Choosing a Display Starts With the Question
A quantitative data set can be displayed in several reasonable ways, but each display emphasizes different information. A graph that makes the overall shape easy to see may hide the individual observations. One that preserves every value may become crowded when the data set is large. The right choice depends on what you want a reader to learn.
Before choosing, identify the quantitative variable and its units, then ask two questions: Do I need to see the exact observed values, or mainly the overall pattern? And how many observations are there, and how many distinct values do they take? The answers help distinguish among dotplots, histograms, stemplots, and boxplots.
These displays are all for quantitative data, but they do not answer exactly the same questions. Dotplots and stemplots can show individual observations. Histograms emphasize how observations are distributed across a scale. Boxplots compress a distribution into a compact summary, which is particularly useful when comparing several groups.
What Each Display Makes Visible
| Display | Most useful when | What it emphasizes | Main limitation |
|---|---|---|---|
| Dotplot | The data set is small or moderate in size, especially when values repeat | Individual observations, repeated values, gaps, and clusters | Can become crowded with many observations or distinct values |
| Histogram | The data set is large or has many distinct values | Overall shape and how observations are distributed across intervals | Exact values are grouped rather than shown individually |
| Stemplot | The data set is small enough to display individual values legibly | Individual numerical values in an ordered display | Can be awkward for large data sets, decimals, or values with many digits |
| Boxplot | A compact summary is needed, especially to compare groups | Median, quartiles, and spread | Does not show individual values, clusters, or gaps clearly |
A discrete count, such as the number of calls received in an hour, takes separated countable values. A continuous measurement, such as time or mass, can in principle take any value within a range, even if it is recorded to a fixed precision. These distinctions guide the choice but do not dictate it. A discrete variable with many possible values can be displayed well in a histogram, while a modest set of measurements can be shown in a dotplot or stemplot.
Similarly, “small” and “large” are not fixed sample-size cutoffs. A display becomes unsuitable when its details are hard to read or when it fails to answer the question. A dotplot of 40 observations may still be clear; a stemplot of 40 values with many digits may not be. Think about the display’s readability and purpose rather than applying a rigid rule.
Worked Examples
Worked Example: A Small Set of Discrete Counts
Suppose a hypothetical clinic records the number of reminder calls made to each of 12 patients before an appointment. The counts are:
\(0, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 5\).
The variable is a discrete count, and there are only 12 observations. If the question is, “How many patients required each number of calls?”, a dotplot is a strong choice. Each dot can represent one patient, so repeated values are immediately visible: there are three patients at 1 call and four at 2 calls. A dotplot also retains each count, making values such as 0 and 5 easy to identify.
A stemplot could also retain these values, but it would provide little advantage for such a short list of small integers. A histogram is possible, but grouping the counts into intervals would discard detail that is easy to display directly. A boxplot would give a compact summary but would hide the frequencies at individual counts. Thus the dotplot best matches the question and the size of the data set.
Worked Example: A Large Set of Continuous Measurements
Imagine a hypothetical quality-control exercise recording the fill volume, in millilitres, of 240 bottles. The measurements range from 496.2 mL to 503.8 mL and include many distinct decimal values. The question is whether the volumes are concentrated near the target or spread broadly across the range.
A histogram is a suitable first choice. With 240 observations and many distinct measurements, plotting one dot for each bottle or listing the values in a stemplot could be crowded. The histogram groups measurements into intervals and makes the broad concentration and spread easier to see.
This choice fits the question, but it has a trade-off: the graph shows how many measurements fall in each interval, not the exact volume of every bottle. If a particular bottle’s recorded value matters, the raw data are needed. A boxplot would also offer a compact view of location and spread, but it would show less about the distribution’s detailed shape than a histogram.
Worked Example: Preserving Exact Values in a Small Measurement Set
Suppose a hypothetical group of 10 runners records the time, in minutes, required to complete a short course:
\(12.4, 12.8, 13.1, 13.1, 13.5, 13.7, 14.0, 14.2, 14.6, 15.0\).
If the purpose is to retain the exact recorded times while arranging them so that their order and repetition can be inspected, a stemplot is a reasonable choice. The set is small, and the values have a consistent precision to one decimal place, so the display can keep the observed values visible without a long list.
A dotplot would also preserve individual times and may be preferable if a visual number line is the main goal. A histogram is less attractive for this small set if the exact values matter, because it would group the times into intervals. A boxplot could summarize the center and spread, but it would not reveal which specific times were observed. The stemplot is not automatically best for every small data set; here it is useful because the values are compact and exact-value detail matters.
Worked Example: Comparing Several Groups With Boxplots
A hypothetical instructor records quiz scores, in points out of 20, for three sections with 80 students in each section. The question is whether the sections differ in typical score and spread, rather than which student earned each particular score.
Side-by-side boxplots are a practical choice. Each boxplot provides a compact summary of a section’s median, quartiles, and spread, allowing the three groups to be compared on the same score scale. With 80 scores per group, separate dotplots might be busy, especially when scores repeat heavily; a boxplot uses much less space.
The summary comes at a cost. Boxplots do not display every score and may conceal features such as multiple clusters or gaps between scores. If those features are central to the question, histograms or dotplots for the sections may be more informative. Here, the stated aim is a compact comparison of typical values and spread, so boxplots are well matched to the task.
Why Grouping Loses Detail
A histogram is useful precisely because it compresses data: it reports how many observations fall in each interval rather than listing all the observations. This makes a large data set easier to scan, but means different raw data sets can produce the same histogram. Once values are grouped, their exact positions within each interval cannot be recovered from the bar counts.
For example, consider the same two intervals, \([0,5)\) and \([5,10)\). The data sets below have two observations in each interval:
Both produce histogram counts of 2 in \([0,5)\) and 2 in \([5,10)\). Yet their observations are not the same. The bin counts preserve the number of values in each interval but not their exact locations inside those intervals. This is why a histogram is appropriate when grouped shape matters more than exact values, and a dotplot or stemplot is preferable when individual values need to remain visible.
A boxplot makes an even more compact summary. It retains selected information about center and spread but does not encode every value or every feature of shape. The display is not defective for omitting detail; that is the trade-off that makes it compact. Choose it when the summary is the point, not when the goal is to inspect all observations.
A Practical Selection Process
Decide whether the goal is to inspect exact observations, understand overall shape, or compare a compact summary. Keep the variable’s units clear.
For a small or moderate data set, a dotplot or stemplot can remain readable. For many observations or many distinct values, a histogram or boxplot may communicate more clearly.
Choose a dotplot or stemplot when individual values matter, a histogram when grouped shape matters, and a boxplot when a concise summary or group comparison matters.
Ask whether grouping obscures exact values, whether a compact summary hides clusters, or whether an individual-value display would be too crowded to read.
Common Mistake: Choosing by Variable Type Alone
A common mistake is to reason, “These are measurements, so I should use a histogram,” or “These are counts, so I should use a dotplot.” Variable type is relevant, but it does not settle the choice. A small set of measurements may be clearest in a dotplot or stemplot, and a count variable with many observations and many possible values may be easier to summarize in a histogram. The correction is to consider the number of observations, the number of distinct values, the display’s readability, and the question being asked.
Another mistake is to assume that a more compact graph is always better. A boxplot can compare several groups efficiently, but it should not be chosen if the important question concerns exact values, clusters, or gaps. Conversely, an individual-value display can preserve detail but may overwhelm a reader when the data set is large. The best display is the one that makes the relevant information visible at a useful level of detail.
Check Your Understanding
For each question, explain which display would be a reasonable choice and what information it would emphasize or omit.
- A data set contains 14 daily counts of emails received, with several repeated counts. You want to see which exact counts occurred and how often.
- A laboratory records 500 measurements of the same continuous quantity, and the goal is to inspect the distribution’s broad shape.
- You have 12 measurements recorded to one decimal place and want an ordered display that retains every value. Which displays could work?
- Three large groups are being compared on a quantitative outcome. The main interest is typical value and spread, not individual observations. What display is suitable, and what detail might it hide?
- Explain why two different sets of measurements can produce the same histogram.