# Graphical Representations of Summary Statistics for One Quantitative Variable

> AP Statistics · AP Stats 2024-2026
> Source: https://www.owlsprep.com/study/ap-statistics-u10-graphical-representations-of-summary-statistics/

This module walks through constructing, interpreting, and comparing four core graphical displays for univariate quantitative data, aligned to AP Statistics exam requirements.

**Prerequisites:** [Calculate five-number summary for quantitative data](https://www.owlsprep.com/study/ap-statistics-u10-five-number-summary-and-outliers/); [Define shape, center, spread for univariate distributions](https://www.owlsprep.com/study/ap-statistics-u10-describing-distributions/)

## Learning objectives

- Select appropriate graphical displays for univariate quantitative data sets
- Construct and interpret dotplots, stemplots, histograms, and boxplots from raw or summary statistics
- Compare distributions across groups using aligned graphical representations
- Identify outliers and shape features directly from graphical summary plots

## Dotplots and Stemplots: Raw Data Visualization

These two displays preserve all individual raw data values, making them ideal for small to medium data sets where you need full transparency of observations.

**Stemplot (Stem-and-Leaf Plot)** — A display that splits each data point into a shared leading stem digit and unique trailing leaf digit to group values while retaining full precision

*Notation:* Stem = leading digits, Leaf = trailing digit

*Example:* Data points 23, 27, 31, 32 have stems 2 and 3, with leaves [3,7] and [1,2] respectively

**Worked example:** Construct a stemplot for the following 12 test scores: 62, 65, 71, 74, 75, 77, 80, 82, 86, 91, 93, 98

1. Identify the leading tens digit as the stem, and trailing units digit as the leaf
2. List all unique stems from 6 to 9 in ascending order on the left side of the plot
3. Assign each score's units digit as a leaf to its corresponding stem, sorting leaves in ascending order
4. Add a key to explain notation, e.g. 6 | 2 = 62 points

> **Exam Tip**
>
> AP graders accept stemplots with split stems (e.g. two rows for stem 7, one for leaves 0-4 and one for 5-9) to improve visibility of distribution shape

**Check your understanding**

Test your understanding of raw data displays:

1. What is the largest value in a stemplot with stem 9 and leaves 2, 4, 7?

   *Why:* The stem 9 represents the tens place, and 7 is the units digit.

## Histograms: Binned Frequency Visualization

**Histogram** — A bar chart for quantitative data where the x-axis is split into consecutive non-overlapping bins, and bar height represents the frequency or relative frequency of observations in each bin

*Example:* Bin 0-10 has 7 observations, so its bar rises to 7 on the y-axis

**Worked example:** Construct a histogram for 20 student commute times with bins 0-10, 10-20, 20-30, 30-40, 40-50 and frequencies 5, 7, 4, 3, 1

1. Draw a continuous x-axis labeled 'Commute Time (minutes)' spanning 0 to 50
2. Draw a y-axis labeled 'Number of Students' spanning 0 to 8 to accommodate the maximum frequency of 7
3. Draw adjacent bars for each bin with height equal to its frequency, no gaps between bars
4. Confirm no bars overlap and all data points fall into exactly one bin

> **Common Mistake**
>
> If you use unequal bin widths, you must scale the y-axis to density, not raw frequency, to avoid distorting the relative area of each bin

**Exam command terms**

AP exam questions use specific command terms for histogram tasks:

- **Describe the shape** — You must reference skewness, modality, and outliers explicitly *(The distribution is right-skewed with one outlier near 50 minutes)*

## Boxplots: Summary Statistic Visualization

**Boxplot** — A compact display built exclusively from the five-number summary that clearly marks outliers using the 1.5*IQR rule

*Notation:* Box spans Q1 to Q3, line at median, whiskers extend to farthest non-outlier points

**Derivation:** Construct a boxplot from a five-number summary

*Starting from:* min = 12, Q1 = 18, median = 25, Q3 = 32, max = 47, outlier at 58

1. Draw a number axis covering the full range of values including the outlier
2. Draw a rectangular box from Q1 = 18 to Q3 = 32, with a vertical line at the median 25
3. Calculate IQR = Q3 - Q1 = 14, so 1.5*IQR = 21. Upper fence = Q3 + 21 = 53
4. Draw whiskers from Q1 down to the minimum 12, and from Q3 up to 47, the largest value below the upper fence
5. Plot the outlier 58 as a separate isolated point beyond the upper whisker

*Conclusion:* The final boxplot clearly shows the right skew from the outlier and the central 50% of data

> **Exam Mnemonic**
>
> Use SOCS to describe any distribution: Shape, Outliers, Center, Spread

## Cross-Distribution Comparison Using Aligned Graphs

**Comparing methods**

Choose the right display for group comparison based on sample size and required detail:

- **Side-by-side dotplots** — Best for 2 groups with small n < 30
  - Pros: Preserves all individual data points
  - Cons: Clutters for large n

- **Side-by-side boxplots** — Best for 3+ groups of any size
  - Pros: Easy to compare centers and spreads at a glance
  - Cons: Hides individual point detail

**Check your understanding**

Test your comparison skills:

1. Which display is best for comparing 5 different class test score distributions?

   - Side-by-side boxplots
   - Dotplots
   - Stemplots

   *Why:* Boxplots are compact enough to fit 5 aligned displays on a single axis for easy comparison.

## Common pitfalls

- **Wrong:** Using unequal bin widths in a histogram and counting raw frequency on the y-axis
  - Why it fails: This distorts the relative area of bins, making the distribution shape look incorrectly skewed
  - Correct: Use equal bin widths, or scale the y-axis to density if bins are unequal
- **Wrong:** Adding gaps between bars of a histogram to separate bins
  - Why it fails: Histogram bars represent continuous ordered data, gaps incorrectly signal no data in that range
  - Correct: Leave no gaps between adjacent histogram bars, except for empty bins with zero frequency
- **Wrong:** Claiming a boxplot shows the mode of the data set
  - Why it fails: Boxplots only show quartiles, not individual data points or peak frequency
  - Correct: Use a dotplot or stemplot to identify mode, note that mode cannot be confirmed from a boxplot
- **Wrong:** Ignoring outliers when comparing two side-by-side boxplots
  - Why it fails: Outliers represent extreme values that impact spread and context of the data, and AP rubrics deduct points for omitting them
  - Correct: Explicitly reference outliers as a separate feature when comparing distributions
- **Wrong:** Splitting a stemplot stem into only 2 parts for a wide data range
  - Why it fails: This over-aggregates data and hides shape features like bimodality
  - Correct: Split stems into 5 or 2 parts as needed to get 6-15 total stems for clear shape visibility

## Cheatsheet

| Display Type | Best Use Case | Shows Exact Data Values? | Identifies Outliers Easily? | AP Exam Common Command Term |
| --- | --- | --- | --- | --- |
| Dotplot | Small n < 50 quantitative data | Yes | Yes | Construct / Describe |
| Stemplot | Small to medium n < 200 quantitative data | Yes | Yes | Construct / Compare |
| Histogram | Large n > 100 continuous quantitative data | No | No | Interpret / Describe Shape |
| Boxplot | Comparing 3+ groups of data | No | Yes | Compare Distributions |

## What's next

Mastering these graphical representations is the foundation for all future inferential and comparative statistics work on the AP exam. You will use these displays to justify assumptions for t-tests, chi-squared tests, and regression analysis later in the course, as well as to earn full points on free response questions that ask you to compare two distributions — a question type that appears on nearly every AP Stats exam. To build on this knowledge, practice applying your interpretation skills to formal distribution description frameworks, then move to calculating numerical summary statistics like mean, standard deviation, and interquartile range to pair with your graphical analysis. These paired skills will ensure you never lose points for incomplete description on exam day.

---

From [OwlsPrep](https://www.owlsprep.com) — free study guides for A-Level, IB, AP and IGCSE, written against the official syllabus. Canonical page: https://www.owlsprep.com/study/ap-statistics-u10-graphical-representations-of-summary-statistics/