Read Each Stack as a Count
In Building a Dotplot Step by Step, you learned that each dot represents one observation and that equal values are stacked vertically. Now use those stacks to answer questions about the data: how often a value occurs, which value occurs most often, and how many observations are above a stated cutoff.
The number of dots stacked over a value is its frequency, or count. For example, if a stack above 12 minutes contains four dots, then four observations have a value of 12 minutes. Count every dot in the stack, not just the top dot or the number of spaces between dots.
A dotplot displays individual observations, while a frequency table organizes their counts. You can move between the two: count the dots above each value to fill in a frequency table, or use a table to check that you have read each stack accurately. As in the earlier tutorial Reading a Bar Chart of Counts, the tallest display corresponds to the greatest count. In a dotplot, however, the relevant height is the number of dots in a stack over a numerical value.
A question about a cutoff asks you to combine frequencies from the values that meet the stated rule. Pay close attention to the wording. “Above 20” means greater than 20, so a stack at exactly 20 is not included. “At least 20” means 20 or greater, so it does include the stack at 20. “Below 20” excludes 20, while “20 or below” includes it.
Worked Example: Find a Frequency and the Most Common Value
Worked Example: Find a Frequency and the Most Common Value
A fictional group of hikers records the number of minutes each person spent stretching before a walk. The dotplot below represents 16 people. Each column is a stack above the value named at the bottom; the dots are shown horizontally here to make the counts easy to inspect.
| Stretching time (minutes) | 10 | 15 | 20 | 25 | 30 |
|---|---|---|---|---|---|
| Dots in stack | ●● | ●●●● | ●●●●●● | ●●● | ● |
| Frequency | 2 | 4 | 6 | 3 | 1 |
Question 1: How many hikers stretched for 15 minutes? Count the dots in the stack over 15. There are four dots, so the frequency of 15 minutes is 4. In context, four of the hikers in this group stretched for 15 minutes.
Question 2: What was the most common stretching time? Compare the stack heights. The greatest frequency is 6, and it belongs to 20 minutes. Therefore, the most common stretching time was 20 minutes. The mode is a value, 20 minutes—not the frequency, 6.
Check the total. Add the frequencies: \(2+4+6+3+1=16\). This matches the 16 hikers stated in the example, supporting the reading of the stacks. If the sum did not match, we would check whether a dot had been overlooked or counted twice.
Count Across a Cutoff Carefully
To find how many observations are above a cutoff, first locate that cutoff on the horizontal axis. Then include only the stacks at values that satisfy the wording. When a cutoff is itself one of the plotted values, decide whether that stack is included by checking for words such as “greater than,” “above,” “at least,” or “or more.”
For example, if the plotted values are 5, 10, 15, 20, and 25, the values above 15 are 20 and 25. The value 15 is not above 15. In contrast, the values at least 15 are 15, 20, and 25. Once you identify the correct stacks, add their frequencies to get the requested count.
Worked Example: Count Observations Above a Cutoff
Worked Example: Count Observations Above a Cutoff
A fictional school club records how many minutes its members travel to a meeting. The dotplot represents 16 members. How many have a travel time above 20 minutes? How many have a travel time of at least 20 minutes?
| Travel time (minutes) | 5 | 10 | 15 | 20 | 25 | 30 |
|---|---|---|---|---|---|---|
| Dots in stack | ● | ●● | ●●●● | ●●●●● | ●●● | ● |
| Frequency | 1 | 2 | 4 | 5 | 3 | 1 |
Above 20 minutes: “Above” means strictly greater than 20, so do not include the five members at exactly 20 minutes. The values above 20 are 25 and 30 minutes. Add those stack counts: \(3+1=4\). Therefore, 4 members have travel times above 20 minutes.
At least 20 minutes: “At least” includes 20 itself. Add the frequencies at 20, 25, and 30 minutes: \(5+3+1=9\). Therefore, 9 members have travel times of at least 20 minutes.
Check both answers. The count above 20, which is 4, is smaller than the count at least 20, which is 9, because the second count includes the five members at exactly 20. The total frequency is \(1+2+4+5+3+1=16\), as expected. Neither answer is larger than the group size.
Worked Example: Recognize Tied Modes and Count Above a Value
Worked Example: Recognize Tied Modes and Count Above a Value
A fictional community garden records the number of ripe tomatoes picked from each of 17 plants during one morning. The dotplot is summarized below. Identify the most common number of tomatoes per plant, and find how many plants had more than two ripe tomatoes.
| Tomatoes per plant | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Dots in stack | ● | ●●● | ●●●●● | ●●●●● | ●● | ● |
| Frequency | 1 | 3 | 5 | 5 | 2 | 1 |
Find the most common value. The largest stack height is 5. There are two stacks with that height: the stack at 2 tomatoes and the stack at 3 tomatoes. Thus, the data have two modes, 2 and 3 tomatoes per plant. Reporting only one of these values would leave out a tie.
Count plants with more than two tomatoes. More than two excludes the stack at 2. Include the stacks at 3, 4, and 5 tomatoes, and add their frequencies: \(5+2+1=8\). So 8 plants had more than two ripe tomatoes.
Check the count. There are 17 plants altogether because \(1+3+5+5+2+1=17\). The eight plants with more than two tomatoes are fewer than the total of 17, and the stack at exactly 2 is not part of that count. If the question had asked for at least two tomatoes, the count would instead include the stack at 2: \(5+5+2+1=13\).
Common Mistakes and Full-Credit Communication
- Confusing the mode with its frequency. The mode is the value with the largest count. If 20 minutes has a stack of six dots, the mode is 20 minutes and its frequency is 6.
- Reporting just one mode when there is a tie. Compare all stack heights. If two or more values share the greatest frequency, name every value tied for the greatest frequency.
- Including the cutoff when the question says “above.” Above a cutoff means greater than it. Do not include observations equal to the cutoff. Include that stack only when the wording calls for it, as in “at least” or “or more.”
- Counting the wrong stacks for a range or cutoff. Locate the cutoff on the numerical axis before adding. State which values you included so your reasoning can be checked.
- Mixing up counts and percentages. If asked “how many,” report a count of observations. Do not turn it into a percentage unless the question asks for a proportion or percent.
- Failing to check the total. Add all stack frequencies and compare the sum with the number of observations represented. A mismatch is a cue to recount.
A full-credit response names the requested value or count and makes clear how the stacks were used. For example: “The mode is 20 minutes because its stack has six dots, the greatest frequency. Four members traveled more than 20 minutes because the stacks at 25 and 30 minutes contain three and one dots, and \(3+1=4\). The 20-minute stack is excluded because the question says ‘more than 20.’”
Check Your Understanding
Use the stated stack counts to answer each question. Show the addition when combining frequencies.
- A dotplot has frequencies 2, 5, 3, and 1 at values 4, 6, 8, and 10, respectively. What is the frequency of 6? What is the mode?
- A dotplot shows counts of 1, 4, 4, and 2 at values 0, 1, 2, and 3. Identify every mode.
- A dotplot has frequencies 2, 3, 5, and 1 at values 10, 15, 20, and 25. How many observations are above 15?
- Using the same frequencies, how many observations are at least 15? Explain why the answer differs from the answer to question 3.
- All the stack counts in a dotplot add to 23, but the context says the plot represents 24 people. What should you check before answering questions about the plot?