Match the Evidence, Not Just the Appearance
In What a Boxplot Cannot Show, you saw that a boxplot summarizes selected positions and spread but hides features such as clusters and gaps. Matching a histogram to a boxplot brings the two displays together: use the histogram to judge the distribution’s shape, and use the boxplot to check its center and spread. A match should be supported by evidence from both graphs.
Begin by checking that the displays describe the same quantitative variable, use the same units, and have compatible horizontal scales. Then compare shape, center, and spread. A boxplot does not reveal every detail of shape, so treat its whiskers as clues about spread and possible skewness—not as a replacement for the histogram.
A histogram may appear roughly symmetric when its bars rise toward a central region and then fall in a similar way on both sides. A longer tail toward larger values suggests right-skewness; a longer tail toward smaller values suggests left-skewness. A boxplot may support that impression when one side of the box or one whisker is notably longer, but that clue is not conclusive by itself. As the earlier tutorial on histogram bin widths established, bin choices can affect the visible pattern.
For center, compare the medians: the boxplot marks the median, while the histogram’s concentrations suggest where observations tend to lie. For spread, compare the IQR, represented by the box’s length, and the range from the minimum to maximum, represented by the full extent of a basic boxplot. A longer box means a larger IQR on the same scale; it does not necessarily mean a larger range. The earlier tutorials on boxplot quartiles and spread explain how to read these landmarks.
Use the Five-Number Summary to Check a Candidate
When a question gives several possible boxplots, use their landmarks rather than relying on a quick visual impression. The median should agree with the histogram’s center, and the quartiles and extremes should be plausible for the spread shown. If raw observations are available, calculate the five-number summary using the convention established in Five-Number Summary and Boxplot Construction. If only histogram bins are available, remember that the bins do not usually identify exact quartiles or exact values.
That last point matters. A histogram tells you how many observations fall in intervals, not their exact order within each interval. It may rule out a candidate boxplot—for example, if a quartile would have to lie outside a bin that contains the relevant ranked observation—but it may not provide enough information to choose between two nearby quartile values. State what the displays support without claiming more precision than they show.
Work Through a Complete Match
Worked Example: Match a Right-Tailed Histogram to a Boxplot
A fictional set of 12 students’ practice times, in minutes, is \(2, 3, 3, 4, 4, 5, 6, 7, 8, 12, 16, 19\). A histogram uses the intervals 0 to less than 5, 5 to less than 10, 10 to less than 15, and 15 to 20, including 20. Which candidate boxplot matches?
| Candidate | Minimum | \(Q_1\) | Median | \(Q_3\) | Maximum |
|---|---|---|---|---|---|
| A | 2 | 3.5 | 5.5 | 10 | 19 |
| B | 2 | 3.5 | 5.5 | 10 | 30 |
| C | 2 | 5.5 | 10 | 16 | 19 |
Plan. Count the observations in each histogram bin and calculate the five-number summary from the ordered list. Then compare both the histogram pattern and summary with the candidates.
Do. The bin counts are 5, 4, 1, and 2. They total 12, as they should. Most times are below 10 minutes, and the observations extend toward larger times. This suggests a longer right tail rather than a balanced shape.
For 12 ordered observations, the median is the average of positions 6 and 7: \((5+6)/2=5.5\) minutes. The lower half is positions 1 through 6, so \(Q_1=(3+4)/2=3.5\) minutes. The upper half is positions 7 through 12, so \(Q_3=(8+12)/2=10\) minutes. The minimum is 2 minutes and the maximum is 19 minutes. The resulting five-number summary is \(2,\ 3.5,\ 5.5,\ 10,\ 19\) minutes.
Candidate A agrees with all five values. Candidate B has the wrong maximum, and Candidate C has different quartiles and median. The longer distance from the median to \(Q_3\) than from \(Q_1\) to the median is also consistent with the histogram’s longer stretch toward larger times.
Conclude. Candidate A is the matching boxplot. The histogram supports a right-tailed shape, and the calculated five-number summary confirms the candidate’s center and spread in the context of students’ practice times.
Identical Boxplots Can Match Different Histograms
Matching is not always a one-to-one identification of every detail. Different data sets can share the same five-number summary, as demonstrated in the earlier tutorial on what a boxplot cannot show. Therefore, a boxplot match confirms agreement in the displayed landmarks; it does not prove that two histograms have the same pattern between those landmarks.
Worked Example: Two Histograms, One Boxplot Summary
Two fictional groups of 12 students record the number of pages they read in a study session. Their ordered values are:
Group A: \(0, 1, 2, 4, 5, 8, 10, 12, 15, 17, 19, 20\).
Group B: \(0, 1, 2, 4, 8, 8, 10, 11, 15, 17, 18, 20\).
Each histogram uses seven intervals: 0 to less than 3, 3 to less than 6, 6 to less than 9, 9 to less than 12, 12 to less than 15, 15 to less than 18, and 18 to 21.
| Pages interval | Group A count | Group B count |
|---|---|---|
| 0 to less than 3 | 3 | 3 |
| 3 to less than 6 | 2 | 1 |
| 6 to less than 9 | 1 | 2 |
| 9 to less than 12 | 1 | 2 |
| 12 to less than 15 | 1 | 0 |
| 15 to less than 18 | 2 | 2 |
| 18 to less than 21 | 2 | 2 |
Plan. Find each group’s five-number summary using the same quartile convention. Compare the histogram counts afterward to identify a difference the shared boxplot does not display.
Do. For both groups, the minimum is 0 pages and the maximum is 20 pages. In each ordered list, positions 3 and 4 are 2 and 4, so \(Q_1=(2+4)/2=3\) pages. Positions 6 and 7 are 8 and 10, so the median is \((8+10)/2=9\) pages. Positions 9 and 10 are 15 and 17, so \(Q_3=(15+17)/2=16\) pages. Both five-number summaries are \(0,\ 3,\ 9,\ 16,\ 20\) pages.
The groups therefore match the same basic boxplot. However, the histogram counts differ: Group A has one observation from 12 to less than 15 pages, while Group B has none in that interval. The other bins also show differences in their counts. The boxplot does not display those bin-by-bin patterns.
Conclude. Both groups match a boxplot with minimum 0, \(Q_1=3\), median 9, \(Q_3=16\), and maximum 20 pages. Their histograms are not identical: in particular, Group B has an empty displayed interval from 12 to less than 15 pages. The shared boxplot summary does not rule out this difference.
Compare Two Groups on the Same Scale
When matching several histograms with several boxplots, compare the groups on a common horizontal scale. A boxplot from a group with a higher median should generally pair with a histogram whose observations are centered at larger values. Then check whether the IQR and range agree with the relative widths and spans. Do not compare apparent lengths if the graphs use different scales.
Worked Example: Pair Two Commute-Time Distributions
Two fictional groups record their commute times to school, in minutes. Pair each histogram with the correct boxplot. The shared histogram bins are 0 to less than 5, 5 to less than 10, 10 to less than 15, and 15 to 20, including 20.
Group X’s ordered times are \(5, 7, 8, 9, 10, 11, 12, 13, 14, 15\). Its histogram counts are 0, 4, 5, and 1. Group Y’s ordered times are \(2, 3, 4, 5, 6, 7, 8, 10, 13, 18\). Its histogram counts are 3, 4, 2, and 1.
| Candidate boxplot | Minimum | \(Q_1\) | Median | \(Q_3\) | Maximum |
|---|---|---|---|---|---|
| P | 5 | 8 | 10.5 | 13 | 15 |
| Q | 2 | 4 | 6.5 | 10 | 18 |
Plan. The histogram counts suggest that Group X is centered at higher commute times than Group Y. Verify this with the medians, then compare the IQRs and ranges to confirm the boxplot pairing.
Do. Each group has 10 observations, so the median is the average of positions 5 and 6. For Group X, the median is \((10+11)/2=10.5\) minutes. Its lower five values have median 8, so \(Q_1=8\); its upper five values have median 13, so \(Q_3=13\). Its minimum is 5 and maximum is 15. Its five-number summary is \(5,\ 8,\ 10.5,\ 13,\ 15\).
For Group Y, the median is \((6+7)/2=6.5\) minutes. The median of its lower five values is 4, and the median of its upper five values is 10. Its minimum is 2 and maximum is 18. Its summary is \(2,\ 4,\ 6.5,\ 10,\ 18\). The IQRs are \(13-8=5\) minutes for Group X and \(10-4=6\) minutes for Group Y. Their ranges are \(15-5=10\) minutes and \(18-2=16\) minutes, respectively.
Group X pairs with candidate P, and Group Y pairs with candidate Q. This also fits the histograms: Group X’s counts concentrate in the 10-to-less-than-15-minute interval, while Group Y has more observations at lower times and extends farther overall.
Conclude. Candidate P matches Group X, and candidate Q matches Group Y. Group X has the higher median and smaller range; Group Y has the lower median and larger IQR and range. These comparisons refer to commute times in minutes.
Common Mistakes and AP Exam Tips
- Matching by shape alone. A histogram may look like a plausible match but have a center or spread inconsistent with the boxplot. Check the median, box width, and overall span as well.
- Treating a longer whisker as proof of skewness. Unequal whisker lengths can support a shape impression, but a boxplot does not show the full frequency pattern. Use the histogram to describe the visible shape.
- Assuming the histogram determines exact quartiles. Bins group observations into intervals. Unless the raw values or sufficient rank information are given, a bin count generally cannot establish an exact quartile.
- Ignoring units or scales. A boxplot’s apparent length is meaningful only with its numerical scale. Confirm that both displays concern the same variable and use compatible units and axes.
- Claiming matching summaries mean identical distributions. Identical boxplots show the same five-number summary, not identical observations or histogram counts. The second worked example shows how an empty histogram bin can distinguish groups with the same summary.
- Overstating what a match establishes. A strong explanation says which clues agree—for example, the median and IQR—and names the relevant variable and units. It does not claim the boxplot confirms histogram details it cannot display.
For a clear response, name the histogram and boxplot being paired, then cite shape evidence and at least one center or spread comparison. If several candidates share the same five-number summary, explain that the boxplot alone may not distinguish their histograms.
Check Your Understanding
Use the histogram and boxplot evidence together. Treat any stated intervals and units as part of the information.
- A histogram is concentrated at small values with a tail extending toward larger values. Which side of the boxplot might be longer, and why is that only a clue?
- Two candidate boxplots have the same median, but one has a longer box. What feature of spread differs, and what does the longer box indicate?
- Can a histogram’s bin counts usually establish the exact value of \(Q_1\)? Explain what additional information would help.
- Two groups have identical five-number summaries but different histogram counts. Is either display necessarily incorrect? Explain.
- What should you check before comparing the apparent widths of two boxplots?