Construct two-way tables for categorical data and use relative frequencies to describe association

Learning goal: Understand that patterns of association can also be seen in bivariate categorical data, by displaying frequencies and relative frequencies in a two-way table. Construct a two-way table summarizing data on two categorical variables collected from the same subjects, and use relative frequencies calculated for rows or for columns to describe a possible association between the two variables.

Categorical, not measured. The earlier lessons handled measurement data such as hours and centimeters, where a scatter plot shows the pattern. Many questions instead involve categorical variables, whose values are labels rather than numbers: whether a student has a curfew on school nights, yes or no; whether a student has assigned chores at home, yes or no. Two categorical variables recorded for the same subjects are bivariate categorical data. A scatter plot is the wrong display for "yes" and "no": the values could be coded as numbers, but every student would then land on one of only four points, and what matters, how many students fall in each combination, would be hidden. The display that shows those counts directly is the two-way table.

The data set. Suppose a class of 40 students answers two questions: "Do you have a curfew on school nights?" and "Do you have assigned chores at home?" Each student answers both, so every student is counted exactly once in the table. The numbers are hypothetical and invented for teaching. The responses come to this: 20 students have a curfew and chores; 5 have a curfew but no chores; 6 have no curfew but do have chores; 9 have neither.

Constructing the table. A two-way table has one variable's categories as row labels and the other's as column labels. Put curfew in the rows and chores in the columns. Each inside cell holds the frequency, the count of students who belong to that row's category and that column's category at the same time. The cell for curfew-yes and chores-yes holds 20; curfew-yes and chores-no holds 5; curfew-no and chores-yes holds 6; curfew-no and chores-no holds 9.

The margins hold totals. Row totals: 20 + 5 = 25 students have a curfew, and 6 + 9 = 15 do not. Column totals: 20 + 6 = 26 students have chores, and 5 + 9 = 14 do not. The grand total is 25 + 15 = 40, and it must also equal 26 + 14 = 40. The two ways of reaching the grand total are a check on the arithmetic only: both routes add the same four cells, so if they disagree, a total was added or copied wrongly. They cannot detect a student counted in two cells or a wrong cell entry, because those errors change both routes equally. To catch those, compare the grand total with the number of students who answered (40 here) and, if it differs, go back to the individual records.

Why raw counts can mislead. Twenty curfew students have chores and only six no-curfew students have chores. It is tempting to read that as strong evidence that curfews go with chores. But 25 students have a curfew and only 15 do not, so the curfew group would produce larger counts even if chores were spread evenly. Comparing groups of different sizes needs proportions, not counts.

Row relative frequencies. A relative frequency is a frequency divided by a total. Row relative frequencies divide each cell by its row total, answering the question "of the students in this row, what fraction fall in each column?" For the curfew-yes row: 20 ÷ 25 = 0.80, or 80%, have chores and 5 ÷ 25 = 0.20, or 20%, do not. For the curfew-no row: 6 ÷ 15 = 0.40, or 40%, have chores and 9 ÷ 15 = 0.60, or 60%, do not. Each row's relative frequencies add to 100%, which is a check.

Now the comparison is fair. Among students with a curfew, 80% have chores; among students without a curfew, 40% have chores. The percentage with chores is twice as high in the curfew group. In this class, students who have a curfew also tend to have chores. That is what association between two categorical variables means: the distribution of one variable differs from one category of the other variable to the next.

Column relative frequencies. Dividing by column totals instead asks the reverse question: "of the students with chores, what fraction have a curfew?" Among the 26 students with chores, 20 ÷ 26 is about 76.9% with a curfew and 6 ÷ 26 about 23.1% without. Among the 14 students without chores, 5 ÷ 14 is about 35.7% with a curfew and 9 ÷ 14 about 64.3% without. Rounded to a tenth of a percent, each column still sums to 100%. The association shows again from this direction: students with chores are much more likely to have a curfew than students without chores. Rows and columns give different numbers because they answer different questions; both are valid, and the question decides which to compute.

What no association looks like. In a second hypothetical class of 40, 16 students wear glasses and 24 do not. Of the glasses group, 4 are left-handed and 12 right-handed; of the no-glasses group, 6 are left-handed and 18 right-handed. Row relative frequencies: 4 ÷ 16 = 25% left-handed among glasses wearers, and 6 ÷ 24 = 25% left-handed among the rest. The rows have the same distribution, so knowing whether a student wears glasses tells you nothing about handedness. That is no association. Notice that the raw counts, 4 and 6, differ; only the relative frequencies reveal that the difference comes entirely from group size.

Evidence, not proof. The curfew data are evidence of association in this class: 80% against 40% is a large difference on 40 students. A difference of 52% against 48% would be weak evidence, because the responses of a few students could erase it. And association is not cause. The table cannot show that curfews lead to chores or chores to curfews; a common explanation, such as household rules that come together, is just as consistent with the numbers. The right sentence is: "in this class, students with a curfew tend to have chores, and the two variables appear associated."

Building a table from raw responses. When responses arrive as a list, tally each subject into exactly one cell. Twelve hypothetical students answered whether they bike to school and whether they play an instrument: 1 yes-yes, 2 yes-no, 3 no-yes, 4 yes-yes, 5 no-no, 6 no-yes, 7 yes-yes, 8 no-no, 9 yes-no, 10 no-yes, 11 yes-yes, 12 no-no. The cells are bike-yes and instrument-yes 4, bike-yes and instrument-no 2, bike-no and instrument-yes 3, bike-no and instrument-no 3. Rows total 6 and 6, columns total 7 and 5, grand total 12. Row relative frequencies for instrument-yes: 4 ÷ 6 is about 66.7% among bikers and 3 ÷ 6 = 50% among non-bikers. There is a difference in the direction of association, but on six students per row it is weak evidence: one changed answer moves a row by nearly 17 percentage points.

Worked table with all totals checked: Thirty hypothetical students were asked whether they have a pet and whether they play a school sport. Twelve have a pet, and 9 of those play a sport; 18 have no pet, and 9 of those play a sport. Cells: pet-yes and sport-yes 9, pet-yes and sport-no 12 − 9 = 3, pet-no and sport-yes 9, pet-no and sport-no 18 − 9 = 9. Column totals: sport 9 + 9 = 18, no sport 3 + 9 = 12. Grand total by rows 12 + 18 = 30 and by columns 18 + 12 = 30, so the table is consistent.

Worked row relative frequencies and a conclusion: In that table, 9 ÷ 12 = 75% of pet owners play a sport and 9 ÷ 18 = 50% of non-owners play a sport. The same raw count, 9, is a larger share of the smaller group. The proportion playing a sport is 25 percentage points higher among pet owners, so in this class pet ownership and playing a sport appear associated. The table does not show that pets cause sport or sport causes pets.

Worked column relative frequencies: Using the curfew table's columns: of the 26 students with chores, 20 have a curfew, 20 ÷ 26 ≈ 0.769, or 76.9%; of the 14 without chores, 5 have a curfew, 5 ÷ 14 ≈ 0.357, or 35.7%. The check: 6 ÷ 26 ≈ 23.1% and 76.9% + 23.1% = 100%; 9 ÷ 14 ≈ 64.3% and 35.7% + 64.3% = 100%. Students with chores are far more likely to have a curfew than students without, the same association seen from the rows, expressed from the other direction.

Construct and compute: Draw the curfew and chores table with all four cells, both row totals, both column totals and the grand total. Beneath it write a second copy with row relative frequencies as percentages, and a third with column relative frequencies rounded to a tenth of a percent. Check that every row of the second table and every column of the third sums to 100%. Write one sentence describing the association and one sentence saying what the table cannot show.

Independent performance: Collect data from at least 20 classmates on two yes-or-no questions of your own, such as a curfew on school nights and assigned chores at home, recording both answers for each person so that every subject is counted once. Construct the two-way frequency table with margins and check the grand total two ways and against the number of people you asked. Compute the row relative frequencies and state which row has the higher proportion in one column and by how many percentage points. Decide, with a reason, whether the table shows evidence of association, weak evidence or no association, considering how many people are in each row. Success means the four cells sum to your number of respondents, each row of relative frequencies sums to 100%, the comparison names a percentage difference, and the conclusion uses "tend to" and "associated" rather than "causes".