From a Graph to a Clear Comparison
A side-by-side graph can make differences between groups visible, but a useful description does more than say that one graph is higher or wider. It identifies what was measured, which groups are being compared, and how their distributions differ. It also distinguishes what the graph shows from what it cannot establish.
A careful description usually considers center, spread, shape, and any visually unusual values. You should not force all four into every comparison: a boxplot, for example, gives more direct information about medians and quartiles than about clusters. The kind of graph and the earlier choice of summaries guide what can be said accurately.
Use the variable’s units whenever they apply. “The typical time was higher” is incomplete if readers do not know which time or how much higher. “The median completion time was about 3 minutes higher in Group B” is more specific. When a graph supports only a visual estimate, say “about” or “appears to be”; do not report more precision than the display provides.
A Practical Structure for Describing Two Groups
A useful comparison moves from what the graph displays to what can reasonably be concluded. First name the groups and variable. Then compare the centers, describe the spreads, and note shape or unusual points if the graph shows them. Finish with a limitation or qualification when it helps prevent an overstatement.
Name the groups, the variable, and its units. Make clear whether the values are, for example, minutes per trip or points per quiz.
Use a suitable measure, such as the median or mean, and describe it as a typical value rather than as a value shared by everyone.
Use the IQR or range when those values are available, or describe the overall width of the distributions with appropriate caution.
Report visible skew, clusters, gaps, or isolated points only when the graph supports those observations.
State what this graph suggests about these groups. Do not claim a cause or generalize to a wider population without evidence that supports doing so.
Center and spread should be considered together. A larger median does not mean that every observation in that group is larger. A wider distribution does not mean every observation is farther from the center. A comparison is most informative when it reports the main difference and also notes important overlap or variability.
Reading and Writing About Boxplots
A boxplot shows the median and the first and third quartiles, as well as whiskers and sometimes separately marked points. The middle half of the observations lies between the first and third quartiles. Its width, the interquartile range, is \(IQR=Q_3-Q_1\). A boxplot can support direct comparisons of medians and IQRs. Its whiskers and the placement of the median within the box may also suggest differences in spread or asymmetry, but boxplots do not show every feature of a distribution.
Worked Example: Comparing Battery Life in Two Groups
Suppose side-by-side boxplots summarize battery life, in hours, for 24 hypothetical devices in each of two groups. The Standard group has \(Q_1=7.4\), median \(8.2\), and \(Q_3=9.1\) hours. The Premium group has \(Q_1=8.2\), median \(9.0\), and \(Q_3=9.7\) hours. The Standard plot also marks a separate point at 14.8 hours. Describe the comparison.
Center. The Standard median is 8.2 hours and the Premium median is 9.0 hours. The difference is
Thus, the median battery life in this sample is 0.8 hours higher for the Premium group. This compares the two sample medians; it does not mean that every Premium device lasts longer than every Standard device.
Spread. For the Standard group,
For the Premium group,
The middle half of Standard battery-life values is spread over 1.7 hours, compared with 1.5 hours for Premium devices. These IQRs are similar, with the Standard IQR slightly larger. The boxes overlap from 8.2 to 9.1 hours, so the graph does not show a clean separation between the groups.
Unusual value and limits. The plot marks one Standard observation at 14.8 hours, well above the box and the other plotted values. It is appropriate to describe it as a visually unusual high observation in this graph. The graph alone does not explain why it is high or establish that the measurement is incorrect. Also, the boxplots do not tell us whether either group has multiple clusters.
Contextual summary. In these hypothetical samples, Premium devices have a median battery life 0.8 hours higher than Standard devices, while the spreads of the middle halves are similar. The Standard plot includes a separate high value at 14.8 hours. This description compares the observed groups; it does not establish that the Premium label caused longer battery life.
The arithmetic helps make the wording precise, but the graph still determines what can be claimed. If whiskers have different lengths, describe the visible difference rather than treating whisker length as an IQR. If a plot does not display a numerical summary, a visual estimate should remain approximate.
Reading Shape and Spread in Histograms
Side-by-side histograms make the overall distribution shapes easier to compare. Look for where observations concentrate, how far the tails extend, and whether the distributions appear symmetric or skewed. A histogram groups values into bins, so exact medians or quartiles may not be readable from it. Describe features as approximate when the binning or scale limits precision.
Worked Example: Comparing Wait Times on Two Routes
Suppose two hypothetical samples each contain 40 bus-trip wait times, measured in minutes. In Route A’s histogram, most waits fall between 5 and 20 minutes; the bars are tallest from 5 to 15 minutes, and a sparse tail extends to about 35 minutes. Route B’s histogram has its tallest bars from 10 to 20 minutes, with most waits between 10 and 25 minutes and a shorter tail reaching about 35 minutes. A visual estimate places the medians at about 12 minutes for Route A and about 17 minutes for Route B. Describe what the graphs show.
Center. The estimated median is about 5 minutes higher for Route B:
So the middle wait in the Route B sample appears longer. Because these are estimates read from histograms, “about 5 minutes” is more appropriate than reporting an exact difference.
Spread and shape. Both histograms extend to approximately 35 minutes, and both include a long upper tail relative to the main concentration of values. This suggests right-skewed distributions. Route A has many waits in the 5-to-20-minute region; Route B’s main concentration is shifted toward 10 to 25 minutes. The histograms suggest that Route B’s typical wait is longer, but they do not provide enough detail to claim an exact difference in IQR.
Contextual summary. In these samples, Route B’s median wait appears to be about 5 minutes longer than Route A’s. Both distributions are right-skewed and extend to roughly 35 minutes, although their main concentrations occur in different intervals. These observed differences do not show that the route itself caused longer waits; other conditions could differ, and the histograms alone do not address those explanations.
A histogram’s tail matters when choosing and describing a center. A long tail can pull the mean away from the region where most observations lie, while the median is less affected by extreme values. If the mean is not shown or cannot be obtained from the graph, do not invent it. Describe the visible concentration and shape instead.
Using Dotplots for Small Groups
Dotplots show individual observations, making them particularly useful for small samples. They can reveal repeated values, gaps, clusters, and points that sit apart from the rest. With a short list of observations, it is also possible to calculate a median or range and connect those summaries directly to what is plotted.
Worked Example: Comparing Walking Times in Two Groups
A dotplot shows the walking time, in minutes, for eight hypothetical participants in each of two groups. The Morning group’s values are 18, 20, 21, 22, 23, 24, 25, and 38 minutes. The Evening group’s values are 17, 19, 20, 21, 22, 23, 24, and 26 minutes. Compare their centers and spreads, and describe any unusual feature.
Center. With eight observations, each median is the mean of the fourth and fifth ordered values. For Morning:
For Evening:
The Morning median is 1 minute higher. This is a small difference in the centers of these samples, not a claim that each Morning participant walked longer.
Spread. The range is the maximum minus the minimum. For Morning,
For Evening,
The Morning group has the larger range, but this comparison is strongly influenced by its value of 38 minutes. The other seven Morning values lie between 18 and 25 minutes. The Evening values range from 17 to 26 minutes, so the two groups’ main clusters are similar in location and extent.
Contextual summary. The sample medians differ by 1 minute, with Morning slightly higher, while Morning has a much larger overall range because of a value at 38 minutes. A careful description reports both the full range and the visibly separated point rather than allowing that one value to stand for the whole group.
What a Graph Does Not Establish
A graph describes the data it displays. It does not, by itself, establish that a group difference is important in a wider population, explain why the distributions differ, or show that group membership caused the difference. For example, if one group has higher values than another, the graph alone cannot rule out other differences between the groups or how the observations were collected.
It is also important to distinguish “no obvious difference in this graph” from “the groups are identical.” A display may hide small differences, and samples may vary. Likewise, overlap between two distributions does not erase a difference in their centers. Report the visible comparison, then avoid conclusions that require evidence the graph does not provide.
Common Mistakes
- Reporting only which group is higher. “Route B is higher” does not identify the variable or units. Name the comparison: for example, “the estimated median wait on Route B is about 5 minutes longer.”
- Treating a median as a value for everyone. A higher group median does not mean every observation in that group exceeds every observation in the other group. Compare centers, not all individual values.
- Overstating what the graph measures. A histogram may show general width and shape without revealing an exact IQR. State what is visible and label estimates as approximate.
- Calling a visually separate point an error. A point that stands apart is a feature to describe, not proof that a measurement is wrong or that it should be removed.
- Claiming cause from a group difference. “This group had a higher median” is descriptive. “Being in this group caused higher values” is a causal claim that the graph alone does not justify.
Check Your Understanding
Use the graph features and summaries described in each question to write a contextual comparison.
- Two boxplots show weekly exercise time, in minutes. Group A has a median of 90 and an IQR of 50; Group B has a median of 110 and an IQR of 48. What can you say about their centers and middle-half spreads?
- A histogram has a main concentration near 12 minutes and a long tail toward larger values. What shape does this suggest, and why should you avoid giving an exact median if the graph does not show one?
- A dotplot for one group has a point well separated from its other observations. What is a careful way to describe that point, and what conclusion would go beyond the evidence?
- Two groups have noticeably overlapping boxplots, but their medians differ. Explain why both facts can be included in the same description.
- One group has a higher sample median in a graph. Why does that observation not establish that group membership caused higher values?