From Sleep-Hour Values to a Dotplot
A list of measurements contains the value for each individual, but repeated values can be hard to compare by scanning the list. A dotplot organizes those observations along a numerical scale. Each observation is represented by one dot, and dots for the same value are stacked above one another.
In Why Histograms and Bar Charts Are Different, you learned that a quantitative variable calls for a display that respects numerical values. A dotplot is another way to display a quantitative distribution, especially when the data set is small enough for individual observations to remain visible. Unlike a histogram, a dotplot does not group values into intervals.
This tutorial builds a dotplot from 20 sleep-hour values, from the first scale decision to the final check. The central rule is simple: each observation must appear exactly once. Repeated values are not omitted or placed beside one another; they are stacked in a column over their shared value.
A Step-by-Step Construction Routine
Begin by identifying what one observation represents and what units were recorded. Here, each observation is one person’s reported hours of sleep. The horizontal axis should therefore show sleep duration in hours. The dotplot needs a title or a clear label that tells a reader what group the data describe.
Find the smallest and largest observations, note the recorded precision, and check that there are 20 values. Do not round measurements unless the data were recorded that way.
Use evenly spaced numerical marks that cover the values. Mark values at the precision used in the data, and label the axis with the variable and units.
Count how many observations equal each value. Put that many dots directly above the value, stacking from the axis upward.
Make sure every value has the right stack, the stacks are above the correct scale marks, and the total number of dots is 20.
The scale should make the values easy to locate without suggesting more precision than the data contain. If measurements are whole hours, whole-hour marks are usually appropriate. If values include half hours, the scale must allow readers to locate half-hour values. The axis should extend far enough to include the minimum and maximum, but it does not have to begin at zero: a dotplot places observations at numerical positions rather than using bar lengths measured from a baseline.
For a beginner’s hand-drawn plot, evenly spaced tick marks and legible stacks matter more than decorative detail. Leave enough horizontal space to distinguish neighboring values and enough vertical space for the largest stack. A dot should be centered over its value, rather than shifted left or right to make the drawing look more balanced.
Worked Example: Build a Dotplot for 20 Sleep-Hour Values
Worked Example: Build a Dotplot for 20 Sleep-Hour Values
Suppose a fictional class records the number of hours each of 20 students slept the previous night. The observations, in the order recorded, are:
\(7,\ 8,\ 6,\ 7,\ 5,\ 8,\ 7,\ 6,\ 9,\ 7,\ 8,\ 6,\ 7,\ 5,\ 8,\ 7,\ 6,\ 8,\ 7,\ 9\).
Inspect the values. There are 20 observations. The smallest value is 5 hours and the largest is 9 hours. Each value is a whole number of hours, so a scale marked in one-hour steps from 5 through 9 is suitable.
Count each value. The frequency table organizes the number of observations at each scale position. As in the earlier tutorial Frequency Tables and Relative Frequency, check that the counts add to the number of observations.
| Sleep duration (hours) | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|
| Number of students | 2 | 4 | 7 | 5 | 2 |
The count check is \(2+4+7+5+2=20\), matching the 20 students. Now put one dot in a value’s column for each student with that sleep duration. In the table below, the highest stack is shown first and the dots sit above the bottom axis marks.
| Stack level | 5 hours | 6 hours | 7 hours | 8 hours | 9 hours |
|---|---|---|---|---|---|
| 7 | ● | ||||
| 6 | ● | ||||
| 5 | ● | ● | |||
| 4 | ● | ● | ● | ||
| 3 | ● | ● | ● | ||
| 2 | ● | ● | ● | ● | ● |
| 1 | ● | ● | ● | ● | ● |
| Sleep duration (hours) | 5 | 6 | 7 | 8 | 9 |
The first column has two dots, matching the two students who slept 5 hours. The stack above 7 hours has seven dots, the greatest frequency in this data set. Across all five stacks there are \(2+4+7+5+2=20\) dots. The table’s horizontal values and axis label make clear that the display is about sleep duration, measured in hours. On paper, draw a horizontal numerical axis from 5 to 9 and place the same stacks above the matching tick marks.
Finish with a check. Compare every stack to the frequency table and confirm that no observation has been added twice or left out. The completed dotplot describes these 20 students; it does not, by itself, establish what all students typically sleep.
Scale Choices When Values Include Half Hours
The scale depends on the way measurements were recorded. If the values include half hours, a plot with only whole-hour marks can make the data difficult to place accurately. Do not squeeze a value such as 6.5 between marks without showing how that position should be read. Include half-hour marks, or otherwise make the numerical spacing explicit.
Worked Example: Place Half-Hour Values on the Scale
Worked Example: Place Half-Hour Values on the Scale
A fictional group of 10 students records sleep duration to the nearest half hour. The values are \(6,\ 6.5,\ 7,\ 7.5,\ 7.5,\ 8,\ 8,\ 8,\ 8.5,\ 9\) hours. Explain how to set up the scale and show how many dots belong at each value.
The data are recorded in half-hour increments, so use evenly spaced scale marks at 6, 6.5, 7, 7.5, 8, 8.5, and 9 hours. Label the axis “Sleep duration (hours).” Counting the observations gives one at 6, one at 6.5, one at 7, two at 7.5, three at 8, one at 8.5, and one at 9.
A dotplot would have one dot over each of 6, 6.5, 7, 8.5, and 9; two vertically stacked dots over 7.5; and three vertically stacked dots over 8. The count check is \(1+1+1+2+3+1+1=10\). Because each dot represents one student, the stack of three over 8 hours means that three students in this group reported 8 hours.
A whole-hour-only scale would not show the 6.5-, 7.5-, and 8.5-hour positions clearly. Rounding those values to whole hours would change the data rather than display them. The appropriate solution is to keep the recorded values and provide a scale that can represent them.
Worked Example: Find and Fix Dotplot Construction Errors
Worked Example: Find and Fix Dotplot Construction Errors
A student makes a plot from the 20 sleep-hour values in the first example. The student draws axis marks from 5 through 9 hours, but places all seven observations of 7 hours in a horizontal row beside the 7-hour mark. The student also forgets to label the axis units. What needs to be fixed?
First, the repeated 7-hour observations should be stacked vertically above the 7-hour position. Seven side-by-side marks are not the conventional dotplot stack and can make the display difficult to compare with the other values. The corrected column above 7 should contain seven dots, one above another.
Second, label the horizontal axis “Sleep duration (hours).” A reader should not have to infer whether the values are hours, minutes, or another measurement. The plotted numbers alone do not provide that information.
Finally, the student should count the dots in every stack and check that their total is 20. In this data set the correct stack sizes are 2, 4, 7, 5, and 2 at 5, 6, 7, 8, and 9 hours, respectively. Their sum is 20, so the corrected display represents every observation exactly once.
Common Mistakes and Full-Credit Communication
- Using one dot for a distinct value instead of for each observation. A value that occurs seven times needs seven dots, not one. State or show the number of observations represented by each stack.
- Putting repeated values side by side. Stack equal values vertically above their shared numerical position. This makes frequencies visible as stack heights.
- Choosing a scale that skips data values. Check the minimum, maximum, and recorded precision before drawing. If observations are recorded to half hours, the scale needs to show half-hour positions.
- Forgetting the variable or units. Label the axis with both what is measured and its units, such as “Sleep duration (hours).” Add a title or context identifying the group if it is not already clear.
- Assuming the scale must start at zero. A dotplot’s points are positioned on a numerical axis; the axis can begin near the smallest observation as long as it covers the data and has clear, evenly spaced marks.
- Skipping the total-dot check. Add the stack counts and compare the sum with the number of observations. A mismatch signals that a value may have been missed or counted twice.
For full-credit communication, identify the data and units, describe the scale, and show one dot per observation with repeated values stacked. A clear construction explanation could say: “The values range from 5 to 9 hours, so I used one-hour scale marks from 5 through 9. I stacked 2, 4, 7, 5, and 2 dots over those values, respectively, and checked that the plot contains 20 dots.”
Check Your Understanding
Answer each question by describing the construction choice or showing the count check.
- A set of 12 observations has values 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, and 8. How many dots should be stacked above each value, and how many dots should the finished plot contain?
- A sleep-duration data set includes values of 6.5, 7, and 7.5 hours. What scale marks would help represent these values clearly? What should the axis label say?
- A student’s frequency counts for a 20-observation plot add to 19. What does this suggest the student should check?
- Why should repeated values be stacked vertically rather than represented by only one dot?
- A data set ranges from 5 to 9 hours. Must its dotplot’s horizontal scale start at zero? Explain what the scale does need to include.