One Data Set, Different Views
A graph is not a neutral copy of a data set. Each graph organizes the observations in a different way, so it makes some features easy to see while making other details harder—or impossible—to recover. A dotplot and a stemplot retain the individual measurements. A histogram groups measurements into intervals. A boxplot compresses the distribution into a few landmarks.
Comparing graphs made from the same observations makes these trade-offs concrete. We will use one hypothetical data set throughout: the completion times, in minutes, for 16 people on a practice task. The ordered measurements are:
12, 14, 15, 17, 18, 19, 20, 21, 22, 22, 23, 24, 25, 27, 31, 45
The set includes a repeated value, a relatively separated high value, and observations in between. These features help show what is retained or hidden when the display changes. The examples use the same measurements throughout, so differences among the displays come from the graph type, not from different data.
A Dotplot Keeps the Observations Visible
In a dotplot, observations at the same value are stacked. The horizontal position communicates the value, and the number of dots at that position communicates its frequency. In our data, 22 occurs twice; each of the other observed values occurs once.
Worked Example: Counting the Observations in a Dotplot
Here is a frequency-style rendering of the dotplot. Each bullet represents one dot at the listed value; the values are measured in minutes.
| Value | Dots above the value |
|---|---|
| 12 | • |
| 14 | • |
| 15 | • |
| 17 | • |
| 18 | • |
| 19 | • |
| 20 | • |
| 21 | • |
| 22 | •• |
| 23 | • |
| 24 | • |
| 25 | • |
| 27 | • |
| 31 | • |
| 45 | • |
There are 15 distinct values shown, and the stack at 22 contains two dots. Thus the total number of observations is
This count agrees with the stated sample size, \(n=16\). A dotplot preserves every observed value and makes repetitions visible. It also shows the empty stretches between observed values, including the wide separation between 31 and 45. If many observations crowd together, however, overlapping or tightly packed dots can be difficult to read; spacing or stacking conventions help manage that problem.
A Histogram Groups Values Into Bins
A histogram replaces the exact positions of individual observations with bins. The bin boundaries matter: each observation must be assigned to exactly one bin. For this example, use the equal-width intervals \([10,20)\), \([20,30)\), \([30,40)\), and \([40,50)\). The bracket at the left means the left endpoint is included; the parenthesis at the right means the right endpoint is excluded. For example, a value of 20 belongs to \([20,30)\), not \([10,20)\).
Worked Example: Building a Histogram From the Shared Data
Count the measurements in each bin. The first bin contains 12, 14, 15, 17, 18, and 19, giving a frequency of 6. The second contains 20, 21, 22, 22, 23, 24, 25, and 27, giving a frequency of 8. The last two bins contain 31 and 45, respectively.
| Bin, in minutes | Frequency |
|---|---|
| [10, 20) | 6 |
| [20, 30) | 8 |
| [30, 40) | 1 |
| [40, 50) | 1 |
The histogram would have four adjoining bars with these heights. As a check, the bin frequencies must account for every measurement:
The histogram makes the broad distribution of observations across intervals easy to compare. But it does not show which eight values fall in \([20,30)\), or that 22 occurs twice. A different set of eight observations in that bin would produce the same bar height. The bins also affect what pattern is visible: combining or subdividing intervals can change the display’s apparent detail. Here, the bins conceal the exact gap between 31 and 45 because the two observations appear in separate neighboring bars, even though there are no observations in the intervals between those specific values.
A Stemplot Retains Values in Order
A stemplot divides each measurement into a stem and a leaf. For these two-digit integer times, the tens digit is the stem and the ones digit is the leaf. Leaves are listed in increasing order within each stem; repeated observations appear as repeated leaves.
Worked Example: Making a Stemplot Without Dropping Values
The values in the teens give stem 1 with leaves 2, 4, 5, 7, 8, and 9. The values from 20 through 29 give stem 2 with leaves 0, 1, 2, 2, 3, 4, 5, and 7. The final two values give stems 3 and 4.
Key: \(1|2\) represents 12 minutes. The leaves number \(6+8+1+1=16\), and the repeated leaf 2 on stem 2 records both observations of 22 minutes. The stemplot retains the exact data values, like the dotplot, but arranges them by place value. It can be easier than a dotplot to read when there are many distinct values close together. Its usefulness depends on the measurements having a sensible stem-and-leaf split; with very many digits or highly precise measurements, the display may become awkward or require a stated rounding rule.
A Boxplot Compresses the Distribution
A boxplot trades individual values for a compact summary. Its box marks the first and third quartiles, and a line inside the box marks the median. A modified boxplot extends whiskers to the most extreme observations that are not beyond the usual \(1.5\,IQR\) fences; any observations beyond those fences are shown separately. A boxplot does not show how observations are distributed within the box or along a whisker.
Worked Example: Summarizing the Shared Data With a Modified Boxplot
For this example, use the median-of-halves convention for the quartiles. The overall median is the average of the two middle observations, 21 and 22:
The lower half is 12, 14, 15, 17, 18, 19, 20, 21, so its middle two values give \(Q_1=(17+18)/2=17.5\) minutes. The upper half is 22, 22, 23, 24, 25, 27, 31, 45, so \(Q_3=(24+25)/2=24.5\) minutes. Thus:
The lower and upper \(1.5\,IQR\) fences are \(17.5-1.5(7)=7\) minutes and \(24.5+1.5(7)=35\) minutes. The measurement 45 is beyond the upper fence, while 31 is not. The modified boxplot therefore has a box from 17.5 to 24.5 minutes, a median line at 21.5 minutes, whiskers reaching 12 and 31 minutes, and a separately marked point at 45 minutes.
The boxplot quickly displays the middle half and the locations of the median and marked high observation. It does not reveal that two observations equal 22, or show the exact values 14, 15, 17, and so on. The box and whiskers summarize ranges; they are not a picture of where every measurement lies.
What Each Display Preserves, Emphasizes, and Conceals
The displays answer different visual questions, even when they describe exactly the same measurements.
| Display | Preserves | Emphasizes | Conceals or limits |
|---|---|---|---|
| Dotplot | Each observed value and its frequency | Repeated values, gaps, and individual positions | Can become crowded when values are numerous or tightly packed |
| Histogram | Counts within the chosen bins | How observations are distributed across intervals | Exact values and within-bin arrangement; appearance depends on bin choices |
| Stemplot | Each value, in order, with repetitions | Individual values and their organization by place value | Can be cumbersome for dense or high-precision data |
| Boxplot | Selected summary landmarks and, in a modified plot, flagged points | Median, quartiles, spread of the middle half, and extremes of whiskers | Most details about individual values and distribution within the summary regions |
A useful way to compare displays is to trace what happens to each observation. In a dotplot or stemplot, each observation has a visible mark. In a histogram, each observation contributes to one bin count, but its precise value is lost. In a boxplot, observations contribute to summary quantities, and only a point beyond a fence may be drawn separately. This difference explains why two histograms can match despite different values within bins, and why distinct data sets can have the same boxplot summary.
A Short Check on Information Loss
For bins that cover the entire observed range without overlap, every observation contributes to exactly one bin count. If the bin counts are \(f_1,\ldots,f_k\), then their sum is the sample size:
This follows because the bins partition the observations: none is counted twice, and none is left out. The calculation checks whether the histogram frequencies are consistent with the data set, but it cannot recover the individual values. For example, knowing that eight values lie in \([20,30)\) does not tell us whether those values are 20, 21, 22, 22, 23, 24, 25, and 27, or a different collection in the same interval. Grouping preserves the count per bin, not the arrangement within it.
Choosing a Display for the Question
The best display depends on what a reader needs to learn. To inspect individual values or repeated measurements, a dotplot or stemplot is more informative than a histogram or boxplot. To summarize a distribution compactly, especially when comparing groups, a boxplot can be useful, but it should not be treated as a substitute for a display that reveals individual structure. A histogram is useful when the pattern across intervals is the main interest, provided the bin boundaries and widths are clear.
In practice, two displays can complement each other. A histogram can show the broad grouping while a dotplot or stemplot preserves specific observations. A boxplot can make a compact summary easy to compare while another graph provides detail that the boxplot omits. Graphs should be read as different views of the same data, not as competing claims about what the data are.
Common Mistake
A common mistake is to assume that displays that look similar preserve the same information. For example, someone might argue, “The histogram and boxplot both show where most observations are, so the exact measurements can be read from either one.” That is not correct. The histogram records counts by bin, and the boxplot records summary landmarks; neither gives every observation’s value. If exact values or repeated measurements matter, use a dotplot or stemplot when the data size and measurement scale make those displays readable.
Check Your Understanding
Use the distinctions in this tutorial to answer the following questions.
- In the shared data set, why does the dotplot have 15 distinct value positions but represent 16 observations?
- What information about the two observations equal to 22 is visible in the stemplot but not recoverable from the histogram bins used here?
- Explain why the four histogram frequencies must add to 16 when the bins partition all observations.
- Which display would you choose to retain every individual value in an ordered arrangement, and what is one limitation of that display?
- What information does the modified boxplot show about 45 minutes, and what individual-value detail does it omit?