Why Matching Scales Matter
In Comparing Two Dotplots, you practiced comparing distributions displayed on shared axes. Histograms also show how quantitative data are distributed, but they group observations into intervals called bins. A bar’s height shows the number of observations in its bin. To compare two histograms fairly, check that they use the same horizontal scale, the same bin boundaries, and a comparable vertical scale.
A display can make two groups look more different—or more alike—than they are if the axes do not match. For example, the same horizontal distance might represent 5 minutes in one histogram and 10 minutes in the other. Or the tallest bar in each plot might reach the top of its own vertical axis even though the bars represent different counts. Matching scales let you compare positions and heights directly.
A histogram’s peak is the bin or bins with the tallest bar. Describe a peak using its interval, not as an exact value: a bar covering 10 to less than 15 minutes identifies the interval where the largest count occurs, not the most common exact time. If two or more bars tie for the greatest height, report all of those peak intervals.
Two histograms overlap in the sense that both groups have observations in some of the same value intervals. With matching bins, you can identify those intervals by finding where both groups’ bars have positive heights. That is a useful comparison of the displayed distributions, but it does not show that any particular individual in one group has the same value as an individual in the other group.
A Method for Comparing Two Histograms
Start by checking the axes and the bins. Confirm that the variable and units match, that equal horizontal distances represent equal amounts, and that the bars refer to the same intervals. Then check the vertical axes: a count of 10 should have the same height in both displays. If one plot uses a count scale and the other uses a different count scale, comparing apparent bar heights directly can mislead.
Next, describe the peaks and overlap. Name each group’s tallest bin or tied tallest bins, then identify intervals where both groups have observations. Look beyond the peaks as well: are the bars concentrated in similar regions, or does one group have more observations in intervals where the other has few or none? As in Comparing Shapes of Two Distributions, describe the patterns the histograms support rather than relying on a vague impression.
Sample sizes matter. If both groups have the same number of observations and the vertical axes show counts on the same scale, counts are directly comparable. If sample sizes differ, a group may have taller bars simply because it contains more observations. In that situation, report the sample sizes and consider comparing the percentage of each group in corresponding bins. Make clear whether you are describing counts or percentages.
Verify the variable, units, horizontal scale, bin widths, bin boundaries, and vertical count scale.
Identify the tallest bin or tied tallest bins for each group, describing each peak as an interval.
Find the intervals with positive counts in both groups and note where the groups’ bar heights differ.
Name the variable and groups, cite the relevant intervals and counts or percentages, and avoid claims about exact values or individuals that the histograms cannot show.
Worked Example: Comparing Peaks and Shared Intervals
Worked Example: Comparing Peaks and Shared Intervals
A fictional school program records how long students in two groups spend traveling to school, in minutes. The histograms use the same bins and count scale. The table gives the height of each bar.
| Travel time (minutes) | Group East | Group West |
|---|---|---|
| 0 to less than 5 | 2 | 5 |
| 5 to less than 10 | 8 | 12 |
| 10 to less than 15 | 15 | 14 |
| 15 to less than 20 | 10 | 7 |
| 20 to less than 25 | 5 | 2 |
Check the display. Both histograms measure travel time in minutes, use the same five-minute bins, and show counts on the same vertical scale. Each group has 40 students: East’s counts sum to \(2+8+15+10+5=40\), and West’s sum to \(5+12+14+7+2=40\). Counts can be compared directly.
Compare peaks. Group East’s tallest bar is the 10-to-less-than-15-minute bin, with 15 students. Group West’s tallest bar is also that interval, with 14 students. Thus, both groups have the same peak interval, though East’s bar is one student taller.
Compare overlap. Both groups have positive counts in every displayed interval, so their histograms overlap across all five bins. The bar heights differ: West has more students in the first two bins, while East has more in the last three bins.
Conclude in context. “For these students, both groups have a peak in the 10-to-less-than-15-minute travel-time interval, with 15 students in East and 14 in West. The groups have observations in every shared interval, but West has higher counts below 10 minutes and East has higher counts from 10 to less than 25 minutes.”
The comparison reports what is visible in the grouped counts. It does not claim, for example, that the students represented by matching intervals have identical travel times.
Worked Example: Unequal Sample Sizes and Percentages
Worked Example: Unequal Sample Sizes and Percentages
A fictional community center compares the number of minutes participants spend using a fitness area during a visit. Both histograms use identical ten-minute bins and the same horizontal values, but the groups have different sample sizes.
| Visit time (minutes) | Group A count | Group B count |
|---|---|---|
| 0 to less than 10 | 3 | 12 |
| 10 to less than 20 | 12 | 18 |
| 20 to less than 30 | 10 | 18 |
| 30 to less than 40 | 5 | 12 |
| Total | 30 | 60 |
Check the sample sizes. Group A has 30 observations and Group B has 60. The count scale can show Group B’s bars as taller, but that alone does not mean its participants are more concentrated in those intervals. To compare the distributions’ relative patterns, calculate the percentage in each bin for each group.
For Group A, the percentages are \(3/30=10\%\), \(12/30=40\%\), \(10/30\approx33.3\%\), and \(5/30\approx16.7\%\). For Group B, they are \(12/60=20\%\), \(18/60=30\%\), \(18/60=30\%\), and \(12/60=20\%\). These percentages make the different group sizes explicit.
Compare peaks and overlap. Group A’s peak is the 10-to-less-than-20-minute bin, containing 40% of its sample. Group B has tied peaks in the 10-to-less-than-20 and 20-to-less-than-30-minute bins, each containing 30% of its sample. Both groups have observations in all four intervals, so the histograms overlap across the displayed range.
Conclude in context. “For these visits, Group A has a single peak in the 10-to-less-than-20-minute interval, where 40% of its participants fall. Group B has tied peaks in the 10-to-less-than-20 and 20-to-less-than-30-minute intervals, each containing 30% of its participants. Both groups have observations in all four intervals; Group B’s larger raw counts partly reflect its twice-as-large sample.”
The peak comparison here uses percentages because the question concerns the pattern within each group. If the question were how many people were recorded in a bin, the raw counts would answer it. State which comparison you are making.
Worked Example: Why Both Histograms Need the Same Vertical Scale
Worked Example: Why Both Histograms Need the Same Vertical Scale
A fictional gardening club records the number of seedlings that sprout in two batches. Each batch contains 50 trays. The bins and counts are shown below.
| Seedlings per tray | Batch A count | Batch B count |
|---|---|---|
| 0 to less than 10 | 5 | 10 |
| 10 to less than 20 | 10 | 15 |
| 20 to less than 30 | 20 | 15 |
| 30 to less than 40 | 10 | 5 |
| 40 to less than 50 | 5 | 5 |
Suppose Batch A’s histogram has a vertical axis marked from 0 to 20 trays, while Batch B’s axis is marked from 0 to 15 trays. Each histogram’s tallest bar reaches the top of its own plot. Looking only at those plot heights might make the peaks appear equally tall, even though the counts are 20 for Batch A and 15 for Batch B.
Use a shared scale. If both vertical axes run from 0 to 20 trays with the same tick marks, the height difference is clear. Batch A peaks in the 20-to-less-than-30-seedlings bin with 20 trays. Batch B has tied peaks in the 10-to-less-than-20 and 20-to-less-than-30 bins, with 15 trays in each.
Describe overlap. Both batches have positive counts in all five intervals. Their bars are tallest in nearby, shared intervals, but Batch A has more trays in the 20-to-less-than-30 bin, while Batch B has more in the 0-to-less-than-10 and 10-to-less-than-20 bins. Both have five trays in the highest interval.
Conclude in context. “With the same scale, Batch A has a single peak of 20 trays in the 20-to-less-than-30-seedlings interval. Batch B has tied peaks of 15 trays in the two intervals from 10 to less than 30 seedlings. Both batches have trays in every interval, but their counts differ across several bins.”
The horizontal bins and vertical scale both matter. Matching only one axis does not make a visual comparison fully fair.
What Overlap and Peaks Do—and Do Not—Show
A bin is an interval, not a single value. If both groups have bars in the 10-to-less-than-15-minute interval, the histograms show that both groups include observations somewhere in that interval. They do not show which exact values those observations have. In particular, overlapping bars do not prove that the two groups share identical observations, and non-overlapping bars do not establish that the groups’ populations never have similar values.
A peak also describes a bin, not an exact most common value. The tallest bar might contain many different observed values, and the histogram does not reveal how those values are distributed inside the bin. Bin width and bin placement can affect the visible pattern: changing boundaries can shift counts between bars and may make a distribution appear to have a different number of peaks. When comparing two histograms, matching boundaries are essential; when describing a single histogram, avoid treating its particular peak pattern as the only possible way to group the data.
Keep counts distinct from percentages. Counts answer how many observations fall in a bin. Percentages answer what share of a group falls there. If sample sizes differ, a larger count may reflect a larger sample rather than a larger share. State the sample sizes and the basis of your comparison when that distinction matters.
Common Mistakes and AP Exam Tips
- Skipping the scale check. Confirm that the same horizontal distance represents the same values and that the vertical count scales match before comparing bar heights.
- Comparing bars with different bin boundaries. A count in one group’s 10-to-20 interval is not directly comparable to a count in another group’s 15-to-25 interval. Use corresponding intervals.
- Calling a peak an exact value. Say “the peak is the 10-to-less-than-15-minute bin,” not “the most common time is 12 minutes,” unless the data provide that exact information.
- Ignoring tied peaks. If two bars share the greatest height, report both peak intervals rather than selecting one.
- Equating overlap with identical observations. Shared occupied bins mean both groups have values in those intervals; they do not identify matching individuals or exact values.
- Comparing raw counts without considering sample size. When group sizes differ, explain the counts or compare within-group percentages. Do not mistake a larger count for a larger proportion.
- Using “the distributions are the same” because the peaks match. Matching peak intervals do not establish matching counts, spread, shape, or percentages in the other bins. Describe only the features that actually match.
A full-credit comparison names the variable and groups, identifies peak intervals accurately, describes overlap using the common bins, and supports claims with counts or percentages. It also avoids turning grouped bars into claims about exact observations. For example, “Both batches have observations in all five seedling intervals, but Batch A’s highest bar is 20 trays in the 20-to-less-than-30 interval, while Batch B has tied peaks of 15 trays in each of the two intervals from 10 to less than 30” is specific and supported.
Check Your Understanding
Use the shared bins and values in each question to make a careful comparison.
- Group Pine has counts 2, 9, 14, and 5 in four consecutive equal-width bins. Identify its peak bin by position and count. What additional detail would you need to name the interval?
- Two histograms have positive counts in the same three bins. What does that tell you about overlap, and what does it not establish about exact observations?
- Group A has 20 observations and Group B has 40. Both have 12 observations in the same bin. Compare the counts and the within-group percentages in that bin.
- One histogram uses a vertical scale from 0 to 10 and another uses a scale from 0 to 20. Why should you not compare the apparent heights without checking the numbers?
- A histogram has two bars tied for the tallest height. How should you describe its peak?