One Graph, Several Groups
In Side-by-Side Boxplots Compared, you practiced comparing two groups using a shared numerical scale. The same approach works when one graph displays three, four, or more groups. A single graph lets you scan the groups together, rank them on a particular feature, and see where ties or close comparisons occur.
“Rank the groups” needs a clear target. You might rank the groups by median to compare their centers, by IQR to compare the widths of their middle halves, or by range to compare their full spans. Those rankings can differ. A group with the greatest median is not necessarily the group with the greatest IQR or range.
Before ranking, check that the plots use the same quantitative variable, units, and numerical scale. Then identify the feature named in the question. For medians, compare the lines inside the boxes. For IQRs, compare the box widths from \(Q_1\) to \(Q_3\). For ranges, compare the distance from each actual minimum to its actual maximum, taking plotted outliers and the boxplot convention into account.
A graph may not label every summary value. If you estimate from tick marks, describe the ranking as visual or approximate when the differences are small. Do not report more precision than the scale supports. If two medians appear equal, or too close to distinguish reliably, say they are tied or similar based on the display instead of inventing a precise order.
A Routine for Ranking Groups
A useful technique is to make a short ranking table as you read the graph. Record the group names in the requested order and the relevant values, if the graph provides enough detail. This helps prevent switching features midway—for example, ranking by median at first and then accidentally using box width to decide which group comes next.
Confirm the variable, units, shared numerical scale, group labels, and how whiskers and plotted points are defined.
Decide whether the question asks about center (median), middle-half spread (IQR), or full span (range).
Compare the same feature for every group, working from greatest to least or least to greatest. Keep tied groups together.
Name the variable and groups, give values or differences when supported, and describe ties or close estimates honestly.
A ranking is a compact summary, not a complete description of every distribution. The earlier tutorial Why Comparing Distributions Needs All Four Features explains why shape, center, spread, and unusual features can all matter. A question asking specifically for median order does not require a full description of every boxplot, but a broader conclusion should not pretend that the median alone describes everything.
Worked Example: Ranking Medians for Four Classes
Worked Example: Ranking Medians for Four Classes
A fictional school displays parallel boxplots of scores on the same 100-point mathematics quiz for students in four class periods. The plots share one axis. Reading the median lines gives these approximate values.
| Class period | Approximate median score |
|---|---|
| Period 1 | 74 |
| Period 2 | 81 |
| Period 3 | 78 |
| Period 4 | 81 |
Question: Rank the class periods from greatest to least median quiz score.
Read the requested feature. The question asks for medians, so compare the lines inside the boxes—not the box widths or whiskers. The displayed values are estimates, so the ranking should be presented as approximate.
Order the groups. Periods 2 and 4 both have an approximate median of 81 points. Period 3’s median is about 78, and Period 1’s is about 74. Thus, the order is Periods 2 and 4 (tied), then Period 3, then Period 1.
Compare the amounts. The estimated median for Period 3 is \(81-78=3\) points below the tied medians for Periods 2 and 4. Period 1’s estimated median is \(78-74=4\) points below Period 3’s.
Conclude in context. “The approximate median quiz scores rank highest for Periods 2 and 4, which are tied at about 81 points, followed by Period 3 at about 78 and Period 1 at about 74. These are comparisons of the class medians, not claims that every student in a higher-ranked class scored more than every student in a lower-ranked class.”
The shared scale makes the estimated differences interpretable in points. The ranking does not establish that class period caused any difference, nor does a graph of these groups alone justify a conclusion about students beyond those represented.
Ranking Spread Requires Choosing a Measure
A request to rank groups by “spread” can be ambiguous. The IQR compares the widths of the middle halves; the range compares the full spans from minimum to maximum. As in Comparing Spread Using IQR and Standard Deviation, name the measure instead of treating spread as one single quantity. A group can rank higher by IQR and lower by range.
For IQR rankings, read \(Q_1\) and \(Q_3\) for each group and calculate \(Q_3-Q_1\). For range rankings, use the actual minimum and maximum. In a modified boxplot, whiskers may stop short of plotted outliers, so the whisker endpoints alone may not be the true minimum and maximum. Follow the plot’s key or description before ordering groups by range.
Worked Example: The IQR Ranking Differs From the Range Ranking
Worked Example: The IQR Ranking Differs From the Range Ranking
A fictional community garden compares the heights, in centimeters, of seedlings from four planting beds. The display provides the five-number summaries, and all listed endpoints are actual minimums and maximums.
| Bed | Minimum | \(Q_1\) | Median | \(Q_3\) | Maximum |
|---|---|---|---|---|---|
| A | 12 | 18 | 23 | 30 | 44 |
| B | 15 | 19 | 25 | 29 | 36 |
| C | 10 | 16 | 22 | 31 | 39 |
| D | 14 | 20 | 26 | 32 | 40 |
Question: Rank the beds from greatest to least spread, first by IQR and then by range.
Calculate each IQR. Bed A’s IQR is \(30-18=12\) cm. Bed B’s is \(29-19=10\) cm. Bed C’s is \(31-16=15\) cm. Bed D’s is \(32-20=12\) cm.
The IQR ranking is Bed C first, Beds A and D tied next, and Bed B last. Beds A and D have the same IQR, so the graph does not support ranking one above the other on the width of the middle half.
Calculate each range. Bed A’s range is \(44-12=32\) cm. Bed B’s is \(36-15=21\) cm. Bed C’s is \(39-10=29\) cm. Bed D’s is \(40-14=26\) cm.
The range ranking is Bed A first, then Bed C, Bed D, and Bed B. Bed A has the largest full span even though Bed C has the largest IQR. The rankings differ because the IQR describes the middle half, while the range depends on the two endpoints.
Conclude in context. “For seedling heights, Bed C has the greatest IQR at 15 cm, Beds A and D tie at 12 cm, and Bed B has the smallest IQR at 10 cm. By range, Bed A ranks first at 32 cm, followed by Beds C, D, and B. Thus, Bed C has the widest middle half, while Bed A has the greatest full span.”
This conclusion ranks the observed distributions on two specified measures. It does not label the bed with the larger value as universally “more variable”; it explains which aspect of spread is greater.
Close Comparisons, Ties, and Overlap
When several groups appear in one display, some features may be easy to order while others are nearly indistinguishable. Use the axis carefully and avoid turning a rough visual difference into an exact numerical claim. If two boxplot lines are very close, it may be more accurate to say their medians appear similar than to assign them confidently to consecutive ranks.
Ties are meaningful results, not a failure to rank. If two groups have the same displayed median, report them as tied on median. If their IQRs differ, they may still rank differently by IQR. It is often helpful to state the ranking rule explicitly: “tied by median” is clearer than “tied,” which could mean tied on any feature.
Overlapping boxes mean that the intervals from \(Q_1\) to \(Q_3\) share some numerical values. They do not show exactly how many observations overlap, and they do not mean the groups have the same individuals or identical distributions. Likewise, a higher median does not mean every observation in that group is higher than every observation in another group. These cautions are especially important when turning a visual ranking into a sentence about people or outcomes.
Worked Example: Rank Centers Without Ignoring Spread
Worked Example: Rank Centers Without Ignoring Spread
A fictional recreation program displays boxplots of weekly cycling distances, in kilometers, for participants in three training groups. The summaries below are read from the graph.
| Group | \(Q_1\) (km) | Median (km) | \(Q_3\) (km) |
|---|---|---|---|
| Steady | 12 | 18 | 24 |
| Mixed | 10 | 20 | 30 |
| Long-ride | 16 | 22 | 28 |
Question: Rank the groups by median weekly distance, and compare their IQRs as a separate feature.
Rank the medians. Long-ride has a median of 22 km, Mixed has a median of 20 km, and Steady has a median of 18 km. From greatest to least median, the order is Long-ride, Mixed, then Steady. The difference between the highest and lowest medians is \(22-18=4\) km.
Compare IQRs. Steady’s IQR is \(24-12=12\) km. Mixed’s IQR is \(30-10=20\) km. Long-ride’s IQR is \(28-16=12\) km. Mixed has the greatest IQR; Steady and Long-ride tie for the smallest IQR. The middle half of weekly distances spans 8 km more in Mixed than in either of the other groups: \(20-12=8\) km.
Conclude in context. “The median weekly cycling distance ranks highest in Long-ride at 22 km, then Mixed at 20 km, and Steady at 18 km. However, Mixed has the widest middle half, with an IQR of 20 km compared with 12 km for both other groups. The group ranked highest by median is not the group with the greatest IQR.”
The boxplots can support both comparisons, but the conclusions answer different questions. The median ranking compares centers; the IQR ranking compares the spread of the middle half. Neither one gives a complete account of the shape or individual values.
Common Mistakes and AP Exam Tips
- Ranking the wrong feature. A question about median order calls for comparing median lines. Do not use box widths to decide which median is greatest.
- Leaving “spread” undefined. State whether you are ranking IQRs or ranges. These measures can produce different orders.
- Forcing a strict order when there is a tie. If values are equal or indistinguishable at the graph’s scale, report a tie or an approximate similarity. Do not invent extra precision.
- Ignoring the common scale. A ranking is not reliable if the groups use different scales. Check the axis before comparing visual positions or lengths.
- Using whiskers as actual endpoints automatically. Check whether outliers are plotted separately and whether the boxplot is modified before calculating a range.
- Overstating the result. A ranking of sample medians does not show that every value in one group exceeds every value in another, and a descriptive graph alone does not establish a cause.
For a strong response, name the variable, state the order and direction, identify the feature used to rank the groups, and include values or differences when the graph supports them. If two groups tie, say what they tie on. If the medians, IQRs, and ranges give different rankings, report that rather than collapsing them into one vague statement.
Check Your Understanding
Use the stated summaries as if they were read from one parallel boxplot display.
- Three classes have median quiz scores of 68, 75, and 75 points. Rank them from greatest to least median, including any tie.
- Three groups have \(Q_1\) and \(Q_3\) values of 8 and 20, 11 and 25, and 10 and 22 minutes. Find each IQR and rank the groups from greatest to least IQR.
- A modified boxplot has whiskers from 6 to 19 and a plotted low outlier at 2. What information would you need before ranking this group by range against another group?
- Two groups have overlapping boxes, but one median is higher. What does the overlap tell you about the middle-half intervals, and what does it not establish about individual observations?
- A group ranks first by median but third by IQR. Write a sentence explaining why these rankings are not contradictory.