A Compact Summary Can Hide the Shape
In Reading a Boxplot’s Quartiles and Spread, you learned to read the minimum, quartiles, median, and maximum from a boxplot. Those landmarks help describe center and spread, but they do not show every feature of the data. In particular, a boxplot does not show whether observations form separate clusters or whether there is an empty interval between them.
A histogram groups quantitative observations into numerical intervals. It can make concentrations and empty intervals visible, although what it shows depends partly on the chosen bin width and endpoints, as discussed in Choosing Class Width and Number of Bins. Comparing a boxplot with a histogram of the same data makes the difference clear: the boxplot summarizes selected positions, while the histogram displays how observations are distributed across intervals.
What the Boxplot Keeps—and What It Leaves Out
The minimum, \(Q_1\), median, \(Q_3\), and maximum preserve important information about the ordered values. For example, the box edges locate the middle half of the observations, and the median marks the middle position. But many different data sets can share those same five values. A boxplot based on that summary gives those different data sets the same basic visual landmarks.
That compression means that the lengths of the box and whiskers cannot tell you where observations are concentrated within each quartile section. Nor can they tell you whether there is a long empty stretch between two values. A long section can indicate a wide numerical span for that rank section, as in the earlier boxplot tutorial, but it does not show whether observations fill that span evenly or bunch at its ends.
A histogram offers a different view. Each bar represents the observations in an interval; the pattern of bar heights can show concentrations, multiple peaks, or bins with no observations. A bin with zero count is evidence of an empty interval under that binning. However, a histogram’s appearance can change when you change the bin width or starting point. A gap smaller than a bin may be hidden, while a different set of bin boundaries may make a pattern easier to see.
Compare Both Displays for the Same Observations
The following fictional data sets record the time, in minutes, that two groups of 20 students spent on a practice activity. They are invented examples, not results from a real study. To compare the displays, use the same histogram intervals for both groups: \(0\) to less than \(5\), \(5\) to less than \(10\), and so on, with the final interval including 30.
Worked Example: A Histogram Reveals a Large Gap
Group A’s ordered times are \(0, 1, 1, 2, 2, 3, 4, 4, 5, 5, 25, 25, 26, 26, 27, 28, 28, 29, 29, 30\). Find its five-number summary, then use the histogram-bin counts to describe a feature that the boxplot hides.
Plan. For these 20 ordered values, find the median by averaging positions 10 and 11. Find \(Q_1\) by averaging positions 5 and 6 in the lower half, and \(Q_3\) by averaging positions 15 and 16 in the upper half. Then count the observations in each stated interval.
Do. The minimum is 0 minutes and the maximum is 30 minutes. The first quartile is \((2+3)/2=2.5\) minutes. The median is \((5+25)/2=15\) minutes. The third quartile is \((27+28)/2=27.5\) minutes. Thus, the five-number summary is \(0,\ 2.5,\ 15,\ 27.5,\ 30\) minutes. The quartile calculations check against the values at positions 5 and 6, 10 and 11, and 15 and 16 in the ordered list.
Count the observations in each bin. There are 8 from 0 to less than 5, 2 from 5 to less than 10, none from 10 to less than 15, none from 15 to less than 20, none from 20 to less than 25, and 10 from 25 through 30.
| Time interval (minutes) | Group A count |
|---|---|
| 0 to less than 5 | 8 |
| 5 to less than 10 | 2 |
| 10 to less than 15 | 0 |
| 15 to less than 20 | 0 |
| 20 to less than 25 | 0 |
| 25 to 30, including 30 | 10 |
The histogram has observations concentrated near the low and high ends, with three consecutive empty bins between 10 and 25 minutes. The boxplot records the same five-number summary, but its box and whiskers do not draw those empty intervals or show the two concentrations.
Conclude. Group A’s five-number summary describes its quartiles and spread, but the histogram reveals a clear gap from 10 to less than 25 minutes and concentrations on either side. The boxplot alone would not justify claiming that the times form two clusters.
Different Data Can Have the Same Boxplot
The most important comparison is not simply that the two graph types look different. It is that data sets with different internal patterns can produce the same five-number summary. In that case, their basic boxplots are identical even though their histograms are not.
Worked Example: Same Five-Number Summary, Different Histogram
Group B’s ordered times are \(0, 1, 1, 2, 2, 3, 7, 10, 12, 14, 16, 18, 20, 23, 27, 28, 28, 29, 29, 30\). Compare Group B’s boxplot summary and histogram with Group A’s.
Plan. Use the same positions and quartile convention as in the first example, then tally Group B’s values in the same intervals. Comparing the resulting summary and counts will separate what the boxplots share from what the histograms show.
Do. Group B’s minimum and maximum are 0 and 30 minutes. Its first quartile is \((2+3)/2=2.5\) minutes, its median is \((14+16)/2=15\) minutes, and its third quartile is \((27+28)/2=27.5\) minutes. These calculations use the same ordered positions as Group A. So both groups have the five-number summary \(0,\ 2.5,\ 15,\ 27.5,\ 30\) minutes.
Group B’s counts in the intervals are 6, 1, 3, 2, 2, and 6. For example, 10, 12, and 14 fall in the interval from 10 to less than 15; 16 and 18 fall from 15 to less than 20. Each interval count comes from tallying the stated observations once.
| Time interval (minutes) | Group A count | Group B count |
|---|---|---|
| 0 to less than 5 | 8 | 6 |
| 5 to less than 10 | 2 | 1 |
| 10 to less than 15 | 0 | 3 |
| 15 to less than 20 | 0 | 2 |
| 20 to less than 25 | 0 | 2 |
| 25 to 30, including 30 | 10 | 6 |
The shared five-number summary means the groups have the same basic boxplot. But their histogram counts differ: Group B has observations in every interval, whereas Group A has no observations in three consecutive intervals. Group B therefore does not have the same large gap shown by Group A’s histogram. The histogram counts also show that Group B’s observations are spread through the middle intervals rather than being separated by an empty stretch.
Conclude. The boxplots match in their five landmarks, but the histograms show different patterns between those landmarks. The identical boxplots do not imply identical distributions.
Be Careful About What a Histogram Shows
A histogram is useful for looking for modality and gaps, but it is still a grouped display. It does not show every individual value in the way a dotplot does, and a bar’s height only gives the count or relative frequency for its bin. As covered in Choosing Class Width and Number of Bins, narrower bins can reveal detail but may emphasize small fluctuations; wider bins can smooth away detail or combine a gap with observations on either side.
For example, Group A’s empty bins from 10 to less than 25 are clear with the intervals used above. If wider bins combined observations from both sides of an empty stretch, the displayed bars might not make that gap as apparent. Before describing a histogram, check its scale, bin boundaries, and bin width. When the conclusion matters, consider whether a different reasonable binning would change the apparent pattern.
Worked Example: What Can You Conclude From a Boxplot Alone?
A student sees the boxplot summary \(0,\ 2.5,\ 15,\ 27.5,\ 30\) for practice times and says, “The data have two modes and no one spent between 10 and 25 minutes.” Decide whether the boxplot supports both parts of the claim, and state what display would help.
Plan. Check whether either claim follows from the five-number summary. The first is about modality; the second is about whether observations occupy a particular interval. These are features that require information beyond the five landmarks.
Do. The summary gives the minimum, quartiles, median, and maximum. It does not give the frequency of values within each quartile section, so it cannot establish how many peaks the distribution has. Nor does it give every value between the quartile landmarks: the median of 15 minutes does not mean an observation occurred at every nearby time, or that some interval is empty.
The histograms in the first two examples demonstrate why. Both groups have the stated summary, yet Group A has zero observations in the intervals from 10 to less than 25 and Group B has observations in each of those intervals. A histogram using clearly stated bins—or, when individual observations matter, a dotplot—would provide evidence about concentrations and gaps that the boxplot does not display.
Conclude. The boxplot alone supports neither the claim of two modes nor the claim of no observations from 10 to 25 minutes. To investigate those claims, examine a display that shows more of the distribution’s internal detail, and check how the chosen intervals affect the appearance.
Common Mistakes and AP Exam Tips
- Claiming a boxplot shows a mode. A boxplot marks quartiles and extremes; it does not show the frequency pattern needed to identify peaks. Use a histogram or dotplot to investigate modality.
- Treating the median as a description of every value near it. The median is a measure of position, not a guarantee that observations fill nearby intervals. A gap can include the median’s location or occur elsewhere, depending on the data.
- Assuming matching boxplots mean matching distributions. The same five-number summary can come from different lists of observations. Compare histograms if the question concerns clusters, gaps, or concentrations within quartile sections.
- Calling any low histogram bar a gap. A gap requires an interval with no observations. A short bar has observations; a zero-height bin has none within that bin.
- Ignoring the binning. State the interval when identifying a histogram gap, and remember that another bin width or starting point can change how clearly a feature appears.
- Overstating what the histogram proves. A histogram shows counts in intervals, not exact locations of every observation. Describe the visible pattern using the displayed bins rather than claiming more precision than the graph provides.
For a full-credit comparison, identify what the boxplot summarizes, cite the relevant histogram pattern or bin counts, and describe the difference in context. If a graph does not display enough information to answer a question about modality or gaps, say so directly rather than guessing from the quartile marks.
Check Your Understanding
Use what each display shows—and what it leaves out—to answer the questions.
- Two groups have identical five-number summaries. Does that prove their histograms will have the same bin counts? Explain.
- A histogram has counts of 4, 2, 0, 0, and 5 in five consecutive bins. Which bins show a gap, and what should you include when describing it?
- Why can a boxplot with a long section between \(Q_1\) and the median not tell you whether observations are evenly spread across that section?
- How might increasing a histogram’s bin width affect the visibility of a gap?
- A boxplot is the only display provided, and a question asks whether the data are bimodal. What can you conclude, and what additional display would help?