AP Statistics - Unit 1 - Topic 1.5
Graphs for Quantitative Data Dotplots, Stem-and-Leaf Plots, and Histograms
Quantitative graphs show the distribution of numerical data. In AP Statistics, your job is to choose a useful graph, keep the number line in order, and make the frequency pattern easy to read.
Lesson Overview
Start by deciding whether the variable is numerical, then choose the graph that makes the pattern clearest.
Definition: Distribution
A distribution shows what values a variable takes and how often those values occur.
Important Idea
Quantitative graphs must keep values in natural numerical order from smallest to largest.
AP Notation
Frequency is a count. Relative frequency = frequency / n, where n is the number of observations.
Simple Example
If 4 of 20 students spend 30-39 minutes on homework, that bin has frequency 4 and relative frequency 4 / 20 = 0.20.
Key Definitions
These terms show up constantly when the exam asks you to construct or read a graph.
Quantitative Variable
A variable with numerical values that represent measured or counted amounts, usually with units.
Dotplot
A graph that places a dot for each observation above its value on a number line. Repeated or nearly repeated values are stacked.
Stem-and-Leaf Plot
A graph that splits each value into a stem and a leaf so the original data values are still visible.
Histogram
A graph that groups values into ordered intervals called bins and uses bar heights to show frequency or relative frequency.
Graph Toolbox
The same data can be displayed in more than one way. The graph choice changes what is easiest to see.
Bars touch because the number line is continuous across intervals.
Key: 3 | 2 means 32.
AP Exam Skill Builder
Skill 3.A is about making the graph, not just naming it.
Build a Quantitative Graph
When an AP question gives you raw numerical data, work in this order.
Which Graph Should I Use?
- Dotplot: best for a small or medium data set when individual observations matter.
- Stem-and-leaf: best when you want to show the shape and keep exact data values.
- Histogram: best for many values or measurements spread across intervals.
- Relative-frequency histogram: best when comparing data sets with different sample sizes later in the course.
Worked AP-Style Examples
Practice turning raw values into a graphing decision and an accurate display.
Eight quiz scores are 6, 7, 5, 6, 8, 6, 10, and 5. Construct a dotplot.
Solution: Draw a number line from 5 to 10. Put two dots above 5, three dots above 6, one dot above 7, one dot above 8, no dot above 9, and one dot above 10.
Twenty commute times are grouped into 0-9, 10-19, 20-29, and 30-39 minutes with counts 3, 8, 6, and 3.
Solution: Make a histogram with four equal-width bins. The bar heights are 3, 8, 6, and 3. The bars touch because the bins cover adjacent intervals on a number line.
Create a stem-and-leaf plot for 21, 23, 28, 30, 32, 34, 34, 41, and 46.
Solution: Use tens digits as stems and ones digits as leaves. Write stems 2, 3, and 4. Leaves are 1 3 8, then 0 2 4 4, then 1 6. Add a key: 3 | 4 means 34.
Common AP Mistakes
These are small details, but they can cost points when the task is to construct a display.
Using the Wrong Graph Type
A bar chart is for categorical variables. A histogram, dotplot, or stem-and-leaf plot is for one quantitative variable.
Forgetting Units
Minutes, inches, dollars, points, or degrees make the variable meaningful. Label the axis with context and units.
Unequal Histogram Bins
For beginning AP Statistics work, use equal-width intervals unless the problem clearly gives a different setup.
Gaps Between Histogram Bars
Histogram bars usually touch because adjacent intervals sit next to each other on a number line.
Losing the Original Data
A stem-and-leaf plot should let a reader reconstruct the values. Always include a key.
Changing the Story with Bins
Changing histogram bin widths can make the same data look different. Choose intervals that are sensible and clear.
Flashcard Deck
Use the deck to check the vocabulary and construction rules before taking the quiz.
Multiple-Choice Practice
Answer one question at a time. You will get instant feedback and a review at the end.
FRQ-Style Practice
Prompt: A class collects data related to dotplots, stem-and-leaf plots, and histograms. Write a free-response answer that uses the correct vocabulary and statistical reasoning.
- Identify the variable(s), population/sample, or study design feature requested.
- Choose or describe the appropriate table, graph, summary statistic, sampling method, or design decision.
- Write one contextual interpretation that uses statistical language rather than a vague everyday claim.
Scoring focus: Credit depends on precise vocabulary, context, and a justified choice or description.
Calculator and Technology Check
Output to read: Calculator or spreadsheet output gives n = 48, mean = 16.2, median = 15.4, IQR = 4.8, and one flagged high value.
How to interpret it: For dotplots, stem-and-leaf plots, and histograms, connect the output to the context: compare resistant and nonresistant summaries, mention units, and decide whether the flagged value changes the story.
Source note: Aligned to AP Statistics Course and Exam Description, Effective Fall 2026.