Start With the Variable
In Distinguishing Population, Sample, and Parameter Questions, you practiced identifying the group a numerical summary describes. Before choosing a summary or graph, take one step back: identify the variable being summarized and its type. A good display makes the values easier to understand without implying that the data say something they do not.
As you learned in Categorical Versus Quantitative Variables, a categorical variable places individuals into groups, while a quantitative variable records a count or measurement with meaningful numerical differences. That distinction guides the choice of display. A list of product types is not summarized like a list of delivery times, even if both variables have several possible values.
The first decision is not “Which graph looks familiar?” It is “What kind of information does this variable record?” The table below links variable type to suitable summaries and displays.
| Variable type | Useful summary | Suitable display | What it helps show |
|---|---|---|---|
| Categorical | Count or proportion in each category | Frequency table or bar graph | How many or what fraction of individuals are in each category |
| Quantitative, with a small set of values | Numerical summaries such as a mean or median, when useful | Dotplot | Individual values, repeated values, gaps, clusters, and overall shape |
| Quantitative, especially with many values | Numerical summaries such as a mean or median, when useful | Histogram | The shape of the distribution across intervals of values |
| Quantitative | Five-number summary, including the median and quartiles | Boxplot | Center and spread, and a visual comparison of distributions |
A frequency table lists categories alongside their counts or proportions. A bar graph displays those category summaries with separated bars. The separation matters: categories are distinct groups, not consecutive numerical intervals.
A dotplot, histogram, or boxplot is for quantitative data. A dotplot places a mark for each observed value, stacking marks when values repeat. A histogram groups quantitative values into intervals called bins and shows how many observations fall in each interval. A boxplot represents a distribution with five landmarks: the minimum, first quartile, median, third quartile, and maximum.
A Decision Process for Choosing a Display
Use this process when a question asks you to describe or display a variable. First identify the variable, not just the column heading or the way its values are written. Then choose a display that preserves the information you need.
State what information was recorded about each individual, as practiced in What Is a Variable in Statistics.
Decide whether the values represent categories or meaningful numerical counts or measurements. A number written as a label can still be categorical, as in Spotting Numerical Variables That Are Really Categorical.
For categories, count individuals or find proportions. For quantitative data, consider numerical summaries that describe values, such as a mean or median.
Use a dotplot to retain individual quantitative values, a histogram to group values into intervals, or a boxplot to emphasize center and spread.
The best choice also depends on the question. If the goal is to see every observed value in a modest-sized data set, a dotplot is informative. If there are many measurements and the goal is to see where values concentrate, a histogram can make the overall shape easier to see. If the goal is to compare centers and spreads across groups, boxplots can give a compact view. A display is useful when its emphasis matches the question.
Worked Example: Favorite After-School Activity
A school club asks 24 students to name their favorite after-school activity. The responses are 9 sports, 7 music, 5 art, and 3 reading. Choose a suitable summary and display, and calculate the proportions.
Identify and classify: The variable is each student’s favorite activity. It is categorical because each response names a group, not a measured numerical amount.
Choose: Summarize the categories with counts and proportions. A frequency table or bar graph is appropriate; a dotplot, histogram, or boxplot is not, because the activity names are not quantitative values.
Calculate: Divide each category count by the total of 24 students. For sports, the proportion is \(9/24=0.375\). For music, it is \(7/24\approx0.292\). For art, it is \(5/24\approx0.208\). For reading, it is \(3/24=0.125\). These rounded proportions add to 1.000.
| Favorite activity | Count | Proportion |
|---|---|---|
| Sports | 9 | 0.375 |
| Music | 7 | 0.292 |
| Art | 5 | 0.208 |
| Reading | 3 | 0.125 |
| Total | 24 | 1.000 |
Interpret: In this group of 24 students, 9 named sports as their favorite activity, which is 37.5% of the group. A bar graph could show the same information with one separated bar for each activity. The counts preserve the number of students; the proportions make the categories easier to compare as shares of the group.
Dotplots and Histograms for Quantitative Data
For quantitative data, a dotplot and a histogram both show how values are distributed, but they do not show the same details. A dotplot keeps each observed value visible. This works well when there are not too many observations and individual values, repeated values, gaps, or clusters matter.
A histogram trades some detail for a compact view. Its bins group nearby values together, so the exact observations are not visible. The chosen bin width and boundaries affect the pattern shown. A histogram should have equal-width bins when comparing frequencies across intervals in a standard display. Be clear about which interval includes a boundary value, especially when values fall exactly on a bin edge.
A histogram is not a bar graph with a different name. The bars represent intervals along a quantitative scale, so neighboring bins touch. In a categorical bar graph, the bars represent separate categories and are separated by gaps. This distinction follows from the meaning of the variable, not just the appearance of the picture.
Worked Example: Daily Orders at a Food Cart
A food cart records the number of online orders it receives on each of 10 days: 6, 7, 7, 8, 8, 8, 9, 10, 10, and 12. Choose a display for showing individual values, and then group the same data into four histogram intervals.
Identify and classify: The variable is the number of online orders per day. It is quantitative and discrete, because it is a count that takes whole-number values.
Dotplot: A dotplot preserves each day’s count. The table below gives the number of dots to place above each value.
| Orders | Dots |
|---|---|
| 6 | • |
| 7 | •• |
| 8 | ••• |
| 9 | • |
| 10 | •• |
| 11 | — |
| 12 | • |
There are 10 dots altogether, matching the 10 recorded days. The dotplot makes the repeated value of 8 and the gap at 11 easy to see.
Histogram: Use the intervals from 6 to less than 8, from 8 to less than 10, from 10 to less than 12, and from 12 to less than 14. Their counts are 3, 4, 2, and 1, respectively.
| Interval of orders | Count |
|---|---|
| 6 to less than 8 | 3 |
| 8 to less than 10 | 4 |
| 10 to less than 12 | 2 |
| 12 to less than 14 | 1 |
The first interval contains 6, 7, and 7, so its count is 3. The second contains 8, 8, 8, and 9, so its count is 4. The third contains 10 and 10, so its count is 2. The last contains 12, so its count is 1. The counts total \(3+4+2+1=10\), as they should.
Choose and explain: Use the dotplot if the question emphasizes individual daily counts, repeated values, or the gap at 11. Use the histogram if the goal is a compact view of how the counts are distributed across intervals. The histogram’s bins combine information: it shows how many days fall in each interval, but not the exact values within that interval.
Boxplots for a Compact Summary
A boxplot is also used for quantitative data, but it focuses on a different set of features. It marks the median and the quartiles, which divide ordered data into sections, along with the minimum and maximum. The box extends from the first quartile to the third quartile; a line inside it marks the median. The lines, or whiskers, extend to the minimum and maximum in a basic boxplot.
The difference between the third quartile and the first quartile is the interquartile range, or IQR. It describes the spread of the middle half of the data. In a boxplot, the box’s length represents that IQR on the measurement scale. A boxplot does not show every individual observation, so it may conceal clusters, gaps, or repeated values that a dotplot reveals.
Worked Example: Screen-Free Reading Time
A family records the number of hours spent reading for pleasure during one week by eight participants: 2, 3, 4, 4, 5, 6, 8, and 12 hours. Find the five-number summary and decide whether a boxplot is suitable.
Identify and classify: The variable is hours of reading per participant during the week. It is quantitative, so a boxplot is an appropriate display if a compact summary of center and spread is useful.
Find the median: The values are already in order. With eight observations, the median is the average of the fourth and fifth values: \((4+5)/2=4.5\) hours.
Find the quartiles: Using the median-of-halves method, the lower half is 2, 3, 4, 4. Its median is \((3+4)/2=3.5\) hours, so \(Q_1=3.5\). The upper half is 5, 6, 8, 12. Its median is \((6+8)/2=7\) hours, so \(Q_3=7\).
The minimum is 2 hours and the maximum is 12 hours. Therefore the five-number summary is 2, 3.5, 4.5, 7, and 12 hours. The IQR is \(7-3.5=3.5\) hours. A boxplot would extend from 2 to 12 hours, with a box from 3.5 to 7 hours and a median line at 4.5 hours.
Interpret: The median reading time in this group was 4.5 hours. The middle half of the recorded times extends from 3.5 to 7 hours, a spread of 3.5 hours. A dotplot would be preferable if the aim were to keep all eight individual times visible; the boxplot offers a shorter summary of center and spread.
Common Mistakes and AP Exam Tips
- Choosing a graph from the values’ appearance instead of their meaning. A set of digits may be category codes rather than measurements. Check whether numerical differences have a meaningful interpretation before treating the variable as quantitative.
- Using a mean or median for category labels. The mean of “sports,” “music,” and “art” is not meaningful. Summarize a categorical variable with counts or proportions.
- Calling a histogram a bar graph. A histogram’s bars represent adjacent quantitative intervals and touch. A categorical bar graph uses separated bars for distinct categories.
- Claiming that a boxplot shows the whole data set. It summarizes selected landmarks, not every observation. If individual values or gaps matter, a dotplot may be more informative.
- Forgetting what a proportion’s denominator is. State the group total when explaining a proportion. For example, “9 of the 24 students, or 37.5%, named sports” identifies both the count and the whole group.
- Leaving out the reason for the display choice. A strong answer names the variable type and explains why the display fits the purpose. For example: “The variable is quantitative, and a dotplot is useful here because the data set is small and the individual values matter.”
On an AP Statistics question, a full-credit response does more than name a graph. Identify the variable as categorical or quantitative, connect that type to the display, and state what the display will help the reader see. If the question asks for a summary as well, use counts or proportions for categories, and choose an appropriate numerical summary for quantitative data.
Check Your Understanding
For each situation, identify the variable type and choose a suitable summary or display. Briefly explain why it fits.
- A library records each visitor’s reason for coming: borrowing books, using a computer, attending an event, or another reason. What summary and display would you use?
- A gardener records the number of tomatoes harvested from each of 12 plants. Would a dotplot or histogram preserve individual plant counts more clearly? Explain.
- A transit office records the time, in minutes, that 200 buses take to complete a route. Which display could show the distribution across intervals?
- A sports trainer wants a compact summary of the quantitative jump distances recorded for athletes. Which display and summary should the trainer consider?
- A survey records each respondent’s postal-code digits. Should the office use a histogram of those digits? Explain using the meaning of the variable.