Look for Concentrations and Empty Intervals
In Describing a Histogram With Real Numbers, you practiced reading value ranges and counts from grouped bars. A dotplot preserves the individual observations, so it can make small-sample features especially clear: several values close together, a stretch with no observations, or one value set apart from the rest.
As in Reading Values and Counts From a Dotplot, each dot represents one observation, and stacked dots show repeated values. Before describing a display, check its numerical scale and the variable’s units. Then scan along the axis: where do observations gather, where are there no dots, and does any dot sit noticeably apart from its neighbors?
A gap is not just a spot where there are fewer dots than elsewhere. It is a stretch containing no observations. Describe its location using the values on both sides: for example, “There are no recorded times from 6 through 10 minutes; one cluster is from 2 to 5 minutes and another is from 11 to 14 minutes.” That wording identifies the empty interval and the neighboring concentrations instead of saying only that the graph “has a gap.”
Clusters are regions, not necessarily single values. A cluster may include several distinct values, with some values repeated more often than others. In Unimodal, Bimodal, and Multimodal Distributions, you learned to describe peaks as regions of concentration rather than treating every change in frequency as a new peak. Apply the same restraint to dotplots: a small rise and fall in stacked dots does not automatically create separate clusters.
Describe What the Scale Supports
The distance between observations matters. A gap from 5 to 6 units is not necessarily as visually important as a gap from 5 to 20 units. Consider the overall scale and the spacing of the observations on both sides. If the display shows a conspicuous empty stretch between two regions of concentration, it is reasonable to describe two clusters separated by a gap. If there is only a small spacing difference, describe it more modestly.
The recorded precision also matters. If the measurements are whole numbers, you might say that no recorded values are 6 through 10. If the variable is measured to the nearest tenth, a gap between 5.2 and 7.8 has a different meaning. Describe the empty interval at the level supported by the axis and data. A dotplot cannot show unrecorded measurements or establish why a region is empty.
In a small sample, a single observation can stand apart from a cluster. Describe that position accurately: “One observation, at 24 minutes, is separated from the other times, which range from 8 to 13 minutes.” Do not jump from “separated” to “definitely an outlier.” As covered in Identifying Outliers and Unusual Features, a dotplot can show an observation that appears unusual, while the 1.5 IQR rule provides a separate method for flagging possible outliers. Even a flagged value should be described carefully and in context.
Worked Examples
Worked Example: Two Clusters of Waiting Times
A fictional community center records how many minutes 11 visitors wait before joining a scheduled activity. The dotplot’s observations are \(2, 3, 3, 4, 4, 5, 11, 12, 12, 13, 14\). Describe the clusters and the gap.
| Wait time (minutes) | Number of dots |
|---|---|
| 2 | 1 |
| 3 | 2 |
| 4 | 2 |
| 5 | 1 |
| 6–10 | 0 |
| 11 | 1 |
| 12 | 2 |
| 13 | 1 |
| 14 | 1 |
State. The quantitative variable is visitors’ waiting time, measured in minutes. The dotplot shows two regions with several observations and an empty interval between them.
Plan. Group nearby observations into concentrations, then use the scale to name the full span of each cluster and the values with no observations. Avoid inferring what caused the pattern.
Do. Six visitors have waiting times from 2 to 5 minutes, and five have waiting times from 11 to 14 minutes. There are no observations from 6 through 10 minutes. The repeated values at 3, 4, and 12 contribute to the concentrations but do not change the clusters’ ranges.
Conclude. The waiting times form one cluster from 2 to 5 minutes and another from 11 to 14 minutes, with no recorded waiting times from 6 through 10 minutes. The dotplot supports describing two concentrations in this group of visitors; by itself, it does not explain why the waiting times are separated.
Worked Example: One Time Separated From the Others
A fictional group of nine students records how many minutes it takes each person to finish a short puzzle. The recorded times are \(8, 9, 9, 10, 10, 11, 12, 13, 24\). Describe the main concentration, the empty interval, and the observation at 24 minutes.
| Time (minutes) | Number of dots |
|---|---|
| 8 | 1 |
| 9 | 2 |
| 10 | 2 |
| 11 | 1 |
| 12 | 1 |
| 13 | 1 |
| 14–23 | 0 |
| 24 | 1 |
State. The puzzle-completion times are concentrated between 8 and 13 minutes, while one recorded time is much farther to the right.
Plan. Report the range containing the main concentration, locate the empty interval, and identify the separated observation with its units. Call the observation isolated if appropriate, but do not label it an outlier solely from its position.
Do. Eight of the nine times range from 8 to 13 minutes. No recorded times fall from 14 through 23 minutes, and the remaining time is 24 minutes. Thus, 24 minutes is separated from the cluster by an empty interval.
Conclude. The dotplot shows a cluster of puzzle times from 8 to 13 minutes, no times from 14 to 23 minutes, and one isolated time of 24 minutes. The display supports calling 24 minutes an isolated observation. To decide whether it is flagged by the 1.5 IQR rule, we would need to calculate the quartiles and fences, as in Applying the 1.5 IQR Rule for Outliers.
Worked Example: A Short Gap Between Concentrations
A fictional environmental club records the number of plastic pieces found along several equal-length sections of a streambank. The counts are \(3, 3, 4, 4, 5, 8, 9, 9, 10, 10, 13\). Describe the concentrations and gaps, taking care not to overstate what a short gap means.
| Plastic pieces per section | Number of dots |
|---|---|
| 3 | 2 |
| 4 | 2 |
| 5 | 1 |
| 6–7 | 0 |
| 8 | 1 |
| 9 | 2 |
| 10 | 2 |
| 11–12 | 0 |
| 13 | 1 |
State. The quantitative variable is the number of plastic pieces found per section. The display has concentrations among the smaller counts and the middle counts, plus one higher count.
Plan. Identify the groups of nearby values and list the empty integer counts between them. Then describe the prominence of the gaps cautiously: a gap exists whenever no observations occur in an interval, but its importance depends on the full display and scale.
Do. One concentration runs from 3 to 5 pieces per section, and another runs from 8 to 10. No sections have counts of 6 or 7. There are also no counts of 11 or 12, before the single observation of 13.
Conclude. The dotplot has concentrations from 3 to 5 and from 8 to 10 pieces, separated by the empty counts 6 and 7. A single section has a count of 13, with 11 and 12 absent. The display supports reporting these gaps and concentrations; whether the count of 13 is meaningfully unusual should be judged from the whole pattern and, if needed, another method for flagging possible outliers.
Turn the Display Into a Careful Description
A strong description tells the reader what the dots show, not what you imagine happened. Name the group and variable, state the units, report the cluster ranges, and give the values in any important gap. When a lone observation stands apart, state its value and position relative to the others. This makes the description specific enough to check against the display.
Be precise about whether you mean an observed value or a range. “The cluster is from 8 to 13 minutes” means that the observations in that concentration run from 8 through 13; it does not mean that every value in that range occurred. If a gap is between clusters at 5 and 11 minutes, say which recorded values are absent rather than implying that 5 or 11 is missing.
A gap can suggest a question, but it does not answer it. Two clusters might reflect different conditions, schedules, or subgroups, but a dotplot alone does not identify a cause or prove that distinct populations produced the data. In context, you can say the display “suggests two concentrations” or “shows two clusters.” Avoid claiming that the observations belong to two known groups unless the data collection provides that information.
Likewise, an isolated value is not automatically an error. It could be a valid observation, a recording mistake, or a person with a different experience. The dotplot does not tell you which. Describe the visible feature first; investigate the data or use a stated outlier procedure before making stronger claims.
Common Mistakes and AP Exam Tips
- Calling every low-frequency region a gap. A gap contains no observations. A region with a few dots is a low concentration, not an empty interval.
- Describing a gap without its location. “There is a gap” is vague. A clearer answer names the values or interval with no observations and the clusters on either side.
- Treating every uneven stack as a new cluster. Small samples naturally have ups and downs in dot counts. Identify broad regions of concentration rather than interpreting every change as a separate peak.
- Calling a separated dot a definite outlier. Say it is isolated or appears unusual in the dotplot. A full-credit response does not claim that it is a confirmed outlier unless a relevant rule or additional information supports that conclusion.
- Inventing a cause or subgroup. Two clusters may prompt further questions, but the display alone does not prove why the pattern occurred. Describe what is shown and keep explanations tentative.
- Leaving out context or units. “The values are 8 to 13” is incomplete. Identify what was measured, for whom or what, and in which units.
- Ignoring the scale. The numerical distance between dots helps determine whether an empty interval is prominent. Use the axis rather than judging spacing from the page alone.
Check Your Understanding
Use the dotplot values and context in each question. Distinguish what the display shows from what it does not establish.
- A dotplot of delivery times has a cluster from 18 to 22 minutes, no observations from 23 through 30 minutes, and another cluster from 31 to 35 minutes. Describe the clusters and gap in context.
- A dotplot has eight observations between 4 and 9 seconds and one observation at 19 seconds, with no values from 10 through 18 seconds. What is a careful way to describe the observation at 19 seconds? What conclusion should you avoid making from the dotplot alone?
- In a dotplot of daily bird counts, values 6 and 8 occur, but no value of 7 occurs. Is 7 a gap? Explain what additional context about the scale and full pattern would help determine whether it is a prominent feature.
- A small-sample dotplot shows frequencies that rise from one value to the next and then fall slightly, with no empty interval. Does every rise and fall identify a separate cluster? Explain.
- Two clusters appear in a dotplot of student travel times. Can you conclude from the display alone that the students belong to two different neighborhoods? Explain.