Frequency analysis

Which means of transportation do college students use most to get to school? Which is more common, the subway or the bus? And if some students didn’t answer the survey, how should the percentages be calculated?

In cases like these, the method that summarizes the number and proportion of people in each response category is frequency analysis. It goes beyond simply counting responses and lets you see how much of all respondents each category accounts for.

Where descriptive statistics broadly summarize distributions, including those of numeric variables, with the mean, median, standard deviation and so on, frequency analysis is particularly suited to finding the counts and proportions of categorical responses. To compare the relationship between two different categorical variables, you can also look at a crosstab analysis.

It is often used in surveys, education research, user research and customer profiling. In this article we’ll use a survey on how students commute to cover building frequency tables, percentages and handling missing values, reading graphs, cautions with multiple-response questions, how frequency analysis differs from other analyses, and how to report results.

Frequency analysis counts the number of observations in each category and, as needed, calculates percentages, cumulative frequencies and cumulative percentages. In a single-choice survey each respondent falls into one category, but in a multiple-choice survey one person can fall into several categories, so the basis of the calculation has to be different.

An average alone cannot describe the distribution of responses for commuting modes like subway, bus and walking. A frequency table shows at a glance which categories were chosen often and which were rare, and it also lets you check how many people didn’t respond.

① Organize the response categories

Set up five categories for commuting mode: subway, bus, walking, car and bicycle. If the same meaning was entered in several different ways, standardize it before counting.

② Count the frequency of each category

Count the number of students who chose each commuting mode. If 90 students chose the subway, the frequency for subway is 90.

③ Decide the denominator for the percentages

Based on all 252 respondents, the subway percentage is 90 ÷ 252 × 100 ≈ 35.7%. Based on the 240 students left after excluding missing responses, the valid percentage is 90 ÷ 240 × 100 = 37.5%.

④ If needed, check the order and cumulative percentages

For ordered categories such as a 1–5 satisfaction rating, cumulative percentages are useful. For nominal categories with no natural order, such as commuting mode, adding cumulative percentages has little interpretive meaning.

We asked 252 hypothetical college students to choose the one commuting mode they mainly use. 240 answered the question and 12 did not respond. The following table and graph are a hypothetical example for teaching.

Valid percentages are calculated with the 240 students left after excluding non-responses as the denominator. Because the percentages in the table are rounded to one decimal place, they may differ slightly from the totals.

When reading results, always check what was used as the denominator next to each percentage. In particular, when there are missing responses, the percentage and the valid percentage differ.

In this example, 12 of the 252 students did not report their commuting mode. When describing all respondents, you should also report the 12 non-responses (4.8%). To describe only the distribution of students who actually chose a commuting mode, use percentages based on the 240 valid responses.

For nominal data such as commuting mode, a bar chart is easy to read. You can sort the bars starting from the most common, the subway, or arrange them in the order of the answer choices. A pie chart shows shares at a glance, but it is hard to compare when there are many categories or the differences in proportions are small.

The axis of a bar chart usually starts at 0, and labeling the graph with frequencies or percentages helps reduce exaggerated visual differences.

For example, satisfaction from 1 (very dissatisfied) to 5 (very satisfied) is ordered, so you can add up the shares of responses from 1 to 3 to see the share of “neutral or below.” For subway and bus, on the other hand, there is no statistically defined order of which category comes first, so there is no reason to interpret cumulative percentages.

In a separate hypothetical survey, 120 students were asked to choose all their reasons for using the library. 75 chose individual study, 60 team projects and 45 borrowing books, for a total of 180 selections. Because one person could choose several items, the respondent-based percentages can add up to more than 100%.

The percentage of respondents is the number of people who chose an item divided by 120, and the percentage of responses is the number of selections divided by the total of 180. The two percentages answer different questions, so the results table must always show the denominator.

Were responses with the same meaning split into several categories?

Counting “subway” and “metro” as separate categories can misrepresent how many people actually use it. Check the coding rules before the analysis.

Whom does the sample represent?

If you surveyed only students taking a class in one particular department, it is hard to generalize to the distribution of commuting modes among all college students.

Did you state the denominator and how missing values were handled?

The number of respondents can differ from question to question, so show the number of valid responses and report missing responses separately.

Did you add up multiple responses as if they were single responses?

With multiple choice, it is natural for respondent-based percentages to total more than 100%. First check whether the question allows more than one choice.

Frequency analysis is generally used to describe data rather than to test hypotheses. To claim differences between groups or associations between variables, you need additional analyses suited to the study design.

A report should give the question analyzed, how the categories were defined, the total sample size, the number of missing responses, the number of valid responses, and the frequency and percentage of each category. For a multiple-response question, you must state separately whether percentages are based on respondents or on responses.

Frequency analysis is especially useful as the starting point of a study. Checking which categories are rarely observed also helps you judge whether later crosstab analyses and tests can be applied.

In the commuting example, the subway was the most common with 90 users, but its share was 35.7% based on all 252 students and 37.5% based on the 240 students left after excluding non-responses. Presenting frequencies and percentages together and keeping missing responses separate is what makes an accurate description of the same data possible. A bar chart helps you compare the distribution across categories quickly.

How the results are calculated also depends on the structure of the question. For ordered satisfaction responses you can interpret cumulative percentages, but for nominal categories such as commuting mode they generally have no meaning. In multiple-response questions where several items can be chosen, the respondent-based percentages can total more than 100%. Frequency analysis is a basic step for describing distributions; to check differences between groups or statistical associations, you need to add a crosstab analysis or an appropriate test.