Put the Boxplots and the Table Together
A side-by-side boxplot makes it possible to compare two distributions visually on the same scale. A summary table adds exact values that can support your written comparison. The task is to use both sources of evidence to explain how the distributions are alike and different—not simply to copy numbers from the table.
In Writing a Full Comparison Using Summary Statistics, you practiced selecting statistics and connecting their differences to context. Here, the boxplots help you assess center, spread, overlap, and any marked outliers. The table lets you report exact summaries. As in Side-by-Side Boxplots Compared, compare medians and IQRs directly when those are appropriate; do not assume a boxplot reveals every feature of a distribution.
A boxplot displays the median as a line inside the box, the first and third quartiles as the box edges, and whiskers that extend toward the smaller and larger observations. Some displays mark outliers separately. In a modified boxplot, whiskers typically reach the most extreme values that are not classified as outliers, while outliers appear as individual points. Check the plot’s key or accompanying description before interpreting whisker endpoints.
The box extends from \(Q_1\) to \(Q_3\), so its width represents the IQR. A five-number-summary table gives the minimum, \(Q_1\), median, \(Q_3\), and maximum; these values let you report exact locations and calculate the IQR and range. A separate table may also provide means and standard deviations. Those statistics can be useful, but select a matching pair that suits the distributions, as explained in Matching Measures: Mean With Standard Deviation, Median With IQR.
A boxplot does not show all the raw observations. It usually does not reveal modality, clusters, or gaps, and it gives only limited evidence about shape. Unequal whisker lengths or unequal distances from the median to the box edges can suggest asymmetry, but they do not prove a particular detailed shape. Use a histogram or a supplied description if the question asks about features that the boxplot cannot show.
A Plan for an Exam-Style Response
Before writing, identify what the question asks you to compare. Then use the same feature for both groups: for example, median with median or IQR with IQR. If the plot indicates skewness or marks outliers, the median and IQR are often suitable. If the distributions are reasonably symmetric without strong outliers, the mean and standard deviation may be suitable. Use evidence from the displays to explain your choice.
Name the groups, the variable being measured, and its units.
Compare medians for center, box widths for IQR, and marked outliers or endpoints where relevant. Check exact values against the table.
Use matching statistics. Subtract in a stated order and report the difference with units.
Describe the important similarities and differences, and avoid interpreting the display as evidence of a cause.
A useful response usually compares center and spread, then includes any important supported detail about overlap or unusual values. You do not need to mention every value. The goal is a focused paragraph in which a reader can tell what the numbers mean.
Worked Example: Compare Typical Commute Times
Worked Example: Compare Typical Commute Times
A fictional student survey records the number of minutes students spend commuting to school. The question gives side-by-side boxplots on a shared scale and this table of values read from the plots and accompanying summaries. Neither boxplot has a separately marked outlier.
| Commute method | Sample size | Minimum | \(Q_1\) | Median | \(Q_3\) | Maximum | Mean | Standard deviation |
|---|---|---|---|---|---|---|---|---|
| Walk or bike | 32 | 6 | 12 | 18 | 27 | 46 | 20.5 | 9.6 |
| Bus | 38 | 8 | 16 | 23 | 34 | 58 | 26.1 | 12.8 |
Read and select. The median commute for students who walk or bike is 18 minutes; for students who take the bus, it is 23 minutes. The boxplots show longer upper than lower whiskers in both groups, suggesting some right-skewness. The median and IQR are therefore useful summaries for this comparison. Calculate each IQR from the quartiles:
The median difference, in the order bus minus walk or bike, is \(23-18=5\) minutes. The IQR difference is \(18-15=3\) minutes. Thus, the bus group has the higher median, and its middle half of commute times is somewhat more spread out.
Write the comparison. “Among the surveyed students, those who take the bus had a higher median commute time than those who walk or bike: 23 minutes compared with 18 minutes, a difference of 5 minutes. The middle half of bus commute times also had a slightly greater spread, with an IQR of 18 minutes compared with 15 minutes, a difference of 3 minutes. The boxplots suggest right-skewness in both groups, and neither plot marks an outlier. These summaries describe the surveyed students; they do not show that using a particular commute method causes a longer commute.”
The paragraph uses exact table values while also referring to evidence visible in the boxplots. The sample sizes provide context, but they are not a measure of commute time and do not change the subtraction.
Worked Example: Use the Mean and Standard Deviation
Worked Example: Use the Mean and Standard Deviation
A fictional school garden club compares the mass, in kilograms, of tomatoes harvested from two greenhouse sections during sampled harvests. The side-by-side boxplots use a shared scale. Both look approximately symmetric, and neither has a marked outlier.
| Greenhouse section | Sample size | Minimum | \(Q_1\) | Median | \(Q_3\) | Maximum | Mean (kg) | Standard deviation (kg) |
|---|---|---|---|---|---|---|---|---|
| North | 24 | 13.0 | 16.5 | 18.2 | 19.9 | 23.1 | 18.4 | 2.6 |
| South | 28 | 14.2 | 18.1 | 20.0 | 21.9 | 25.7 | 20.1 | 2.9 |
Choose and calculate. The boxplots show no marked outliers and appear reasonably symmetric, so comparing means and standard deviations is reasonable. In the order South minus North, the difference in mean harvested mass is \(20.1-18.4=1.7\) kg. The standard deviation difference is \(2.9-2.6=0.3\) kg. The South section has both the higher mean and slightly greater standard deviation.
Write the comparison. “For the sampled harvests, the South greenhouse section had a greater mean tomato mass than the North section: 20.1 kg compared with 18.4 kg, a difference of 1.7 kg. The harvest masses in the South section also had slightly more variability, with a standard deviation of 2.9 kg compared with 2.6 kg in the North section, a difference of 0.3 kg. The boxplots appear reasonably symmetric and show no marked outliers, supporting the use of means and standard deviations. These results summarize the sampled harvests and do not establish why the sections differ.”
The table also gives medians and quartiles, but including every statistic would distract from the main comparison. The boxplots provide a useful check on whether the mean and standard deviation are sensible choices; the table supplies the exact values needed to quantify those comparisons.
Worked Example: Interpret an Outlier and Overlap
Worked Example: Interpret an Outlier and Overlap
A fictional community program records the number of books borrowed by participants during a month in two reading groups. The accompanying side-by-side modified boxplots share a scale. The plot for Group X marks one unusually high value at 90 books. Its upper whisker ends at 68 books; the marked point lies beyond the whisker. Group Y has no marked outlier.
| Reading group | Sample size | Minimum | \(Q_1\) | Median | \(Q_3\) | Maximum | Mean (books) | Standard deviation (books) |
|---|---|---|---|---|---|---|---|---|
| X | 30 | 22 | 36 | 42 | 49 | 90 | 45.0 | 14.0 |
| Y | 35 | 25 | 39 | 46 | 53 | 66 | 46.2 | 8.1 |
Compare center and spread. Since Group X has a marked high outlier, the median and IQR provide a useful resistant comparison. Group X’s median is 42 books; Group Y’s is 46 books. In the order Y minus X, the median difference is \(46-42=4\) books. The IQRs are \(49-36=13\) books for X and \(53-39=14\) books for Y, so Group Y’s IQR is \(14-13=1\) book greater. The middle-half spreads are similar.
The box portions overlap from 39 to 49 books, because that is the shared interval between the two boxes. This is evidence that the middle-half intervals overlap; it does not mean every value in that interval occurred in both groups. Group X’s maximum of 90 is the separately marked outlier, not the upper whisker endpoint of 68. The five-number summary reports the actual maximum, while the modified boxplot uses the whisker to show the most extreme non-outlier.
Write the comparison. “The median number of books borrowed was 46 in Group Y and 42 in Group X, so Group Y’s median was 4 books higher. The IQRs were similar: 14 books for Group Y and 13 for Group X, a difference of 1 book. The boxes overlap from 39 to 49 books, while Group X’s boxplot marks a high outlier at 90 books; Group Y has no marked outlier. Because of the outlier in Group X, the medians and IQRs give a useful comparison of the groups’ centers and middle-half spreads. These summaries describe the participants in the two samples.”
This example shows why it matters to understand what each display element represents. The maximum in the table and the whisker endpoint do not conflict: a modified boxplot can show the outlier as an individual point beyond the whisker.
Common Mistakes and AP Exam Tips
- Describing each group but never comparing them. “The median is 42 in X and 46 in Y” gives values, but a full comparison states that Y’s median is 4 books higher.
- Reading the shared scale incorrectly. Check the axis labels and units before reading a value. A boxplot on a shared scale supports direct visual comparison; plots with different scales may not.
- Calling the maximum the whisker endpoint. In a modified boxplot, a separately marked outlier can lie beyond a whisker. Check the plot’s symbols and reconcile them with the table.
- Choosing a statistic without considering the plot. A strong outlier or skewness often makes median and IQR more appropriate than mean and standard deviation. Explain your choice using evidence provided.
- Claiming a shape the boxplot cannot establish. Boxplots do not show modality, detailed clusters, or gaps. Describe only the features visible or stated in the prompt.
- Confusing overlap with equality. Overlapping boxes do not mean the distributions are identical. State which intervals overlap and keep the claim at that level.
- Leaving out context and units. “The difference is 4” is incomplete. Say which group’s median number of books is higher and by how many books.
- Making a causal claim. A comparison of groups does not, by itself, show that group membership caused the observed difference. Describe the data rather than inventing an explanation.
For full-credit communication, name the groups and variable, give accurate comparisons supported by the plot or table, quantify important differences with units, and interpret unusual features precisely. If the information needed to discuss shape or outliers is not present, say so rather than guessing.
Check Your Understanding
Use the boxplot evidence and summary values in each question. State both the comparison and its context where possible.
- Two groups have medians of 31 and 27 minutes and IQRs of 12 and 9 minutes. Which group has the higher median, and which has the greater middle-half spread? Give both differences with units.
- A modified boxplot has an upper whisker ending at 54 points and a separate dot at 81 points. A five-number summary lists 81 as the maximum. Explain why these displays are consistent.
- A table gives means and standard deviations, and the boxplots show strong right-skewness with a high outlier in one group. Which matching pair of summaries would usually be more appropriate for comparing center and spread? Why?
- Two boxplot boxes overlap from 18 to 25 minutes. What can you conclude about the middle-half intervals, and what can you not conclude about every observation?
- Write one comparison sentence for two groups with medians of 46 and 42 books, naming which median is higher and by how much.