupied positions. How many observations are represented, and what calculation checks your answer?
- What does a gap between occupied positions tell you about the displayed data, and what does it not prove?