1.3 - Tabular Representation and Summary Statistics for One Categorical Variable

Overview

When a variable is categorical, the first thing to do is count. This topic shows how to organize categorical data in frequency tables and relative frequency tables, how proportions, percentages, and ratios give the same information in different forms, and how to use counts and proportions to support or reject a claim.

Frequency tables

A frequency table shows the number of observational units in each category of a categorical variable. The count for a category is its frequency, and the frequencies add up to the total number of units.

To build one:

List every category.

Count (tally) the units in each category.

Check that the counts add up to the total.

Relative frequency tables

A relative frequency table shows the proportion of observational units in each category:

\text{relative frequency} = \frac{\text{frequency of the category}}{\text{total number of observational units}}

Each relative frequency is between 0 and 1, and together they add up to 1. After rounding, the sum might come out as 0.999 or 1.001, which is fine. A sum like 0.85 means a category or a count is missing.

Relative frequencies are often more useful than counts, because they show how big each category is compared to the whole, and they let you compare groups of different sizes.

Example 1: Building both tables

The 30 students in a homeroom were asked, "How do you usually get to school?" Here are their answers:

Car, Bus, Car, Walk, Car, Bike, Bus, Car, Other, Walk, Bus, Car, Car, Bike, Bus, Walk, Car, Bus, Car, Other, Bike, Bus, Car, Walk, Bus, Car, Bike, Bus, Walk, Car

Make a frequency table and a relative frequency table.

Step 1: Count the answers in each category. Then check the total: 11 + 8 + 5 + 4 + 2 = 30.

Step 2: Divide each count by 30. For example, for Car:

\frac{11}{30} \approx 0.367

Here are both tables in one, with percents added:

How students get to school	Frequency	Relative frequency	Percent
Car	11	0.367	36.7%
Bus	8	0.267	26.7%
Walk	5	0.167	16.7%
Bike	4	0.133	13.3%
Other	2	0.067	6.7%
Total	30	1.001 (rounding)	100.1% (rounding)

Answer: Car is the most common way these 30 students get to school (11 students, or about 36.7%), followed by bus (8 students, about 26.7%). Only 2 students, about 6.7%, use some other way. The relative frequencies add to 1.001 only because each one was rounded to three decimal places.

Proportions, percentages, and ratios

Proportions, relative frequencies, percentages, and ratios all give the same information. They're different ways of writing "this part out of the whole."

Proportion (relative frequency): $\frac{11}{30} \approx 0.367$

Percentage: multiply the proportion by 100, so $0.367 \times 100 = 36.7\%$

Ratio, part to whole: 11 out of every 30 students ride in a car.

Ratio, part to part: car riders to bus riders is 11 to 8, or 11 : 8.

Be careful with part-to-part ratios. If the ratio of car riders to everyone else is 11 : 19, the proportion who ride in a car is

\frac{11}{11 + 19} = \frac{11}{30}

You add the parts to get the whole before dividing.

You can also work backward from a proportion to a count:

\text{count} = \text{relative frequency} \times \text{total}

Example 2: The same information in different forms

In a survey of 250 students, the ratio of students who own a pet to students who don't is 3 : 2. Write the proportion who own a pet as a decimal and a percent, and find how many surveyed students own a pet.

Step 1: Turn the part-to-part ratio into a proportion. Out of every

3 + 2 = 5

students, 3 own a pet:

\frac{3}{3 + 2} = \frac{3}{5} = 0.6

Step 2: Write it as a percent:

0.6 \times 100 = 60\%

Step 3: Find the count:

0.6 \times 250 = 150

Answer: 3 out of every 5, a proportion of 0.6, or 60% of the surveyed students own a pet. That's 150 of the 250 students. All four forms say the same thing.

Using counts and proportions to justify claims

Counts and relative frequencies reveal information you can use to justify claims about a variable in context. To check a claim:

Find the counts or proportions that the claim is about.

Compare them to exactly what the claim says.

State whether the data support the claim, quote the numbers, and say it in context.

Watch the wording. These words have exact meanings:

Majority means more than half, so a proportion greater than 0.5.

Most common means the category with the highest frequency (the mode). It can be far below half. In Example 1, Car is the most common answer, but it's only 0.367 of the homeroom, not a majority.

Twice as many or 3 out of 10 are claims you check by division.

When you compare groups of different sizes, compare relative frequencies, not counts. Topic 1.4 shows this with graphs. And remember that data from a sample describe that sample. You can only generalize to a larger population when the sample was chosen at random (Topics 1.10 and 1.11).

Example 3: Checking claims about a survey (AP-style)

A city council randomly surveyed 1,250 residents and asked, "What should be the council's top priority next year?" The results:

Top priority	Frequency
Roads and transit	410
Housing costs	385
Schools	255
Public safety	200
Total	1,250

(a) Make a relative frequency table.

(b) A council member says, "A majority of the surveyed residents chose roads and transit." Do the data support this claim?

(c) Another member says, "More than 6 out of 10 surveyed residents chose either roads and transit or housing costs." Do the data support this claim?

Solution

(a) Divide each count by 1,250:

Top priority	Relative frequency
Roads and transit	0.328
Housing costs	0.308
Schools	0.204
Public safety	0.160
Total	1.000

(b) No. Roads and transit was the most common choice, but its relative frequency is

\frac{410}{1250} = 0.328

which is less than 0.5. About 1 in 3 surveyed residents chose roads and transit, not more than half.

\frac{410 + 385}{1250} = \frac{795}{1250} = 0.636

That's 63.6% of the surveyed residents, which is more than 6 out of 10 (60%).

Common mistakes

Dividing by the wrong total. A relative frequency is the category count over the total for the whole data set.

Confusing "most common" with "majority." The most common category can be well under half.

Comparing counts across groups of different sizes. Use relative frequencies.

Misreading part-to-part ratios. A 3 : 2 ratio means 3 out of 5, not 3 out of 2.

Worrying about sums like 1.001. That's rounding. A sum far from 1 is a real error.

Making claims without numbers. Say "0.328 is less than 0.5," not just "it's not a majority."

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