Read Two Dotplots on a Shared Scale
In Comparing Outliers and Unusual Features, you learned to assess unusual values relative to each group’s own distribution. Now we will compare two small samples displayed as dotplots on shared axes. A dotplot places one dot for each observation, stacking dots when values repeat. With both groups on the same numerical scale, you can compare where their observations fall and how they are distributed.
A useful comparison does more than say one group is “higher” or “more spread out.” It names the variable and groups, points to evidence in the dotplots, and describes how the distributions differ or are similar. Depending on the display, relevant features include center, spread, shape, clusters, gaps, overlap, and unusual values. As in Why Comparing Distributions Needs All Four Features, do not stop at one feature when the question calls for an overall comparison.
Before comparing, check that both displays show the same quantitative variable and units. Confirm that the tick marks represent the same values and that neither group has been shifted or rescaled. Shared axes make horizontal positions comparable; if the scales differ, apparent differences in center or spread could be misleading.
A Practical Method for Comparing Dotplots
Begin with the overall location of the dots. A group whose dots tend to lie farther to the right has generally larger observed values. To describe center more specifically, compare the medians of both groups or the means of both groups, as appropriate. Use the same measure for each group. The earlier tutorials Choosing Mean or Median to Describe Center and Comparing Centers Using Medians and Means explain how to choose and interpret those summaries.
Next, compare spread: how much the observations vary. A dotplot shows spread directly through the horizontal extent and the concentration of its dots. You can compare ranges, or compare IQRs when quartiles are available. Do not compare one group’s range with the other group’s IQR. Comparing Spread Using IQR and Standard Deviation explains why matching measures matter.
Then look at shape and details. Are dots concentrated in one region, or are there distinct clusters? Is there a gap? Does one group have a more balanced pattern, or a tail extending toward larger or smaller values? Is a value separated from the rest? The tutorials Describing a Dotplot With Gaps and Clusters and Comparing Shapes of Two Distributions explain these features. When comparing, name which group displays the feature.
Finally, consider overlap. If both groups have dots across some of the same values, their observed ranges overlap. That does not mean the distributions are identical. Conversely, separated dotplots describe the samples shown; by themselves, they do not establish that every member of one broader population has a larger value than every member of another.
Verify that the variable, units, and horizontal scale are the same for both groups. Notice whether the sample sizes differ.
Describe which group tends to have larger values, and compare the medians or means if those summaries are useful and available.
Look at the horizontal extent, concentration, shape, clusters, gaps, and any values that stand apart.
Name the variable and both groups, state the important similarities or differences, and support them with evidence from the dotplots.
Worked Example: Similar Spread but Different Centers
Worked Example: Similar Spread but Different Centers
A fictional school compares the number of minutes students in two groups spend walking to school. The dotplots use the same horizontal scale. The observations are listed below; each repeated value represents a stack of dots at that position.
Group East: 4, 5, 5, 6, 6, 7, 7, 8, 8, 9 minutes.
Group West: 7, 8, 8, 9, 9, 10, 10, 11, 11, 12 minutes.
State. Compare the distributions of walking times for the two groups, describing their centers, spreads, and overlap.
Plan. Both groups measure walking time in minutes, and the dotplots share a scale. Compare medians with medians and ranges with ranges, then describe the visible overlap and pattern.
Do: center. Each sample has ten values, so its median is the average of the fifth and sixth ordered observations. For East, those values are 6 and 7: \((6+7)/2=6.5\) minutes. For West, they are 9 and 10: \((9+10)/2=9.5\) minutes. West’s sample median is \(9.5-6.5=3\) minutes greater.
Do: spread and overlap. East’s range is \(9-4=5\) minutes. West’s range is \(12-7=5\) minutes, so the ranges are equal. Both groups have observed values from 7 through 9 minutes, so their ranges overlap over that interval. In both dotplots, the counts rise toward the middle values and then taper toward the ends.
Conclude in context. “For these students, Group West’s walking times have a median 3 minutes greater than Group East’s. The groups have the same range of 5 minutes and overlapping observed values from 7 to 9 minutes; both dotplots show a similar concentration around their centers.”
This comparison distinguishes location from spread. West’s values tend to be larger, but the ranges are equal. It also describes the overlap rather than implying that every West student walks longer than every East student.
Worked Example: Similar Centers but Different Spread
Worked Example: Similar Centers but Different Spread
A fictional recreation center compares the number of minutes two groups take to complete a short indoor course. The shared-axis dotplots correspond to these samples:
Group A: 12, 13, 14, 15, 16, 17, 18, 19, 20 minutes.
Group B: 8, 12, 14, 16, 16, 16, 18, 20, 24 minutes.
Compare centers. Each group has nine observations, so the median is the fifth value. Group A’s median is 16 minutes, and Group B’s median is also 16 minutes. Their sample medians are equal.
Compare spread. Group A’s range is \(20-12=8\) minutes. Group B’s range is \(24-8=16\) minutes. Thus, Group B’s range is \(16-8=8\) minutes greater. Group A’s dots occupy consecutive values from 12 through 20, while Group B has observations farther from its median, including values at 8 and 24.
Compare shape and concentration. Both dotplots are balanced around 16 minutes: values occur at similar distances on either side of that center, with Group B having three observations at 16. Group B also has a wider horizontal extent. Equal medians do not mean the full distributions are alike.
Conclude in context. “The two groups have the same median course time of 16 minutes, but Group B’s times are more spread out: its range is 16 minutes, compared with 8 minutes for Group A. Group B’s dotplot includes more widely separated times, although both groups are centered at 16 minutes.”
The range uses the full span of each sample, so this comparison is supported by the endpoints. If the question instead emphasizes the middle half, compare IQRs for both groups. Choose the feature that matches the question and the evidence available.
Worked Example: Same Median, Different Clusters and a Gap
Worked Example: Same Median, Different Clusters and a Gap
A fictional parks program records how many minutes participants spend on a short activity at two locations. The dotplots share an axis and show:
Location A: 10, 11, 11, 12, 12, 13, 13, 14, 14, 15 minutes.
Location B: 8, 8, 8, 9, 9, 9, 16, 16, 17, 17, 18, 18 minutes.
Compare centers. Location A has ten observations. Its median is the average of the fifth and sixth values: \((12+13)/2=12.5\) minutes. Location B has twelve observations. Its median is the average of the sixth and seventh values: \((9+16)/2=12.5\) minutes. The medians are equal.
Compare patterns. Location A’s dots form one concentration from 10 to 15 minutes. Location B’s dots form two clusters, one from 8 to 9 minutes and another from 16 to 18 minutes. No Location B observations fall from 10 through 15 minutes, creating a gap in that interval. The dotplots therefore have different patterns even though their medians match.
Compare spread. Location A’s range is \(15-10=5\) minutes. Location B’s range is \(18-8=10\) minutes. Location B’s range is \(10-5=5\) minutes greater. The wide gap and separate clusters help explain why a single center does not capture the full appearance of that sample.
Conclude in context. “Participants at both locations have a median activity time of 12.5 minutes, but the dotplots differ substantially in pattern. Location A has one concentration from 10 to 15 minutes, whereas Location B has clusters at 8–9 and 16–18 minutes with a gap from 10 through 15 minutes; Location B also has the greater range.”
This description reports the median without letting it hide the clusters and gap. It also states the interval with no Location B observations, rather than saying vaguely that the values are “spread out.”
Sample Size, Overlap, and What the Plot Supports
A dot represents an observation, so the total number of dots shows the sample size. If sample sizes differ, compare the patterns and values, but avoid treating a taller stack as proof of a higher rate or proportion. For example, a group with more observations may have more dots at a value simply because it contains more observations overall. A comparison of counts should make the sample sizes clear when they matter.
Describe overlap carefully. Two samples can have overlapping values even when one has a greater median. They can also have the same median while differing in spread or shape, as the examples show. A statement such as “the groups are the same” is too broad when only one feature matches. Specify that their medians are equal, or that their ranges are similar, and then mention other visible differences.
Dotplots display the observed values, not every possible value and not the entire population. A gap means no observations in that sample lie in the stated interval. An isolated dot may be described as standing apart, but use the earlier guidance in What Outliers Mean in Context before labeling it an outlier. Do not infer a cause from the graph alone.
Common Mistakes and AP Exam Tips
- Ignoring the axes. First verify the groups share the same variable, units, and scale. A visual comparison is not fair if one axis is stretched or uses different values.
- Reporting only one group’s features. A comparison must say how the other group is positioned or patterned. “Group A is spread out” does not explain whether it is more or less spread out than Group B.
- Comparing mismatched summaries. Compare median with median or mean with mean, and range with range or IQR with IQR. State the measure and units when giving a numerical difference.
- Equating equal centers with equal distributions. Matching medians do not rule out different spreads, clusters, gaps, or unusual values. Describe other important evidence in the dotplots.
- Confusing overlap with equality. Shared values or overlapping ranges do not mean the samples have the same center or shape. State what overlaps and what differs.
- Overstating what a sample shows. A dotplot describes the observations displayed. Do not turn a sample comparison into a claim about every person in broader groups unless the study design supports that conclusion.
- Using vague words without evidence. Replace “higher,” “weird,” or “more spread out” with a specific comparison, such as “the median is 3 minutes greater” or “there is a gap from 10 through 15 minutes.”
A strong AP response names the variable and both groups, compares relevant features directly, and supports the claims with values or visible patterns. For example: “The groups have the same median of 12.5 minutes, but Location B has two clusters and a gap from 10 through 15 minutes, while Location A has one concentration from 10 to 15 minutes.” This is specific, contextual, and limited to what the dotplots show.
Check Your Understanding
Use the values and descriptions to write comparisons that are specific and supported.
- Group North has values 3, 4, 4, 5, 6, and Group South has values 5, 6, 6, 7, 8. Find both medians and compare them.
- Group A has a range of 12 minutes and Group B has a range of 7 minutes. Which sample has the greater range, and by how much?
- Two dotplots have the same median, but one has a gap and two clusters. Why is “the distributions are identical” not supported?
- A dotplot for one group has more dots stacked at a value, but that group also has twice as many observations. Why should you be careful when comparing the counts?
- Write one sentence comparing two groups whose observed values overlap but whose medians differ. Include the variable, both groups, and a relevant unit.