What Makes a Comparison a Comparison?
A description can accurately report features of two groups and still fail to compare them. For example, “Group A is centered near 8, and Group B is centered near 11” gives two facts, but it leaves the reader to work out the relationship. A stronger comparison says that Group B’s center is higher than Group A’s, and supports that statement with the values.
As in Why Comparing Distributions Needs All Four Features, a thorough comparison can consider shape, center, spread, and unusual values. The goal is not to recite four separate checklists. It is to explain the similarities and differences that matter, using evidence from the display or summaries. The earlier tutorial Using Comparison Words: Greater, Less, Similar introduced words such as “greater,” “less,” and “similar.” Here, we focus on common ways a comparison can go wrong and how to repair it.
A useful habit is to put both groups into the same sentence. Name the feature, say how the groups relate, and give evidence. For example: “The median growth for seedlings in Tray B is 3 centimeters greater than the median for Tray A, 11 centimeters compared with 8 centimeters.” That sentence identifies the feature, comparison, groups, and context.
Mistake 1: Describing Each Group Separately
A common weak response gives a description of one group, then a description of another, without stating what the reader should notice between them. Even if both descriptions are accurate, the response may not answer a question asking how the distributions compare.
Instead, connect the groups feature by feature. You might say that one group has a higher median, while the other has a smaller IQR; or that both groups have similar centers but different shapes. Do not assume that listing the two values automatically communicates the comparison.
Worked Example: Comparing Seedling Growth
A fictional greenhouse team compares the growth of seedlings in Tray A and Tray B after three weeks, measured in centimeters. Dotplots use the same horizontal scale. Tray A’s distribution is roughly symmetric, with a median of 8 centimeters and an IQR of 3 centimeters. Tray B’s distribution is right-skewed, with a median of 11 centimeters and an IQR of 5 centimeters. Tray B also has one unusually high observation at 24 centimeters; Tray A has no comparably distant value.
Identify what the weak description misses. “Tray A is roughly symmetric and centered at 8 centimeters. Tray B is right-skewed and centered at 11 centimeters” reports features separately. It does not explicitly say which tray has the higher center, or how the spreads compare.
Compare the centers. Subtract the medians in the stated order: \(11-8=3\) centimeters. Tray B’s median seedling growth is 3 centimeters greater than Tray A’s median growth.
Compare the spreads. Compare IQR with IQR: \(5-3=2\) centimeters. Tray B’s IQR is 2 centimeters greater than Tray A’s, so the middle half of its growth observations covers a wider interval.
Compare shape and unusual values. Tray A’s distribution is roughly symmetric, while Tray B’s is right-skewed. Tray B has an unusually high growth value of 24 centimeters, whereas Tray A has no comparably distant observation.
Write the comparison. “Tray B’s median growth is 3 centimeters greater than Tray A’s (11 centimeters versus 8 centimeters), and its IQR is 2 centimeters greater (5 centimeters versus 3 centimeters). Tray A’s distribution is roughly symmetric, whereas Tray B’s is right-skewed and includes an unusually high observation at 24 centimeters.” This response connects the groups directly and supports each comparison with evidence.
The example does not claim that every seedling in Tray B grew more than every seedling in Tray A. A larger median describes a difference in the distributions’ centers, not a ranking of every individual observation. Likewise, a larger IQR describes a wider middle half, not necessarily a wider range. Choose wording that matches the statistic or feature you actually observed.
Mistake 2: Omitting Comparison Words or Evidence
A sentence can name both groups and still be unclear if it never says how they relate. “Group P has an IQR of 4 hours and Group Q has an IQR of 7 hours” gives the reader two values. Add comparison language to make the conclusion explicit: “Group Q’s IQR is 3 hours greater than Group P’s.”
Comparison words also need to match the evidence. “Similar” does not mean exactly equal, and “greater” must refer to a particular feature. A statement such as “Group Q is greater” is incomplete: greater in median, IQR, mean, or something else? When making a numerical comparison, show the relevant values or calculation and include the variable’s units.
Worked Example: Revising a Vague Comparison
A fictional recreation center compares the number of minutes people spend on a climbing wall during a visit. For Center North, the median is 18 minutes and the IQR is 8 minutes. For Center South, the median is 21 minutes and the IQR is 8 minutes. The displays show roughly similar shapes and no clear outliers.
Evaluate the weak statement. “North has 18 and 8; South has 21 and 8. South is more.” The values lack units, do not identify which one is the median or IQR, and “more” does not say what feature is greater. In fact, the two IQRs are equal, so South is not more spread out by this measure.
Compare the centers. The difference in medians is \(21-18=3\) minutes. Center South’s median visit to the climbing wall is 3 minutes longer than Center North’s median visit.
Compare the spreads. The IQR difference is \(8-8=0\) minutes. The two groups have the same IQR, so the widths of their middle halves are equal.
Write the comparison. “The median time on the climbing wall is 3 minutes greater at Center South than at Center North (21 minutes versus 18 minutes). The IQR is 8 minutes at both centers, so their middle-half spreads are equal. The displays show roughly similar shapes and no clear outliers.” This gives an explicit comparison while distinguishing the feature that differs from the feature that is the same.
“Similar” is often appropriate for visual features such as shape, but say what appears similar. For numerical summaries, report the values when available. If you say the IQRs are equal, for instance, show that both are 8 minutes. If two values are close but not identical, “similar” may be more appropriate than “equal,” depending on the context and the precision of the summaries.
Mistake 3: Comparing Counts in Unequal Groups
A graph that uses counts shows how many observations fall in each category or interval. If the groups have different sample sizes, raw counts can mislead: the larger group may have more observations in a category simply because it contains more people or items overall. To compare how common a category is within each group, compare the relative frequencies—the proportions or percentages of each group in that category.
This matters when comparing bar charts or histograms that display counts. Check whether the groups have the same sample size before interpreting the heights as evidence that a category is more common in one group. When sample sizes differ, percentages or relative-frequency displays make the within-group comparison clearer.
Worked Example: Comparing Commute-Time Categories
A fictional school surveys 40 students who walk to school and 80 students who take a bus. Commute times are sorted into the same three intervals for both groups. The counts are shown below.
| Commute time | Walkers (40 students) | Bus riders (80 students) |
|---|---|---|
| 0 to less than 10 minutes | 18 | 20 |
| 10 to less than 20 minutes | 14 | 36 |
| 20 to less than 30 minutes | 8 | 24 |
Spot the problem with raw counts. In the first interval, there are 20 bus riders and 18 walkers. A count-only statement might say more bus riders have commutes under 10 minutes. But the groups differ in size: there are twice as many surveyed bus riders as walkers. Compare the shares within each group instead.
Calculate relative frequencies. For the first interval, the walker proportion is \(18/40=0.45\), or 45%, and the bus-rider proportion is \(20/80=0.25\), or 25%. For the second interval, the proportions are \(14/40=0.35\), or 35%, for walkers and \(36/80=0.45\), or 45%, for bus riders. For the third interval, the proportions are \(8/40=0.20\), or 20%, for walkers and \(24/80=0.30\), or 30%, for bus riders. Each set of percentages totals 100%.
Compare in context. Although the raw count is slightly higher for bus riders in the under-10-minute interval, the percentage of walkers in that interval is greater: 45% compared with 25%. Bus riders have greater percentages in both the 10-to-under-20-minute interval (45% versus 35%) and the 20-to-under-30-minute interval (30% versus 20%). These percentages compare the distributions within the two surveyed groups more fairly than the counts do.
Keep the denominator attached to the group it describes: walkers’ counts are divided by 40, and bus riders’ counts are divided by 80. Dividing both counts by the same total, or comparing only the numerators, does not give the share within each group. Also check that the categories or interval boundaries match; percentages from different categories are not a direct comparison of the same feature.
A Quick Audit for a Strong Comparison
Before submitting a response, read it as if the reader cannot see the graph. Can the reader tell which groups are being compared, what feature is under discussion, and what evidence supports the claim? If you have only written a description of each group, add a sentence stating the relationship. If you use “more,” “less,” or “similar,” make clear what is more, less, or similar.
Make clear which populations or samples and which quantitative variable the comparison concerns.
Compare shape with shape, center with center, spread with spread, or unusual values with unusual values.
Use a precise comparison such as greater, less, wider, more right-skewed, or similar.
Support the statement with a graph feature or relevant summaries. Include units for numerical comparisons.
If sample sizes differ, use within-group proportions rather than raw counts to compare how common a category is. Avoid claims about individuals when your evidence describes a distribution summary.
Common Mistakes and AP Exam Tips
- Writing two disconnected descriptions. “A is centered at 12; B is centered at 16” is less direct than “B’s median is 4 units greater than A’s.” Name the comparison, then give the values.
- Leaving out the comparison word. A list of numbers may be correct but does not clearly answer which group is higher, lower, wider, or similar. State the relationship plainly.
- Using “more” without naming the feature. Say “greater median,” “larger IQR,” or “more right-skewed,” rather than “Group B is more.”
- Comparing raw counts from unequal groups. More observations in one category do not necessarily mean that category is more common within that group. Divide each category count by its own group total and compare the resulting proportions or percentages.
- Using mismatched summaries. Do not compare one group’s IQR with another group’s standard deviation. As covered in Matching Measures: Mean With Standard Deviation, Median With IQR, compare like with like.
- Overstating what a summary shows. A higher median does not mean every observation in that group is higher. An IQR comparison describes the middle-half widths, not the full ranges or each individual value.
A full-credit comparison is direct, specific, and supported. It identifies both groups and the feature, uses accurate comparison language, and reports evidence with units when appropriate. For counts from unequal groups, it compares within-group percentages rather than treating the larger count as proof that a category is more common.
Check Your Understanding
For each question, focus on making the comparison explicit and fair.
- Group A has a median of 14 minutes and Group B has a median of 19 minutes for the same activity. Write a comparison sentence that includes the difference and units.
- A student writes, “The first distribution is left-skewed. The second is symmetric.” What should the student add to make the response explicitly comparative?
- In one group of 30 people, 12 chose option X. In another group of 60 people, 18 chose option X. Calculate each group’s percentage choosing X and identify which group has the greater percentage.
- Why is “Group B has more observations in this interval” not necessarily a fair comparison if Group B’s sample is larger?
- Two groups have the same median but different IQRs. Write a sentence that compares their centers and spreads without implying that the distributions are identical.