1.3 Representing a Categorical Variable with Tables

Categorical variables get organized into tables that count how many cases fall into each category (a frequency table) or what proportion of cases fall into each category (a relative frequency table). Once the data is in a table, you can read off counts, proportions, percentages, or rates and use those numbers to make claims about the data in context.

Why This Matters for the AP Statistics Exam

This is one of the first building blocks in AP Statistics. Frequency and relative frequency tables are how you take a messy list of category responses and turn it into something you can actually read and describe.

On the exam you may be asked to:

• Build a frequency or relative frequency table from raw category data.
• Read values out of a table and convert between counts, proportions, percentages, and rates.
• Describe what a table tells you about a categorical variable and justify a claim using the numbers in context.

Getting comfortable here sets you up for the next topics, where the same categorical data becomes bar graphs (1.4) and, later, two-way tables for comparing two categorical variables (Unit 2).

Key Takeaways

• A frequency table lists each category and the count of cases in it. A relative frequency table lists each category and the proportion of cases in it.
• Relative frequency = (frequency of that category) / (sum of all frequencies). Percentage = relative frequency x 100.
• Proportions, percentages, relative frequencies, and rates all carry the same information, just scaled differently.
• Frequencies add up to the total sample size n. Relative frequencies add up to 1.00 (or close, if rounded), and percentages add up to 100.
• Always describe the data in context: name the variable, the categories, and what the numbers mean for real cases.
• You can combine categories (for example, "very" plus "somewhat") to justify a broader claim.

Frequency Tables

Suppose you survey your AP Statistics class about how stressful being a student feels, using the categories "very," "somewhat," and "none" (not stressful at all). You collect 30 responses. How do you organize them so they make sense?

One way is to "pile" the data by counting how many responses fall into each category. Those counts go into a frequency table, which records the category names and the totals.

A frequency table lists all categories in one column and the number of cases that belong to each category in the next column. Tally marks (for example, ||||) can help you count the raw data. Here is what a frequency table looks like:

Source: Prem S. Mann. Introductory Statistics. John Wiley and Sons Inc. 2010

The variable here is stress on job, which takes on three categories: very, somewhat, and none. The frequency table always reports the sum of the frequencies, which equals the total sample size.

Relative Frequency Tables

You can extend a frequency table using relative frequencies and percentages.

• The relative frequency for a category is its frequency divided by the sum of all frequencies.
• The percentage is the relative frequency multiplied by 100.

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Relative frequency of a category = (frequency of that category) / (sum of all frequencies)
Percentage = relative frequency x 100

A relative frequency table is set up like a frequency table, but it reports proportions or percentages for each category instead of raw counts. Using a relative frequency distribution for stress on job, you might state that 33.3% of employees answered that their jobs are very stressful. You can also combine groups: adding "very" and "somewhat" lets you report that 80% of employees said their jobs are very or somewhat stressful.

The sum of the relative frequencies should be 1.00 (or close to 1.00 if values are rounded). Likewise, the sum of the percentages is always 100 (or close to 100 if rounded).

Source: Basic Mathematics

How to Use This on the AP Statistics Exam

Problem Solving

When you build a table from raw data:

1. List every category.
2. Count how many cases fall in each one (tally marks help).
3. Check that the counts add up to the total sample size n.
4. For relative frequencies, divide each count by n. Confirm they add to 1.00 (or 100% as percentages).

Reading and Describing

When a table is given to you, practice converting between forms. A count of 10 out of 30 is a relative frequency of about 0.333, or 33.3%, or a rate of about 333 per 1,000. They all say the same thing.

When asked to describe or justify a claim, point to specific numbers and tie them to context. For example: "33.3% of the 30 surveyed students rated being a student as very stressful." A vague answer with no numbers or no context is incomplete.

Common Trap

If a question asks for a proportion but the table shows counts, divide by the total yourself. If it asks for a count but the table shows percentages, multiply the percentage (as a decimal) by n. Read carefully so you report the form the question actually wants.

Common Misconceptions

• Frequency and relative frequency are not the same. Frequency is a count. Relative frequency is a proportion (count divided by total). Mixing them up changes your answer.
• Relative frequencies sum to 1, not to n. Counts sum to the sample size; proportions sum to 1.00 and percentages to 100 (allowing for small rounding differences).
• Proportions, percentages, and rates are not different information. They are the same value scaled differently, so a "rate per 1,000" and a percentage describe the exact same data.
• Tables alone do not prove a population claim. A frequency table summarizes the data you collected. It describes your sample and can suggest ideas to test later, but it does not by itself prove something about a larger population.
• Context is required. Reporting "33.3%" without naming the variable, the categories, and the cases is an incomplete description on the exam.

Worked Example

Suppose 20 students answer a survey question asking which study method they use most: flashcards, practice problems, videos, or group review.

A frequency table counts how many students chose each category:

A relative frequency table expresses each count as a proportion of the total sample size:

The relative frequencies add up to 1.00. You could describe the result in context by saying that 40% of the surveyed students chose practice problems as their main study method.

Related AP Statistics Guides

• Unit 1 Overview: Exploring One-Variable Data
• 1.1 Introducing Statistics: What Can We Learn from Data?
• 1.8 Graphical Representations of Summary Statistics
• 1.9 Comparing Distributions of a Quantitative Variable
• 1.4 Representing a Categorical Variable with Graphs
• 1.6 Describing the Distribution of a Quantitative Variable