From a List of Measurements to a Picture
A list of measurements preserves the recorded values, but it can be difficult to see how often values occur or how they are distributed along the numerical scale. A dotplot turns each measurement into a mark on a number line. It is especially useful for a short data set because it shows both the individual values and how many times each value was recorded.
The construction must preserve two things at once: the location of every measurement and the total number of observations. If a value occurs more than once, its dots are stacked vertically at that value. Horizontal position represents the measurement; vertical stacking represents repetition, not a second measured variable.
A Reliable Construction Method
Start by checking that all observations use the same units and that each value is transcribed accurately. Sort the values from least to greatest. Sorting makes repeated values easier to count, but the dots must still be placed at the values on a number line, not at equally spaced list positions.
Write the measurements in increasing order and count how many times each distinct value occurs.
Set a numerical range that includes the smallest and largest observations. Select one constant tick interval that allows every recorded value to be located.
Write the variable and its units, and mark numerical values at equal physical intervals. Equal differences in measurement must correspond to equal distances on the page.
Place one dot above the appropriate value for each observation. Stack repeated observations vertically at precisely the same horizontal position.
Count all the dots and compare that total with the number of recorded observations. Confirm that each distinct value has the correct number of dots.
Tick marks need not always be placed at every possible measurement value. They do, however, need to establish a consistent scale, and every observation must be placed in its correct position relative to that scale. If measurements fall between labeled ticks, the axis can use a smaller tick interval or otherwise show those positions accurately. Do not round a value merely to make plotting easier.
Worked Example: Daily Step Counts
Suppose a hypothetical person records these step counts on seven days: 2,400; 3,100; 2,800; 2,400; 3,500; 3,100; and 2,600 steps. Construct a dotplot.
1. Sort and count. In increasing order, the values are 2,400, 2,400, 2,600, 2,800, 3,100, 3,100, and 3,500. The distinct values have frequencies 2, 1, 1, 2, and 1, respectively. The observations range from 2,400 to 3,500 steps.
2. Choose and label the axis. Use tick positions every 100 steps from 2,400 through 3,500. Because the difference between adjacent tick values is always 100, the columns in the table below represent equal numerical distances.
3. Place the dots. Each dot level is a row: the lower row holds the first dot at a value, and the rows above it hold additional observations with that same value. Blank cells contain no dots.
| Step count (steps) | 2,400 | 2,500 | 2,600 | 2,700 | 2,800 | 2,900 | 3,000 | 3,100 | 3,200 | 3,300 | 3,400 | 3,500 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Dot level 2 | • | • | ||||||||||
| Dot level 1 | • | • | ||||||||||
| Number line | 2,400 | 2,500 | 2,600 | 2,700 | 2,800 | 2,900 | 3,000 | 3,100 | 3,200 | 3,300 | 3,400 | 3,500 |
There are two dots at 2,400, one at 2,600, one at 2,800, two at 3,100, and one at 3,500. No observation was recorded at the other tick values. The total is \(2+1+1+2+1=7\) dots, matching the seven daily measurements. Notice that 2,800 is four 100-step intervals to the right of 2,400, while 3,100 is seven intervals to the right; this spacing preserves their actual numerical positions.
Why One Dot per Observation Matters
A dotplot is a direct encoding of the observations. Each recorded measurement contributes exactly one dot, so repeated values do not disappear: they create a vertical stack. This gives a useful check on the construction. If the data set contains \(n\) observations, the total number of dots must be \(n\).
To see why, let \(f_j\) be the number of observations equal to the \(j\)th distinct value. Each observation belongs to exactly one distinct-value group, so adding the group counts accounts for every observation once and only once. Thus the total number of dots is the sum of the frequencies, which equals the size of the data set:
This is more than a counting convenience. If the dot total is too small, an observation may have been omitted; if it is too large, a value may have been plotted more than once. Comparing each stack with the sorted list can also reveal an incorrect frequency even when the overall dot total happens to match.
Worked Example: Commute Times in Ten-Minute Steps
A hypothetical group records six commute times, in minutes: 20, 40, 20, 60, 30, and 40. Construct a dotplot using tick marks every 10 minutes.
Sort and count. The sorted list is 20, 20, 30, 40, 40, 60. There are two observations at 20 minutes, one at 30, two at 40, and one at 60. The range is 20 to 60 minutes.
Build the number line and plot. Mark 20, 30, 40, 50, and 60 at equal intervals. The 50-minute position is included to keep the scale continuous, although no commute had that recorded time.
| Commute time (minutes) | 20 | 30 | 40 | 50 | 60 |
|---|---|---|---|---|---|
| Dot level 2 | • | • | |||
| Dot level 1 | • | • | • | • | |
| Number line | 20 | 30 | 40 | 50 | 60 |
The columns are equally spaced because each represents a 10-minute increase. Counting the marks gives \(2+1+2+0+1=6\), the same as the six recorded commute times. The empty position at 50 minutes is part of the scale, not a missing measurement that should receive a dot.
Choosing a Scale for Decimal Measurements
The same construction works when measurements include decimals. The tick interval should be small enough to locate each recorded value, while still making the axis readable. For example, measurements recorded to the nearest quarter hour can be displayed with quarter-hour tick intervals. It is important to use the precision actually recorded: a value of 1.25 hours should not be silently changed to 1.3 hours.
Worked Example: Quarter-Hour Readings
Suppose seven hypothetical readings of a device's operating time, in hours, are 1.25, 1.50, 1.25, 1.75, 2.00, 1.50, and 2.25. Construct a dotplot with tick marks every 0.25 hour.
Sort and count. The ordered observations are 1.25, 1.25, 1.50, 1.50, 1.75, 2.00, and 2.25 hours. The frequencies are 2 at 1.25, 2 at 1.50, and 1 at each of 1.75, 2.00, and 2.25.
Plot on the labeled scale. The tick values from 1.25 through 2.25 increase by 0.25 hour each time, so every adjacent column represents the same difference.
| Operating time (hours) | 1.25 | 1.50 | 1.75 | 2.00 | 2.25 |
|---|---|---|---|---|---|
| Dot level 2 | • | • | |||
| Dot level 1 | • | • | • | • | • |
| Number line | 1.25 | 1.50 | 1.75 | 2.00 | 2.25 |
The stacks contain two dots at 1.25 and two at 1.50 hours, followed by one dot at each of 1.75, 2.00, and 2.25 hours. Their total is \(2+2+1+1+1=7\), matching the seven values in the original list. The units in the axis label make clear that the numbers are hours, not minutes or some other measurement.
Common Mistakes and Checks
A frequent mistake is to make the displayed columns equally spaced while using labels that have unequal numerical gaps. For example, if the axis displays 2,400, 2,800, and 3,100 at equal distances, it incorrectly makes the 400-step gap look the same as the 300-step gap. Use a constant tick interval, or clearly construct the axis so its positions preserve the numerical distances. A dotplot is a quantitative display, so its horizontal distances carry meaning.
Another mistake is to plot only one dot for each distinct value. If a sorted list contains a value three times, the correct stack contains three dots at that value. Conversely, do not add a dot at every tick just because a tick is drawn: a dot represents an observation, while a tick only marks the scale.
A final check is to compare the plot against the raw list, not merely to check that the total looks plausible. Confirm that every observation has a dot at its value, that every repeated value has the right stack height, that the axis includes all values, and that the variable and units are labeled. A correct dotplot is a faithful display of the measurements, not a smoothed or rounded version of them.
Check Your Understanding
Answer these questions by referring to the construction steps and examples.
- A list contains the measurements 4, 5, 5, 7, and 8. How many dots should the finished dotplot have, and how many should be stacked at 5?
- Why should equal physical distances on the number line represent equal numerical differences?
- A display has one dot for each distinct value, regardless of its frequency. What information has been lost?
- For measurements recorded in quarter-hour increments, give one suitable tick interval and explain why it works.
- A data set has eight observations, but the plot has seven dots. Name two possible construction errors to check.