Tabular Representation and Summary Statistics for One Categorical Variable

Frequency tables

A categorical variable places each observational unit in a group (see 1.2 Variables). The first thing to do with categorical data is count how many units land in each category. A frequency table lists each category with its count, called its frequency.

To build one from raw responses, list the categories, tally each response exactly once, and check that the counts add up to the number of units. A missing or doubled tally shows up right away as a total that is off.

A student council asked a random sample of 200 students at its school, "Where do you usually do your homework?" Each student gave one answer, so the 200 students are the observational units and homework location is the categorical variable. The first two columns below are the frequency table.

Relative frequency tables

A relative frequency is the proportion of all the units that fall in a category:

For the bedroom, that is $\frac{94}{200}=0.47$. A relative frequency table lists this proportion for every category, as in the third column. The relative frequencies add to 1, because every student is in exactly one category. Once the values are rounded the total can come out at 0.99 or 1.01, and that is fine.

Relative frequencies are what make groups of different sizes comparable. A larger school surveyed 400 students and found 188 who work in their bedrooms. That is twice as many students, but it is the same proportion: $\frac{188}{400}=0.47$.

Proportions, percents and ratios

A proportion, a percent and a ratio carry the same information. Of the 200 students, 26 do their homework at school. That fact can be written as a proportion, $\frac{26}{200}=0.13$, as a percent, 13%, or as a ratio, 13 out of every 100 students.

Use whichever form the question uses. What matters is that both sides of a comparison are in the same form: comparing 13% at one school with "30 students" at another compares nothing.