Why the Value Scale Matters
A count such as the number of absences in a week can take only whole-number values: 0, 1, 2, and so on. A measurement such as body temperature can, in principle, take values between any two reported values. Both are quantitative variables, but their possible values are arranged differently. A useful graph should show that difference without making either variable seem more precise, more continuous, or more categorical than it is.
The distinction matters when choosing a display and setting its horizontal scale. For a small set of distinct count values, a dotplot can show the observations directly above the integer values. For a larger range, a histogram can group counts into intervals. Measurements are also commonly shown with histograms, but their bin boundaries divide a continuous scale rather than separating a list of possible integer values. In either case, the axis labels and bins should make clear what the numbers represent.
Displaying Integer-Valued Counts
For a count with a manageable range, a dotplot is often a direct choice. Place each observation above its value on a number line, stacking observations that have the same value. The horizontal axis should be labelled with the variable and its units or counting period, if relevant. The vertical stacking shows how many observations have each value. Because the values are numbers with an order and a meaningful distance, the count remains quantitative; it is not a set of unrelated categories.
A histogram is useful when a count has many possible values or the data set is large enough that individual dots are crowded. When the purpose is to show the frequency of each integer separately, use bins of width 1 centered on the integers. The boundaries fall halfway between adjacent integers. When a broader overview is more useful, combine neighboring integer values into wider intervals and label those groups explicitly.
Worked Example: Making a Dotplot of Weekly Absences
In a hypothetical class, the number of absences for 12 students during one week is recorded as:
0, 0, 0, 1, 1, 2, 2, 2, 3, 4, 4, 6.
The variable is a count of absences per student for the week. Tallying the observations gives three students with 0 absences, two with 1, three with 2, one with 3, two with 4, none with 5, and one with 6. The frequencies add to 12:
A dotplot places one dot above each student’s count. A compact tally of the plotted dots is:
| Absences per student | Dots to plot | Frequency |
|---|---|---|
| 0 | ••• | 3 |
| 1 | •• | 2 |
| 2 | ••• | 3 |
| 3 | • | 1 |
| 4 | •• | 2 |
| 5 | — | 0 |
| 6 | • | 1 |
On the finished dotplot, the horizontal axis would be labelled “Absences per student during the week” and marked at the integers from 0 through 6. Each dot represents one student; the stack above 2 has three dots. Showing 5 on the axis, even though it has no dot, makes the gap between the observed values 4 and 6 visible. It does not mean that an observation of 5 was recorded.
Bin Boundaries for Discrete Counts
A histogram groups values into intervals. With integer-valued counts, a clear way to place one value in each bin is to center a width-1 interval on each integer. For example, the bin for a count of 3 is \([2.5,3.5)\). It contains the integer 3 and no other integer count. Adjacent bins meet at a boundary, and the convention of including a left endpoint and excluding a right endpoint ensures that a value on a boundary belongs to just one bin.
This placement follows directly from the spacing between consecutive integers. For any integer \(k\), the only integer in the interval from \(k-0.5\) up to, but not including, \(k+0.5\) is \(k\). Thus each observed count goes into exactly one bin. If each observation is tallied once, the frequencies across all bins must add to the total number of observations:
where \(f_k\) is the frequency in the bin centered at integer \(k\), and the sum covers the count values included in the display. This is a useful check on the construction.
Worked Example: Grouping a Larger Range of Counts
In a hypothetical workplace survey, 16 employees report how many scheduled work sessions they missed in a month:
0, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 6, 6, 7, 8, 9.
A separate bin for every value may be less useful for a quick overview, so group the counts into 0–2, 3–5, 6–8, and 9–11 missed sessions. Each interval contains three possible integer counts. With boundaries halfway between the grouped values, the intervals are \([-0.5,2.5)\), \([2.5,5.5)\), \([5.5,8.5)\), and \([8.5,11.5)\). Tallying gives:
| Missed sessions | Counts included | Frequency |
|---|---|---|
| 0–2 | 0, 1, 2 | 6 |
| 3–5 | 3, 4, 5 | 5 |
| 6–8 | 6, 7, 8 | 4 |
| 9–11 | 9, 10, 11 | 1 |
The frequency check is \(6+5+4+1=16\), matching the 16 employees. A histogram can show these four frequencies with bars over the stated intervals. The horizontal axis should identify the variable and units, for example “Missed scheduled sessions in the month,” and the vertical axis should say “Number of employees.” The first mathematical boundary, \(-0.5\), is a bin edge used to place the zero-count group; it does not suggest that negative absences are possible.
Displaying Continuous Measurements
For a continuous measurement, bins represent intervals along the measurement scale. A temperature histogram might use intervals of width 0.5 degrees, for example. The value 36.4°C belongs in the interval from 36.0°C up to, but not including, 36.5°C, while 36.5°C belongs in the next interval. The values in each bin are not restricted to whole numbers or to the printed endpoints.
Use a horizontal axis labelled with the measurement and its units, and show the bin boundaries or interval labels clearly enough that a reader can tell which observations each bar represents. When bins have equal width, bar height can represent frequency. Adjacent histogram bars touch because the intervals are successive parts of a quantitative scale, not separate named categories. Touching bars do not imply that every possible measurement was observed.
Worked Example: Binning Body Temperatures
A hypothetical set of 10 body temperatures, recorded in degrees Celsius, is:
36.2, 36.4, 36.5, 36.7, 36.8, 37.0, 37.1, 37.3, 37.6, 37.8.
Use four equal-width bins: \([36.0,36.5)\), \([36.5,37.0)\), \([37.0,37.5)\), and \([37.5,38.0)\). The first interval contains 36.2 and 36.4, so its frequency is 2. The second contains 36.5, 36.7, and 36.8, so its frequency is 3. The third contains 37.0, 37.1, and 37.3, so its frequency is 3. The last contains 37.6 and 37.8, so its frequency is 2. The resulting frequencies check against the number of temperatures:
Draw four adjacent bars, each spanning 0.5°C, with heights 2, 3, 3, and 2 students, respectively. Label the horizontal axis “Body temperature (°C)” and the vertical axis “Number of observations.” The boundaries assign every recorded value once; they do not claim, for example, that no possible temperature lies between 36.4°C and 36.5°C.
Choosing and Labelling the Display
The essential choice is not “numbers versus non-numbers.” Both examples use numerical variables. Instead, ask what the values mean and how many distinct values are useful to display individually. A dotplot is often clear for a modest number of observations and a short range of discrete counts. A histogram is useful for summarizing a larger collection or a wider range, for either counts or measurements. For measurements, intervals are natural because values can occur throughout the scale; for counts, bins should respect the integer spacing and should be described in terms of the count values they group.
| Feature | Discrete count | Continuous measurement |
|---|---|---|
| Possible values | Whole-number counts, often starting at zero | Values can occur throughout a measurement range |
| Useful display for a small range | Dotplot with integer tick marks | Histogram with clearly stated intervals |
| Bin meaning | A single integer or a group of neighboring integers | An interval on the measurement scale |
| Axis label | Name the counted event and time period, if relevant | Name the measured quantity and units |
A histogram differs from a bar chart of categories. A bar chart compares categories such as transport mode or blood group; the categories are not positions along a continuous numerical scale, so gaps between bars help signal their separation. A histogram displays quantitative intervals in order. Integer counts do not become categorical merely because only whole numbers are possible. If you do choose to show individual counts with a frequency bar display, make clear that the horizontal positions are numerical count values; do not label them as unrelated categories.
Common Mistake: Treating Counts as Continuous Measurements
A common mistake is to use arbitrary decimal bins for a small integer count and then discuss the resulting graph as though fractional counts could occur. For instance, bins such as 0–0.4, 0.4–0.8, and 0.8–1.2 obscure the simple fact that the only possible counts in that region are 0 and 1. The correction is to choose a display that suits the integer scale: use a dotplot, make unit-width bins centered on the integers, or group whole-number counts into clearly labelled ranges.
The reverse mistake is to label continuous measurements only at the few values that happened to appear in the sample, as if those were the only possible values. Instead, label the measurement scale and its units, and state the intervals used. The graph then describes the observed data without making an unsupported claim about what the variable can take.
Check Your Understanding
Answer each question by referring to the value scale and the display choices in this tutorial.
- A small data set records the number of pets owned by each household. Why is the variable quantitative, and what display could show each observed value directly?
- For a histogram of counts centered on each integer, what are the boundaries of the bin centered at 7? Which integer count does it contain?
- A temperature histogram uses intervals \([18,20)\) and \([20,22)\). Which interval contains a temperature of 20°C, and why?
- Explain why adjacent bars in a histogram of measurements touch, while a bar chart of categories commonly has gaps.
- A grouped count histogram has frequencies 4, 8, 5, and 3. What total sample size should those frequencies represent, and what check would you make if they do not add to that total?