AP Statistics - Unit 1 - Topic 1.7

Summary Statistics One Quantitative Variable

A graph gives the picture. Summary statistics give the numbers that describe a typical value, spread, relative position, and possible outliers. The AP move is to calculate carefully and then explain what the numbers mean in context.

Skill 3.B Calculate summary statistics, relative positions, and related numerical results.

Skill 4.A Describe and compare summary statistics for one or more quantitative data sets.

Skill 4.B Justify a claim using statistical calculations and results in context.

Lesson Overview

Use these boxes as your quick AP checklist before you calculate.

What You Summarize

Summary statistics are numbers that describe a quantitative distribution: center, spread, position, and unusual values.

Center

The mean and median both describe a typical value, but they react differently to outliers and skew.

Spread

The range, IQR, and standard deviation describe how far apart the data values are.

AP Habit

Do not stop at the number. Say what it means: "The typical commute time is about 18 minutes," not just \(\text{median} = 18\).

Key Definitions

These are the vocabulary words that show up in AP prompts and scoring guidelines.

Mean

The arithmetic average: add all data values, then divide by the number of values. It is pulled toward extreme values.

Median

The middle value after the data are ordered from smallest to largest. With an even number of values, average the two middle values.

Quartiles

\(Q_1\) marks about the 25th percentile, \(Q_2\) is the median, and \(Q_3\) marks about the 75th percentile.

Percentile

The pth percentile is a value with about p% of the data less than or equal to it when the data are ordered.

Range and IQR

Range uses the maximum and minimum. IQR uses \(Q_3\) and \(Q_1\), so it focuses on the middle 50% of the data.

Standard Deviation

The sample standard deviation, \(s\), is a typical distance of the data values from the sample mean.

Formula + Notation Box

Memorize the meaning, not just the symbols.

Sample Mean \[\bar{x} = \dfrac{\sum x_i}{n}\]

\(x_i\) is one data value. \(n\) is the sample size.

Range \[\mathrm{Range} = \mathrm{maximum} − \mathrm{minimum}\]

This uses only the two most extreme data values, so it is not resistant.

Interquartile Range \[\mathrm{IQR} = Q_3 − Q_1\]

The IQR measures the spread of the middle 50% of the data.

Sample Standard Deviation \[s = \sqrt{\dfrac{\sum (x_i − \bar{x})^2}{n − 1}}\]

The sample variance is \(s^2\). Larger \(s\) means values are typically farther from the mean.

1.5 × IQR Outlier Rule \[\begin{aligned}x < Q_1 − 1.5(\mathrm{IQR})\\\text{or }x > Q_3 + 1.5(\mathrm{IQR})\end{aligned}\]

Values outside these fences are potential outliers.

2 Standard Deviation Rule \[\begin{aligned}x < \bar{x} − 2s\\\text{or }x > \bar{x} + 2s\end{aligned}\]

This is another common rule for flagging possible outliers.

Finding the Five Key Positions

Order the data first. Most mistakes in this topic start with unordered values.

Example ordered data set

Quiz scores: 4, 5, 6, 6, 7, 8, 9, 10, 25

4 5 6 6 7 8 9 10 25

\(Q_1 = 5.5\), \(\text{median} = 7\), \(Q_3 = 9.5\), \(\mathrm{IQR} = 4\), and 25 is a high potential outlier by the 1.5 × IQR rule.

AP Exam Skill Builder

The AP Exam often asks you to calculate, compare, and justify a choice of statistic.

How to Write an AP-Ready Response

Name the statistic. Mean, median, \(Q_1\), \(Q_3\), IQR, range, or standard deviation.
Show enough work. For hand calculations, order the data and write the formula or substitution.
Use units and context. "The IQR is 6 minutes" is stronger than \(\mathrm{IQR} = 6\).
Choose resistant statistics when needed. If a distribution is skewed or has outliers, median and IQR usually describe it better.

Quick Decision Guide

Symmetric, no outliers Mean and standard deviation are usually useful.

Skewed or outliers Median and IQR are usually safer because they are resistant.

Compare two samples Compare center, spread, shape, and outliers in context.

Change units Convert the statistic the same way you convert the data values, except spreads are not affected by adding or subtracting constants.

Worked AP-Style Examples

These model the calculation and the sentence you would write after it.

Example 1 − Center and Spread

Find the mean, median, range, and IQR

Seven students reported these homework times in minutes: 12, 15, 18, 20, 20, 24, 31.

Mean: \(\dfrac{12 + 15 + 18 + 20 + 20 + 24 + 31}{7} = \dfrac{140}{7} = 20\) minutes.

Median: The middle value is 20 minutes.

Range: \(31 − 12 = 19\) minutes.

IQR: \(Q_1 = 15\), \(Q_3 = 24\), so \(\mathrm{IQR} = 9\) minutes.

AP sentence: A typical homework time is about 20 minutes, and the middle 50% of reported times span about 9 minutes.

Example 2 − Outlier Check

Use the 1.5 × IQR rule

For the data 4, 5, 6, 6, 7, 8, 9, 10, 25, use \(Q_1 = 5.5\) and \(Q_3 = 9.5\).

IQR: \(9.5 − 5.5 = 4\).

Lower fence: \(5.5 − 1.5(4) = −0.5\).

Upper fence: \(9.5 + 1.5(4) = 15.5\).

Decision: 25 is above 15.5, so it is a high potential outlier.

AP sentence: The score of 25 is unusually high compared with the rest of the scores by the 1.5 × IQR rule.

Example 3 − Choosing a Statistic

Mean/SD or median/IQR?

A city compares two samples of commute times. One sample is strongly skewed right because a few people have very long commutes.

Answer: Use the median for center and IQR for spread. Long high values can pull the mean upward and inflate the standard deviation, while the median and IQR are resistant.

Example 4 − Unit Conversion

Convert a statistic from inches to centimeters

A sample of plant heights has mean 9 inches and standard deviation 2 inches. Convert to centimeters using \(1\ \text{inch} = 2.54\ \text{cm}\).

Mean: \(9(2.54) = 22.86\ \text{cm}\).

Standard deviation: \(2(2.54) = 5.08\ \text{cm}\).

AP sentence: Changing from inches to centimeters multiplies both the center and spread by 2.54.

Common AP Mistakes

These are small errors that can cost points even when the idea is close.

Forgetting to Order

Quartiles, median, minimum, maximum, and percentiles require data in increasing order first.

Using Range as IQR

\(\mathrm{Range} = \mathrm{max} − \mathrm{min}\). \(\mathrm{IQR} = Q_3 − Q_1\). They answer different spread questions.

Dropping Units

Mean, median, range, IQR, and standard deviation use the same units as the variable. Say "minutes," "dollars," or "points."

Calling Every Large Value an Outlier

Use a rule or clear graph evidence. A value can be the maximum without being an outlier.

Choosing Mean for Skewed Data

When outliers or strong skew are present, explain why median and IQR are more resistant.

Unit Conversion Mix-Ups

Adding a constant changes centers and positions, but it does not change range, IQR, or standard deviation.

Flashcard Deck

Click a card to reveal the answer. Mark it for review or mastery as you go.

Card 1 of 15

Practice-only

Vocabulary

Click the card or press Show Answer when you are ready.

Multiple-Choice Practice

Answer one question at a time. You will get instant feedback and a review at the end.

Question 1 of 10

Final study tip: Before choosing a summary statistic, ask, "Is the distribution skewed or affected by outliers?" If yes, median and IQR usually tell the story more honestly than mean and standard deviation.

FRQ-Style Practice

Prompt: A class collects data related to mean, median, standard deviation, quartiles, IQR, and outliers. 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 \(n = 48\), \(\text{mean} = 16.2\), \(\text{median} = 15.4\), \(\mathrm{IQR} = 4.8\), and one flagged high value.

How to interpret it: For mean, median, standard deviation, quartiles, IQR, and outliers, 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.