Compare Shape, Not Just Bar Heights
In Using Comparison Words: Greater, Less, Similar, you learned to name both groups and the feature being compared. Here the feature is shape: we compare the way two distributions are arranged, especially their skewness and modality. A useful comparison identifies how each distribution looks, then puts the difference into one direct sentence.
Before comparing shape, check that the histograms display the same quantitative variable, use the same horizontal scale, and group values into matching intervals. If the sample sizes differ, the number of observations in each bar can differ simply because one group has more observations. Relative-frequency histograms help compare the shares of observations in corresponding intervals. As in Describing a Histogram With Real Numbers, report a peak as an interval rather than treating its location as one exact data value.
A distribution is right-skewed when its tail extends toward larger values; it is left-skewed when its tail extends toward smaller values. The tail, not the side where most observations are concentrated, determines the direction. A distribution that has no clear longer tail may be described as approximately symmetric.
Modality is about distinct peaks, not about how tall the single tallest bar is. A distribution with one main peak is called unimodal; one with two distinct peaks is bimodal; and one with more than two peaks may be called multimodal. A low region separating concentrations can help show that two peaks are distinct. A small bump or a single unusually tall bar does not automatically establish an additional mode.
Make a Shape Comparison in One Sentence
A clear sentence can describe both skewness and modality while naming the two groups and the variable. For instance: “For [variable], the distribution for [Group A] is [shape and modality], whereas the distribution for [Group B] is [shape and modality].” Use only the descriptions the histograms support. If the histograms do not clearly show skewness for one group, say so rather than guessing.
This is a writing guide, not a calculation. You do not need to force a skewness label and a modality label into every description if the display does not support both. But when you can identify both features, naming them together makes the comparison compact and specific. In context, “whereas” or “while” can make the contrast easy to follow.
Use the same kind of description for each group. For example, do not call one histogram “right-skewed” and describe the other only as “different”; state whether the second is left-skewed, approximately symmetric, or unclear. Similarly, if one group is bimodal, say whether the other is unimodal, also multimodal, or has no clearly distinguishable peaks.
Confirm that both histograms show the same variable and matching bin intervals and horizontal scales. If sample sizes differ, use relative frequencies to compare proportions across bins.
Look for the direction of each distribution’s longer tail. Use “approximately symmetric” when neither side clearly extends farther.
Count distinct concentrations or peaks, looking for a lower region that separates peaks. Do not count every bar as a mode.
Name the variable and both groups, then connect the supported shape descriptions with “whereas” or another clear comparison word.
Histograms are summaries of grouped observations. The appearance of a peak or tail can depend on the intervals used, and changing the bin width can change how many peaks seem visible. Keep conclusions appropriately cautious: write that a histogram “appears bimodal” when the evidence is suggestive, and do not claim that a histogram proves why the shape occurred.
Worked Example: Opposite Skew, One Peak Each
Worked Example: Opposite Skew, One Peak Each
A fictional transit team compares passenger waiting times, in minutes, at two service desks. The histograms use matching five-minute bins. Desk A recorded 50 waits, with counts 20, 14, 8, 5, and 3 from the smallest to largest interval. Desk B also recorded 50 waits, with counts 3, 5, 8, 14, and 20 in those same intervals. Compare the shapes in one sentence.
| Waiting-time interval (minutes) | Desk A count | Desk B count |
|---|---|---|
| 0 to less than 5 | 20 | 3 |
| 5 to less than 10 | 14 | 5 |
| 10 to less than 15 | 8 | 8 |
| 15 to less than 20 | 5 | 14 |
| 20 to less than 25 | 3 | 20 |
Check the totals. Desk A’s count is \(20+14+8+5+3=50\), and Desk B’s count is \(3+5+8+14+20=50\). The groups have the same sample size, so the counts can be compared directly here. Both histograms use the same bins.
Read the tails and peaks. Desk A has its greatest concentration in the lowest interval, and its counts taper toward larger waiting times; it has a tail extending right. Desk B has its greatest concentration in the highest interval, with counts tapering toward smaller waiting times; it has a tail extending left. Each has one clear main peak.
Write the comparison. “For passenger waiting times, the distribution at Desk A is unimodal and right-skewed, whereas the distribution at Desk B is unimodal and left-skewed.”
The sentence names the variable and both desks, contrasts skewness, and states that modality is the same. It does not confuse the tail’s direction with the location of the highest bar.
Use Relative Frequencies When Sample Sizes Differ
When two groups have different sample sizes, raw bar counts are not directly comparable as proportions. A count of 12 in one group might represent a larger share than a count of 15 in another if the first group has far fewer observations. Relative frequency expresses a bin’s count as a fraction of its group’s total.
When bins have equal widths, the relative frequencies show the share of each group in corresponding intervals. Their totals should be 1, or 100% when written as percentages, apart from rounding. For comparing shape, focus on where each group’s observations are concentrated, whether the distribution has distinct peaks, and which way any tail extends.
Worked Example: Comparing Modality With Unequal Samples
Worked Example: Comparing Modality With Unequal Samples
A fictional school garden records the heights of seedlings from two trays, in centimeters. The histograms use the same five-centimeter bins. Tray A has 40 seedlings and Tray B has 50. Compare the shapes, accounting for the different sample sizes.
| Seedling-height interval (cm) | Tray A count | Tray A relative frequency | Tray B count | Tray B relative frequency |
|---|---|---|---|---|
| 0 to less than 5 | 4 | 4/40 = 10% | 5 | 5/50 = 10% |
| 5 to less than 10 | 14 | 14/40 = 35% | 10 | 10/50 = 20% |
| 10 to less than 15 | 3 | 3/40 = 7.5% | 20 | 20/50 = 40% |
| 15 to less than 20 | 15 | 15/40 = 37.5% | 10 | 10/50 = 20% |
| 20 to less than 25 | 4 | 4/40 = 10% | 5 | 5/50 = 10% |
Check totals and proportions. Tray A’s counts total \(4+14+3+15+4=40\); Tray B’s counts total \(5+10+20+10+5=50\). Tray A’s relative frequencies total \(10+35+7.5+37.5+10=100\%\), and Tray B’s total \(10+20+40+20+10=100\%\). Relative frequencies allow a fairer comparison of the groups’ shapes despite their different sample sizes.
Read the shape. Tray A has two high regions, in the 5-to-less-than-10 and 15-to-less-than-20 centimeter intervals, separated by a much lower interval. This suggests a bimodal shape. Tray B has one main peak in the 10-to-less-than-15 centimeter interval, with the neighboring bins lower on both sides; it appears unimodal and approximately symmetric.
Write the comparison. “For seedling heights, the distribution in Tray A appears bimodal, whereas the distribution in Tray B appears unimodal and approximately symmetric.”
The relative frequencies, rather than the raw counts, help show that the two peaks in Tray A are concentrations within that group and not merely a consequence of a larger total sample. The shape description remains cautious because the data are grouped into intervals.
Worked Example: Contrast a Bimodal Shape With a Right Tail
Worked Example: Contrast a Bimodal Shape With a Right Tail
A fictional community center records the time, in minutes, that visitors spend using a set of shared workstations. Two rooms have different numbers of recorded visits. Their histograms use the same ten-minute bins. Room East has counts 3, 14, 4, 15, and 4, for 40 visits. Room West has counts 25, 12, 6, 4, and 3, for 50 visits. Compare their shapes using relative frequencies.
| Time interval (minutes) | Room East count | Room East relative frequency | Room West count | Room West relative frequency |
|---|---|---|---|---|
| 0 to less than 10 | 3 | 3/40 = 7.5% | 25 | 25/50 = 50% |
| 10 to less than 20 | 14 | 14/40 = 35% | 12 | 12/50 = 24% |
| 20 to less than 30 | 4 | 4/40 = 10% | 6 | 6/50 = 12% |
| 30 to less than 40 | 15 | 15/40 = 37.5% | 4 | 4/50 = 8% |
| 40 to less than 50 | 4 | 4/40 = 10% | 3 | 3/50 = 6% |
Check the totals. Room East’s counts add to \(3+14+4+15+4=40\), and Room West’s add to \(25+12+6+4+3=50\). The relative frequencies for Room East total \(7.5+35+10+37.5+10=100\%\); those for Room West total \(50+24+12+8+6=100\%\).
Read the shape. Room East has two separated high regions, in the 10-to-less-than-20 and 30-to-less-than-40 minute intervals, with a lower interval between them; it appears bimodal. Room West has its largest share in the first interval, followed by progressively smaller shares in later intervals, suggesting a right tail. It appears unimodal and right-skewed.
Write the comparison. “For workstation-use times, the distribution in Room East appears bimodal, whereas the distribution in Room West is unimodal and right-skewed, with a tail toward longer visits.”
This sentence compares modality and skewness without claiming that all visits in one room are longer or shorter than visits in the other. The peaks describe concentrations, while the tail describes the distribution’s asymmetry.
Common Mistakes and AP Exam Tips
- Naming skew from the peak’s location. A peak near the low end does not by itself establish left skew. Identify which direction the tail extends, then name the skew.
- Calling the tallest bar the only mode. A second separated concentration may make a distribution bimodal even if one peak is lower. Look for a valley between concentrations, not just the tallest bar.
- Comparing raw counts from unequal samples. A larger count might reflect a larger group, not a greater share in that interval. Use relative frequencies when sample sizes differ and explain the comparison in terms of shape.
- Comparing histograms with mismatched scales or bins. Different intervals can create misleading apparent differences. Check the horizontal scales and bin boundaries before describing shape.
- Overstating what the display proves. A histogram suggests shape; it does not establish the reason for a peak or prove that a group contains separate subpopulations. Use careful wording such as “appears bimodal” when appropriate.
- Leaving out context or one group. “One is skewed and the other has two modes” does not identify the variable or groups. Name both groups and the variable in the comparison sentence.
For a full-credit AP response, make the comparison explicit and supported: name the variable and both groups, describe the direction of skew when it is clear, identify the number of distinct peaks, and use a contrast word such as “whereas.” If a feature is not clear from the histograms, do not invent a label for it.
Check Your Understanding
Use the descriptions below to answer each question. Treat the displays as histograms of the same variable with matching bins.
- Group A has one peak near the low end and a tail toward larger values. Group B has one peak near the high end and a tail toward smaller values. Write one sentence comparing skewness and modality.
- Two histograms show one distinct peak each. What word describes each distribution’s modality? What additional evidence would you need to describe skewness?
- Group C has 30 observations and Group D has 60. Why might relative-frequency histograms be more useful than raw-count histograms for comparing their shapes?
- A histogram has two high regions separated by a noticeably low interval. What modality might it suggest, and why should the description remain cautious?
- Rewrite this weak comparison as a contextual sentence: “One has two peaks, but the other is skewed.” Assume the variable is package-delivery time, Group East appears bimodal, and Group West appears right-skewed and unimodal.