Displaying Distributions: Dotplots, Histograms, Stemplots, and Boxplots
Construct and interpret dotplots, histograms, stemplots (back-to-back for comparison), and boxplots (with and without outlier marking). Choose the most appropriate display for a given context and dataset size.
A lesson of AP Statistics, in Exploring One-Variable Data, about 10 minutes.
Displaying Distributions
By the end of this lesson, you will be able to choose the appropriate graph for a dataset, construct and read dotplots, stemplots, histograms, and boxplots, and identify the shape of a distribution from a graph.
Dotplot
Each data value is represented by a dot above a number line. Best for small datasets (n ≤ 30).
Strengths: shows every individual value; easy to see gaps, clusters, and outliers.
Example: Scores: 72, 75, 75, 78, 80, 82, 85, 85, 88, 92
. . . . . . .. . .
|---|---|---|---|---|---|---|---|
72 75 78 81 84 87 90 93
Stemplot (Stem-and-Leaf Plot)
Stem | Leaf
7 | 2 5 5 8
8 | 0 2 5 5 8
9 | 2
The stem is the leading digit(s); the leaf is the last digit. Shows actual values AND distribution shape.
Back-to-back stemplots compare two groups.
Example: Scores: 72, 75, 75, 78, 80, 82, 85, 85, 88, 92
Key: 7|2 means 72.
Histogram
Groups data into equal-width bins (intervals); bar height = frequency or relative frequency. Bars touch (unlike a bar chart). Best for large datasets.
When to use: quantitative data; want to see the overall shape of the distribution.
Reading a histogram:
- Shape: symmetric, left-skewed, right-skewed, bimodal?
- Center: roughly where is the middle?
- Spread: how wide is the distribution?
- Outliers: isolated bars far from the main body?
Boxplot (Box-and-Whisker)
Displays the five-number summary: Min, Q₁, Median, Q₃, Max.
|---[ box ]---|
Min Q1 M Q3 Max
The box spans Q₁ to Q₃ (the IQR). The line inside the box is the median. Whiskers extend to the smallest/largest non-outlier values. Outliers are plotted individually (beyond 1.5 × IQR from the quartiles).