Let the Question Guide the Display
In Matching Histograms to Boxplots, you used two displays together: a histogram to examine shape and a boxplot to check center and spread. Here, the task is to choose a display in the first place. Dotplots, stemplots, histograms, and boxplots can all represent quantitative data, but they do not preserve or emphasize the same information.
Start by asking what the reader needs to learn. Is it important to see every observation? Is the main goal to examine the distribution’s shape, including clusters and gaps? Are several groups being compared by center and spread? Then consider how much data there are. A display that works well for a dozen observations may become cluttered for hundreds.
There is no single sample-size cutoff that decides which graph to use. A set of 40 distinct measurements may be easier to summarize with a histogram than a dotplot, while 40 measurements taking only four possible values might still be easy to show with stacked dots. The data’s precision, repetition, and purpose all matter.
What Each Display Is Best At
A dotplot places one dot at each observed value, stacking dots when values repeat. As the earlier tutorials on building and reading dotplots explained, this makes individual values and their frequencies visible. Dotplots are often effective for small or moderate data sets, especially when the values are discrete or repeat. They can also make unusual individual values, clusters, and gaps easy to spot. When many observations have numerous different values, however, the dots can crowd together and make the overall pattern difficult to read.
A stemplot separates each observation into a stem and a leaf. With a suitable key, the leaves retain the original values and are arranged in numerical order. A stemplot can show the shape of a small or moderate distribution without grouping observations into intervals. It is most convenient when the values have a useful place-value structure, such as whole-number scores. If measurements have many decimal places, a very wide range, or many observations, choosing stems and fitting all the leaves can become awkward. The earlier stemplot tutorials explain how to choose a key and read the resulting values.
A histogram groups quantitative values into consecutive intervals, or bins. It is a strong choice when the main purpose is to see an overall distribution pattern—such as a central concentration, skewness, clusters, or gaps—particularly for a larger data set. As covered in the earlier tutorials on constructing and choosing histogram bins, the bin widths affect what pattern is visible. A histogram does not show the exact value of every observation; it shows how many observations fall in each interval.
A boxplot displays a five-number summary and gives a compact view of center and spread. It is especially useful when the task is to compare several groups on a common scale. The earlier tutorials on boxplot construction and interpretation explain how to read the median, quartiles, and whiskers. A boxplot does not reveal the full frequency pattern: it may conceal clusters, gaps, and modality. If the question is about those features, choose a dotplot or histogram as well, or instead.
Choose for Individual Values or Distribution Shape
Worked Example: A Small Set of Repeated Measurements
A fictional greenhouse technician records the number of new leaves on each of 13 seedlings: \(2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 7, 8, 9\). The question is, “Which values occur most often, and are there any gaps in the observed counts?” Which display is the best first choice?
Plan. The sample is small, repeated values are present, and the question asks about exact values, frequencies, and gaps. Compare displays by how clearly they preserve those details.
Do. A dotplot is a strong choice because each observation is represented directly, and repeated counts appear as stacks. In this sample, a reader can readily see that 4 leaves is the most frequent value, occurring three times. The observed values include every whole number from 2 through 9, so there are no gaps between those observed counts. A stemplot could also preserve the individual values: its key would make the leaves readable as leaf counts. It may be a reasonable alternative if the goal includes seeing the values in order.
A histogram is possible, but bins would group the exact counts. Depending on the bin boundaries, the graph might combine neighboring values and make it harder to identify which exact count occurred most often. A boxplot would provide a compact summary, but it would not display the frequency at each count or show the individual observations.
Conclude. Use a dotplot as the clearest first display for these 13 seedling counts because it makes repeated values and gaps visible while retaining every observed value. A stemplot is also defensible, but a histogram or boxplot would not answer the exact-frequency question as directly.
Choose a Histogram When Bins Clarify a Larger Sample
Worked Example: Examining a Larger Set of Waiting Times
A fictional community center records how many minutes 48 visitors wait for assistance. To plan staffing, the center wants to know whether waits are concentrated in one range or spread across several ranges. A histogram groups the observations into these intervals:
| Waiting time in minutes | Number of visitors |
|---|---|
| 0 to less than 2 | 3 |
| 2 to less than 4 | 8 |
| 4 to less than 6 | 14 |
| 6 to less than 8 | 11 |
| 8 to less than 10 | 7 |
| 10 to less than 12 | 5 |
Plan. The question is about the pattern across ranges, rather than the exact waiting time of a particular visitor. Check that the proposed bins account for the full sample, then consider which display makes the range pattern easiest to see.
Do. The bin counts total \(3+8+14+11+7+5=48\), matching the number of visitors. A histogram using these consecutive intervals would show a concentration around 4 to less than 8 minutes, where the counts are largest, and fewer visitors in the intervals toward the low and high ends. The histogram makes that overall pattern visible without requiring the reader to inspect 48 separate observations.
A dotplot could show all 48 values, but if many visitors have distinct waiting times, the display may look crowded. A stemplot would also require arranging many leaves and could become difficult to scan. A boxplot would offer a compact summary of center and spread, but it would not show the bin-by-bin concentration that the staffing question asks about.
The interval choices still matter. The earlier tutorial on choosing class width and number of bins explains that narrower or wider bins can change the visible detail. Here, the chosen bins are equal in width and cover the stated range. The graph should label the horizontal axis in minutes and the vertical axis as a count of visitors, so readers can interpret the bars correctly.
Conclude. A histogram is the best first choice for showing where the 48 waiting times are concentrated. It summarizes the pattern across ranges clearly, while the bin counts and scale make the display’s level of detail explicit.
Choose a Boxplot When Compact Group Comparisons Matter
Worked Example: Comparing Delivery Times Across Routes
A fictional courier service records delivery times for 40 packages on each of three routes. The question is, “Which route tends to have longer delivery times, and which route has more spread?” A set of side-by-side boxplots is proposed. The summaries shown are:
| Route | Minimum | \(Q_1\) | Median | \(Q_3\) | Maximum |
|---|---|---|---|---|---|
| North | 18 | 24 | 29 | 35 | 48 |
| Central | 20 | 28 | 34 | 39 | 46 |
| South | 15 | 22 | 27 | 38 | 61 |
Plan. The question asks for comparisons of typical values and spread across three groups, not for the exact time of each package or the detailed shape within a route. A boxplot is a compact way to compare those features on the same time scale.
Do. The median delivery time is largest for the Central route at 34 minutes, compared with 29 minutes for North and 27 minutes for South. The IQRs, found by subtracting \(Q_1\) from \(Q_3\), are \(35-24=11\) minutes for North, \(39-28=11\) minutes for Central, and \(38-22=16\) minutes for South. Thus South has the largest IQR, indicating more spread in the middle half of its delivery times. Its minimum-to-maximum span is also widest, from 15 to 61 minutes.
Side-by-side boxplots make these comparisons quick because the medians and box lengths can be compared directly. But they do not establish whether a route’s times form one cluster or several, or where a gap might occur. If the service also needs to investigate those details, it should look at histograms or dotplots for the route data. A boxplot is well suited to the stated comparison, not every possible question about the distributions.
Conclude. The boxplots are appropriate for comparing typical delivery times and spread across the three routes. The Central route has the highest median time, and the South route has the largest IQR. Those summaries do not, by themselves, describe each route’s detailed shape.
Use More Than One Display When the Question Has Two Parts
Sometimes a task asks about features that no single display shows well. For example, a boxplot may make group medians and IQRs easy to compare, while a histogram or dotplot may reveal whether one group has a gap or two clusters. In that case, using complementary displays is not redundant: each graph answers a different part of the question.
For group comparisons, keep the scales consistent. Boxplots on different horizontal scales can make apparent distances misleading, and histograms with different bin boundaries can make patterns harder to compare. Use a common scale when the purpose is to compare values. As in the earlier tutorial on matching histograms to boxplots, check that displays refer to the same quantitative variable and units before drawing conclusions.
A graph should also fit the precision and range of the measurements. A stemplot might work neatly for integer scores but become cumbersome for values recorded to hundredths. A dotplot may work well for a small sample even when the values are decimals, provided the axis is clear. A histogram can summarize such measurements in intervals, but its appearance depends partly on the bin choices. Make the display’s scale and units clear so the reader knows what is represented.
Common Mistakes and AP Exam Tips
- Choosing only by sample size. “Small means dotplot” and “large means histogram” are useful starting points, not fixed rules. Consider whether values repeat, whether exact values matter, and what the question asks.
- Using a boxplot to describe clusters or gaps. A boxplot summarizes selected positions and spread. It may hide modality and gaps, as the earlier tutorial What a Boxplot Cannot Show explains. Use a histogram or dotplot when those features matter.
- Using a histogram when exact observations are needed. A bar represents an interval, not one exact value. If a question asks which exact value is most common, a dotplot or an appropriate stemplot is usually clearer.
- Assuming a stemplot is always more informative. Stemplots preserve individual values, but a large or awkwardly scaled set can make them hard to read. Choose a display that makes the evidence accessible, not merely one that contains the most detail.
- Comparing groups on incompatible scales. Use common axes and units when comparing center or spread. Apparent lengths are not meaningful comparisons if the scales differ.
- Claiming that one graph answers every question. A strong response connects the display to the task. For example: “A boxplot is appropriate for comparing the groups’ medians and IQRs, but a histogram would be needed to examine clusters or gaps.”
For full-credit communication, name the display and explain why it fits the variable, sample, and purpose. Mention an important limitation if it affects the question. A recommendation supported by a specific feature—such as preserving individual values or comparing medians—is stronger than simply calling a graph “clearer.”
Check Your Understanding
For each situation, choose a display and explain what information it makes easiest to see.
- A researcher has 16 whole-number ratings and wants to identify every rating tied for most common. Which display would you choose, and why?
- A school has several hundred measured commute times and wants to examine the overall shape and possible clusters. Which display is a strong first choice? Name one limitation.
- A report compares the medians and IQRs of four groups. Which display is especially useful, and what feature might it hide?
- Why might a stemplot be convenient for whole-number test scores but awkward for measurements recorded to hundredths?
- A histogram and boxplot are both available, but the question asks whether observations form two clusters. Which display gives more direct evidence, and why?