Representing a Categorical Variable with Tables - AP Statistics Study Guide

Quick Summary

This guide will equip you to master the foundational skill of summarizing categorical data. You will learn how to take a list of qualitative observations and organize them into clear, informative frequency and relative frequency tables. By the end of this lesson, you will be able to construct these tables, calculate proportions and percentages for each category, and use them to describe the distribution of a single categorical variable in context.

Key Concepts

The first step in analyzing data is often to organize and summarize it. When dealing with data that sorts individuals into groups or categories, tables are our primary tool.

• What is a Categorical Variable?

  A categorical variable (also called a qualitative variable) places an individual into one of several groups or categories. Examples include eye color (blue, brown, green), car make (Ford, Toyota, Honda), or student grade level (Freshman, Sophomore, Junior, Senior). The "values" of these variables are labels, not numbers you can perform meaningful arithmetic on.

• Frequency Tables

  The most basic way to summarize categorical data is with a frequency table. This is a simple table that lists each category and the number of individuals that fall into that category. This count is called the frequency.

  • How to Construct a Frequency Table:

    1. List Categories: Identify all the unique categories in your dataset.

    2. Tally: Go through your raw data and make a tally mark for each individual in the appropriate category.

    3. Count (Frequency): Sum the tallies for each category to get the frequency.

    4. Total: Sum the frequencies for all categories to find the total number of observations (often denoted by n).

  • Example: A class of 25 students was asked to name their favorite social media platform. The raw data is:

    Instagram, TikTok, Instagram, Snapchat, TikTok, TikTok, Instagram, X, Snapchat, Instagram, TikTok, Instagram, TikTok, TikTok, Snapchat, Instagram, X, TikTok, Instagram, TikTok, Snapchat, Instagram, TikTok, X, TikTok

  • Frequency Table Construction:

    Platform	Tally	Frequency
    TikTok	IIII IIII I	10
    Instagram	IIII III	8
    Snapchat	IIII	4
    X	III	3
    Total		25

  [Image: A clean, two-column frequency table showing the social media platform data. Column 1 is "Platform" and Column 2 is "Frequency (Count)".]

• Relative Frequency Tables

  While frequencies are useful, they can be hard to compare between groups of different sizes. For instance, knowing 8 students prefer Instagram is only meaningful if we know the total class size. A relative frequency table solves this by showing the proportion or percent of individuals in each category.

  • How to Calculate Relative Frequency:

    The formula is simple and crucial:

    Relative Frequency = (Frequency of the category) / (Total number of observations)

  • Proportions vs. Percentages:

    • A proportion is the decimal form (e.g., 0.40).

    • A percentage is the proportion multiplied by 100 (e.g., 40%).

    AP Statistics questions may ask for either, so be prepared to provide both.

  • Example (continued from above): Let's convert our frequency table to a relative frequency table. The total is n = 25.

    Platform	Frequency	Calculation	Relative Frequency (Proportion)	Relative Frequency (Percent)
    TikTok	10	10 / 25	0.40	40%
    Instagram	8	8 / 25	0.32	32%
    Snapchat	4	4 / 25	0.16	16%
    X	3	3 / 25	0.12	12%
    Total	25		1.00	100%

  [Image: A four-column relative frequency table. Columns are "Platform," "Frequency," "Relative Frequency (Proportion)," and "Relative Frequency (Percent)".]

• The "Sum to 1" Rule

  A critical check for your work is that the sum of all relative frequencies must equal 1 (if in proportion form) or 100% (if in percentage form). Due to rounding in more complex datasets, the sum might be slightly off (e.g., 0.999 or 100.1%). This is usually acceptable, but a sum far from 1 or 100 indicates a calculation error.

• Describing the Distribution

  Once you have a table, you can describe the distribution of the variable. For a categorical variable, this means commenting on which categories appear most and least often. Use the frequencies or relative frequencies to support your description. For our example, a good description would be: "The most popular social media platform among this group of students was TikTok, chosen by 40% of the class. The least popular platform was X, with only 12% of students choosing it."

Key Vocabulary

• Categorical Variable: A variable that places an individual into one of several non-numerical groups or categories.

• Frequency: The count of times a value or category occurs in a dataset.

• Frequency Table: A table that lists the categories of a variable and the frequency (count) for each category.

• Relative Frequency: The proportion or percentage of observations that fall into a specific category, calculated by dividing the category's frequency by the total number of observations.

• Relative Frequency Table: A table that lists the categories of a variable and the relative frequency (proportion or percent) for each category.

• Distribution: For a categorical variable, the distribution consists of the possible categories and their corresponding frequencies or relative frequencies.

Calculator Tech (TI-84)

No major calculator functions are required for this topic. The primary calculations involve division to find relative frequencies, which can be done directly on the calculator's home screen.

For example, to calculate the relative frequency for TikTok from our example (10 out of 25):

1. Type $10$

2. Press the division key $÷$

3. Type $25$

4. Press ENTER

The calculator will display $0.4$. To convert to a percentage, simply multiply by 100.

While you can enter raw categorical data into a list (e.g., using words in L3 if you have the Stat-List-Editor App), the TI-84 does not have a built-in function to automatically generate a frequency table from this text data. This process is almost always done by hand.

How to Show Work on the FRQ

On the AP Exam, you must clearly communicate your understanding. For questions involving tables, this means showing your calculations and interpreting results in context.

Template for Calculating a Relative Frequency:

When asked to find a proportion or percentage, show your work as a fraction.

• Question: "Calculate the proportion of students who chose Instagram."

• Scorable Work: "The proportion of students who chose Instagram is 8/25 = 0.32."

Template for Describing the Distribution of a Single Categorical Variable:

A complete description identifies the most and least frequent categories, using specific numerical values (frequencies or relative frequencies) and including context.

• Structure:

  1. Identify the category with the highest frequency/relative frequency (the mode).

  2. State its value (count or percent) in context.

  3. Identify the category with the lowest frequency/relative frequency.

  4. State its value (count or percent) in context.

• Example FRQ Response:

  "The distribution of favorite social media platforms is centered on TikTok, which was the most popular choice, selected by 10 of the 25 students (40%). The least popular platform was X, which was selected by only 3 of the 25 students (12%)."

Practice Problems

Problem 1:

A local animal shelter recorded the breed of the last 30 dogs admitted. The data are as follows:

Labrador, Poodle, German Shepherd, Labrador, Beagle, Bulldog, Labrador, Poodle, Bulldog, German Shepherd, Labrador, Beagle, Labrador, Mixed Breed, Poodle, German Shepherd, Mixed Breed, Labrador, Bulldog, Beagle, Labrador, Poodle, Mixed Breed, German Shepherd, Labrador, Beagle, Poodle, Mixed Breed, Bulldog, Labrador.

(a) Construct a frequency and relative frequency table for these data.

(b) What proportion of the dogs were Labradors? Show your work.

(c) Write a sentence or two describing the distribution of dog breeds.

Solution:

(a) First, we tally the frequencies for each breed.

• Labrador: 9

• Poodle: 5

• German Shepherd: 4

• Beagle: 4

• Bulldog: 4

• Mixed Breed: 4

• Total (n) = 30

Now, we construct the full table, calculating relative frequency (Frequency / 30).

(b) The proportion of dogs that were Labradors is the frequency of Labradors divided by the total number of dogs.

Proportion = 9 / 30 = 0.30.

(c) The most common breed admitted to the shelter was the Labrador, accounting for 9 of the 30 dogs (30%). The other five listed breeds (Poodle, German Shepherd, Beagle, Bulldog, and Mixed Breed) were much less common and were admitted with roughly equal frequency.

Problem 2:

A survey asked 200 high school students what their primary source of news was. The results are summarized in the partially completed table below.

(a) Calculate the frequency for the "TV News" category.

(b) Complete the "Relative Frequency" column for all categories. Show your work for the "Social Media" category.

(c) A school administrator claims that fewer than 1/5 of students get their news from online newspapers. Do these data support this claim? Justify your answer with a calculation.

Solution:

(a) The relative frequency for TV News is 0.25, which means 25% of the 200 students chose this category.

Frequency = Total × Relative Frequency = 200 × 0.25 = 50 students.

The frequency for TV News is 50.

(b) We calculate the relative frequency for each category by dividing its frequency by the total of 200.

• Social Media: 82 / 200 = 0.41

• TV News: 50 / 200 = 0.25 (given)

• Online Newspapers: 38 / 200 = 0.19

• Friends/Family: 30 / 200 = 0.15

Completed Table:

(c) To evaluate the claim, we first need to express 1/5 as a decimal: 1/5 = 0.20.

The proportion of students who get their news from online newspapers is 38/200 = 0.19.

Because 0.19 is less than 0.20, the data do support the administrator's claim that fewer than 1/5 of students get their news from online newspapers.

Common Mistakes to Avoid

• Confusing Frequency and Relative Frequency: A question might ask for a proportion (relative frequency), but a student provides the count (frequency). Always read the question carefully to see if it asks for a count, proportion, or percentage.

• Using the Wrong Total: When calculating relative frequency, always divide by the grand total of all observations, not the frequency of another category. Double-check your total before you start dividing.

• Forgetting Context: Answers on the AP exam must be in the context of the problem. Don't just say "the mode is 9." Instead, say "the most frequent dog breed was Labrador, with 9 dogs." Context is critical for earning full credit.

• Incomplete Descriptions: When asked to describe a distribution, don't just state the most frequent category. A good description also identifies the least frequent category (or categories) to provide a more complete picture.

• Poor Table Labels: When constructing a table, always include a clear title and label your columns (e.g., "Category," "Frequency," "Relative Frequency"). This makes your work clear and easy to grade.