Why Repeated Values Need Care
A dotplot places observations along a number line, with one dot for each observation. When several observations have the same value, drawing all their dots at the same horizontal position would hide some of them. Stacking the dots vertically makes the frequency at that value visible. A second challenge is choosing number-line ticks: ticks should make the scale readable without implying that observations occur only at labeled positions.
These choices matter even for a small data set. A plot with too few, irregular, or confusing tick marks can make values hard to locate. A plot that piles repeated dots on top of one another can make a group look like a single observation. The goal is to preserve the data faithfully while making the display easy to read.
Choosing Tick Spacing
Begin by identifying the smallest and largest observed values and the units in which the variable is measured. Then choose a tick interval that allows the full range to fit and makes positions straightforward to estimate. Tick intervals are usually regular, such as 1, 2, 5, or 10 units, or 0.1, 0.2, or 0.5 units when the measurements call for finer precision. These are useful possibilities, not a fixed rule: the appropriate interval depends on the data’s scale and range.
A tick interval should not be finer than the display can use clearly, nor so wide that readers struggle to locate the observations. If ticks are labeled every 5 units, for instance, an observation of 12 still belongs at 12, between the labeled ticks at 10 and 15. It should not be moved to 10 or 15. Tick marks are reference points, not bins and not instructions to round the data.
For a small discrete data set, it may be useful to label every possible value in the observed range, particularly when those values are consecutive integers. If the range is wider, labeling every second or fifth value can reduce clutter while leaving enough reference points to locate each dot. Whatever the interval, the spacing must represent equal numerical distances: equal steps of 2 units should occupy equal horizontal distances throughout the axis.
Worked Example: Stacking Repeated Pet Counts
Suppose a hypothetical survey records the number of pets in each of 12 households: 0, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5. Construct a dotplot and check that every household is represented.
Step 1: Count the observations at each value. There is one household with 0 pets, two with 1, three with 2, two with 3, three with 4, and one with 5. The count is zero for no values from 0 through 5 because every integer in that range appears.
Step 2: Choose the axis. The data are whole-number counts from 0 to 5, so labeling each integer gives a compact, clear axis. The tick spacing is 1 pet. Place every dot directly above its count, and put repeated dots in a vertical stack.
| Stack level | 0 pets | 1 pet | 2 pets | 3 pets | 4 pets | 5 pets |
|---|---|---|---|---|---|---|
| 3 | ● | ● | ||||
| 2 | ● | ● | ● | ● | ||
| 1 | ● | ● | ● | ● | ● | ● |
Each column represents one exact number of pets; each dot in that column represents one household. The table’s vertical levels show the stacks, with the lowest dot nearest the number line. In a drawn dotplot, the columns would be positioned above their corresponding values on a horizontal axis.
Step 3: Verify the display. The stack heights are 1, 2, 3, 2, 3, and 1. Their total is \(1+2+3+2+3+1=12\), matching the 12 recorded households. The largest stack is three dots, so no value occurs more than three times.
How Stacking Preserves the Counts
Stacking is not decoration: it is a simple way to show both the location and the frequency of observations. If a value occurs \(f\) times, its column must contain exactly \(f\) dots. Its height makes the count visible, while the horizontal position identifies the value. The dots in a stack need not represent different amounts of the variable; they represent separate observations sharing the same amount.
The reason stacking works can be stated directly. Suppose the distinct observed values are \(v_1,\ldots,v_k\), and let \(f_j\) be the number of observations equal to \(v_j\). Every observation equals exactly one of these distinct values, so grouping observations by value partitions the data set into non-overlapping groups. Therefore, the total number of observations equals the sum of the group counts:
A dotplot that places \(f_j\) dots above each \(v_j\) consequently contains \(n\) dots altogether. This gives a useful check: count the dots in every stack and add the stack heights. If the total differs from the number of observations, a dot has been missed, duplicated, or assigned to the wrong value.
Worked Example: Labeling Every Second Value
Suppose a hypothetical data set records the number of minutes needed for 18 short tasks: 18, 18, 19, 19, 19, 20, 20, 21, 21, 21, 22, 23, 23, 24, 24, 24, 25, 26. The values are close together, and there are several repeats. Choose tick spacing and describe where the dots go.
Step 1: Find the range. The smallest observation is 18 minutes and the largest is 26 minutes, so the range is \(26-18=8\) minutes. Since observations are recorded as whole minutes, each dot must be placed at an integer minute.
Step 2: Select readable ticks. Labeling every minute would work, but ticks every 2 minutes also provide a clear reference across this short range: 18, 20, 22, 24, and 26. The interval between labeled ticks is 2 minutes. Values 19, 21, 23, and 25 lie halfway between neighboring ticks, so their dots go at those halfway positions, not at a labeled tick.
Step 3: Count and stack. The frequencies from 18 through 26 are, respectively, 2, 3, 2, 3, 1, 2, 3, 1, and 1. For example, stack three dots at 19, three at 21, and three at 24. Put one dot at 22, 25, and 26. Every dot remains at its exact minute value even though not every value is labeled on the axis.
Step 4: Check the count. Adding the stack heights gives \(2+3+2+3+1+2+3+1+1=18\), the number of tasks recorded. The even tick spacing helps readers locate observations, while stacking prevents repeated times from being hidden.
When Observations Are Close Together
Repeated values and close-but-different values require different decisions. Repeated values belong at the same horizontal coordinate and must be stacked. Close-but-different values belong at different coordinates, even if the points seem crowded. The horizontal scale should make those differences visible as far as the available display allows; it should not erase them by rounding.
For decimal measurements, choose an axis interval that suits the precision of the recorded data. If measurements are recorded to the nearest tenth, for example, a dot can be placed at 4.3 even if the axis labels appear only at 4.0, 4.2, and 4.4. The position at 4.3 is halfway between 4.2 and 4.4. Always distinguish the precision of the observations from the frequency of the labels.
If the horizontal space is limited, you can reduce the number of printed labels while keeping the numerical spacing regular. You can also extend the axis slightly beyond the smallest and largest observations so end dots do not sit against the edge. Neither choice changes the observations or the distances represented on the scale. If dots still seem too crowded, make the plotted range and scale clearer rather than shifting points or merging values.
Worked Example: Keeping Tenths Distinct
Suppose a hypothetical set of nine measurements, recorded to the nearest tenth of a unit, is 4.1, 4.2, 4.2, 4.3, 4.3, 4.3, 4.4, 4.5, 4.5. Choose useful axis labels and explain how the repeated and nearby values appear.
Step 1: Use the recorded precision. The smallest measurement is 4.1 and the largest is 4.5. Labels at 4.0, 4.2, 4.4, and 4.6 give a regular interval of 0.2 units and leave a little room at both ends. Measurements at 4.1, 4.3, and 4.5 fall halfway between neighboring labeled ticks.
Step 2: Place dots at exact recorded values. Put one dot at 4.1, two stacked dots at 4.2, three stacked dots at 4.3, one dot at 4.4, and two stacked dots at 4.5. In particular, the three measurements of 4.3 share a horizontal position and form a stack; the measurement at 4.4 is a separate dot to their right.
Step 3: Verify the result. The stack heights total \(1+2+3+1+2=9\), matching the nine measurements. Every dot is located at a value present in the data, and the labeled ticks are equally spaced. The plot preserves both the repeated 4.3 measurements and the distinctions between neighboring tenths.
Common Mistakes and a Reliable Check
A frequent mistake is to treat tick marks as the only allowable dot positions. In the previous example, that reasoning would leave nowhere to put 4.1, 4.3, or 4.5, even though all three values were observed. The correction is to place the dots at their actual values between ticks; ticks provide a scale, not a list of permitted data values.
Another mistake is to draw several repeated observations on top of one another. They then look like only one dot, so the graph understates the frequency. Stack repeated observations vertically and count each dot separately. Conversely, do not stack different values merely because they are close: a dot at 4.2 and a dot at 4.3 have different horizontal positions.
Before considering a dotplot finished, check the following:
- The number line covers the full observed range and has regular, clearly interpreted tick spacing.
- Every dot is positioned at its observation’s value, including values between labeled ticks.
- Each repeated value has one vertical stack with the correct number of dots.
- The total number of dots equals the number of observations in the data set.
- Units are stated, and no values have been rounded, omitted, or merged just to make the plot look simpler.
A well-chosen axis and accurate stacks make a dotplot useful both for reading individual values and for seeing which values occur most often. These construction choices support the broader purpose of quantitative graphs: to show the data’s pattern without losing the observations themselves.
Check Your Understanding
Use the ideas in this tutorial to answer the following questions.
- A data set contains the values 2, 2, 3, 5, 5, 5. How many dots should be stacked above each observed value, and how many dots should the plot contain altogether?
- Observations are whole numbers from 12 to 20. If the axis is labeled every 4 units, where should a dot for 14 be placed?
- Why should two observations at 7.2 be stacked, while observations at 7.2 and 7.3 should be placed at different horizontal positions?
- A dotplot’s stack heights add to 15, but the data set contains 16 observations. Give two possible construction errors that could explain the mismatch.
- A measurement data set is recorded to the nearest tenth. Explain why labels spaced 0.2 units apart do not require rounding observations to the nearest 0.2 units.