One Data Set, Different Questions
A quantitative data set can be displayed in more than one useful way. A dotplot preserves each observation, a histogram groups observations into intervals, and a boxplot summarizes the distribution using selected positions. Mixed-practice questions ask you to choose among these displays, build one from data, or decide whether a claim matches what a display actually shows.
In Choosing the Best Display for Quantitative Data, you learned to select a graph according to the task. In Matching Histograms to Boxplots, you practiced comparing shape, center, and spread. Here, you will use those ideas together and add a consistency check: when several displays represent the same data, their sample size, minimum, maximum, and broad distribution pattern should agree wherever those features are visible.
A Quick Method for Mixed Graph Questions
First identify what the question asks for. “How many observations equal 7?” requires exact values, so a dotplot is useful. “How many observations fall between 10 and 20?” may be answered from a histogram if its bins line up with that interval. “Which group has the higher median?” points naturally to boxplots. The graph type alone does not settle every question: check its labels, scale, and level of detail.
Next, check the information the graph retains. A dotplot can show repeated values and isolated observations. A histogram shows how many observations fall in each bin, but not where individual observations lie inside a bin. A boxplot shows the five-number summary, but it does not show every value, peaks, or gaps. These limits are important when you decide whether a claim is supported.
Identify whether it asks about exact values, interval counts, distribution shape, or center and spread.
Use the display that preserves the information required to answer the question.
Confirm the variable, units, group, horizontal scale, and any histogram bin boundaries.
When graphs represent the same observations, compare sample size and any visible minimum, maximum, center, or broad pattern.
A useful cross-check is to count the observations represented. In a dotplot, count every dot. In a histogram, add the frequencies of all bins. A boxplot does not show the sample size unless it is provided separately, so do not try to recover it from the box and whiskers.
Worked Example: Match the Display to the Question
A fictional school club records how many minutes its members spend practicing a new song. The club has a dotplot, a histogram with five-minute bins, and a boxplot of the same practice times. Decide which display is most useful for each question.
Questions. (1) How many members practiced exactly 15 minutes? (2) How many practiced from 10 minutes up to, but not including, 15 minutes? (3) What is the median practice time, and what are the first and third quartiles?
Plan. For each question, identify the type of information requested. Exact values, grouped interval counts, and quartiles are not equally visible in the three displays.
Do. The dotplot is the best choice for question (1), because each dot represents one member and dots at 15 minutes can be counted directly. The histogram is best for question (2), provided one of its bins is exactly \([10,15)\); the bar’s frequency then gives the count in that interval. The boxplot is best for question (3), because its line and box edges mark the median, \(Q_1\), and \(Q_3\).
The displays may still help with other questions, but they do not all preserve the same detail. A histogram bar covering \([15,20)\) would not identify how many observations equal 15. A boxplot’s summary lines would not give the exact frequency at 15 or the count in \([10,15)\).
Conclude. Use the dotplot for the exact-value count, the histogram for the count in a matching interval, and the boxplot for quartiles and median. Before reading the histogram, verify its bin boundaries; an interval question is answerable directly only when the bins match the requested range.
Construct and Cross-Check Displays
When a question asks you to construct a display, keep the raw observations available until you have checked your work. In Building a Dotplot Step by Step and Constructing a Frequency Histogram, you learned the construction details for those graphs. The mixed-practice goal is to connect the displays and check that you have represented every observation exactly once.
For a dotplot, the number of dots must equal the number of observations. For a histogram, the bin intervals must cover the data, each observation must belong to exactly one bin, and the frequencies must add to the sample size. For a boxplot, sort the values and use the course’s quartile convention to find the five-number summary. Then compare the summaries and patterns across displays: for instance, a boxplot’s minimum and maximum should agree with the smallest and largest values in the raw data.
Worked Example: Build Three Displays From One Data Set
A fictional robotics team records the number of minutes 12 students take to complete a practice setup. The ordered times are:
Plan. Make a dotplot by counting repeated values, construct a histogram using the stated bins, and calculate the five-number summary. Then check that the displays account for all 12 students and agree on the data’s endpoints.
Do: dotplot. The counts at values 2 through 12 are shown below. Values without dots have frequency zero.
| Setup time (minutes) | Number of dots |
|---|---|
| 2 | 2 |
| 3 | 1 |
| 4 | 3 |
| 5 | 1 |
| 6 | 1 |
| 7 | 2 |
| 8 | 0 |
| 9 | 1 |
| 10 | 0 |
| 11 | 0 |
| 12 | 1 |
The dot counts sum to \(2+1+3+1+1+2+0+1+0+0+1=12\), so each observation is represented. In an actual dotplot, draw one dot for each count and stack dots vertically at the same value.
Do: histogram. Use the consecutive intervals \([0,3)\), \([3,6)\), \([6,9)\), \([9,12)\), and \([12,15)\). Their frequencies are 2, 5, 3, 1, and 1. For example, \([3,6)\) contains 3, 4, 4, 4, and 5, while \([12,15)\) contains 12. The total is \(2+5+3+1+1=12\). The histogram’s bars should touch because these are consecutive numerical intervals.
Do: boxplot summary. There are 12 ordered observations, so the median is the mean of the sixth and seventh values: \((4+5)/2=4.5\) minutes. The lower half is \(2,2,3,4,4,4\), giving \(Q_1=(3+4)/2=3.5\) minutes. The upper half is \(5,6,7,7,9,12\), giving \(Q_3=(7+7)/2=7\) minutes. The minimum is 2 and the maximum is 12. Thus, the five-number summary is \(2,\ 3.5,\ 4.5,\ 7,\ 12\) minutes.
Check. Both the dotplot and histogram represent 12 observations. The raw data, dotplot, and histogram have a minimum of 2 and a maximum of 12, matching the boxplot summary. The highest histogram frequency is in \([3,6)\), which agrees with the dotplot’s concentration around 3 and 4. The boxplot places the median at 4.5, also consistent with the ordered values. The boxplot does not show the zero frequencies at 8, 10, and 11 or the detailed concentration near 4.
Conclude. All three displays can represent this data set: the dotplot preserves individual times, the histogram groups times into three-minute intervals, and the boxplot summarizes minimum, quartiles, median, and maximum. The agreement in sample size and endpoints is a useful check that the constructions are consistent.
Read What Is Shown, Not What You Wish Were Shown
Mixed questions often include a statement that sounds plausible but goes beyond the graph. For a histogram, read a bar as a count in its full interval, not as a count at one exact value. For a boxplot, read the quartiles and whisker endpoints according to the displayed convention, but do not infer a peak or a gap. For a dotplot, individual values are available, but a visual description still should not claim a cause for a pattern.
Pay attention to boundaries. In the intervals \([5,10)\) and \([10,15)\), a value of 10 belongs to the second interval, not the first. The left endpoint is included and the right endpoint is excluded. This convention prevents an observation at a shared boundary from being counted twice.
Worked Example: Read a Histogram and Evaluate a Claim
A fictional community garden measures the height, in centimeters, of 30 seedlings. Its histogram uses equal-width bins:
| Seedling height (centimeters) | Frequency |
|---|---|
| [5, 10) | 4 |
| [10, 15) | 9 |
| [15, 20) | 11 |
| [20, 25) | 5 |
| [25, 30) | 1 |
A student claims, “Exactly 11 seedlings were 15 centimeters tall, and 15 seedlings were shorter than 15 centimeters.” Evaluate both parts.
Plan. Check the bin boundaries and sum the relevant bin frequencies. Distinguish observations in an interval from observations at one exact value.
Do. The \([15,20)\) bin contains 11 seedlings with heights from 15 centimeters up to, but not including, 20 centimeters. It does not show that all 11 had a height of exactly 15 centimeters. The count shorter than 15 centimeters is the sum of the first two bins: \(4+9=13\), not 15. The total frequency check is \(4+9+11+5+1=30\), matching the stated number of seedlings.
Conclude. Neither part of the claim is supported as written. The histogram shows 11 seedlings in \([15,20)\), not exactly 11 at 15 centimeters, and it shows 13 seedlings shorter than 15 centimeters. Since the data are grouped, the histogram does not reveal the exact heights within each bin.
Common Mistakes and AP Exam Tips
- Choosing a graph by appearance alone. Start with the question. Exact values, interval counts, and quartiles call for different information.
- Using a histogram to answer an exact-value question. A bin count combines all observations in an interval. Do not report that count as the frequency of one value.
- Ignoring interval endpoints. Read the brackets and parentheses. With \([10,15)\), 10 is included and 15 is not.
- Assuming a boxplot preserves the data’s shape. A boxplot can summarize center and spread, but it cannot establish modality or show specific gaps.
- Forgetting the total. Add dotplot counts or histogram frequencies and check that the result equals the stated sample size.
- Calling displays inconsistent when they show different detail. A dotplot can show a gap that a boxplot cannot. Compare only features each display actually represents, such as endpoints or broad concentration.
- Leaving out context and units. Write “11 seedlings were between 15 and 20 centimeters tall,” not simply “the count is 11.”
For full-credit communication, name the display feature you used, report the value or interval accurately, and answer in context. If the display cannot determine the requested detail, say so and explain why. For construction questions, label the variable and units, use a suitable scale or bin boundaries, and verify the total represented. These checks are more reliable than judging whether a graph merely looks right.
Check Your Understanding
For each question, identify the relevant display feature and explain your answer in context where appropriate.
- A question asks how many observations equal 8. Which display is most useful: a dotplot, histogram, or boxplot? Explain why.
- A histogram uses bins \([0,5)\), \([5,10)\), and \([10,15)\). In which bin does an observation equal to 10 belong?
- A histogram’s bin frequencies are 3, 8, 6, and 2. How many observations are represented, and what check could you make if the stated sample size is 19?
- A boxplot shows a median of 14 and quartiles of 10 and 18. Can it show how many observations equal 14? Explain.
- A dotplot and histogram both represent the same 16 observations. The dotplot has 16 dots, but the histogram frequencies sum to 15. What should you check in the histogram construction?