Summary Statistics for Two Categorical Variables

A school district surveys 400 juniors: do you have a part-time job, and how much do you usually sleep on a school night? Ask "what proportion of students with jobs sleep less than 7 hours?" and "what proportion of students who sleep less than 7 hours have jobs?" and you get two different numbers from the same cell of the same table. Every proportion from a two-way table is a count divided by some total, and the whole topic comes down to choosing the right total.

Joint relative frequency: one cell out of everyone

A joint relative frequency is a cell count divided by the grand total. It describes individuals who are in a particular category of both variables, as a share of the whole table:

The proportion of all 400 juniors who have a job and sleep less than 7 hours is $\frac{72}{400} = 0.18$. The word "and" (or a phrase that pins down both variables, like "working students who sleep less than 7 hours, out of all juniors surveyed") is the signal.

Divide every cell by 400 and you get a table of joint relative frequencies. Its six cells add to 1, because each of the 400 students sits in exactly one cell:

Marginal relative frequency: one variable, ignoring the other

A marginal relative frequency is a row total or a column total divided by the grand total. The name comes from where those totals sit: in the margins of the table. It describes one variable alone, as if the other had never been recorded:

Of all 400 juniors, $\frac{160}{400} = 0.40$ have a part-time job, and $\frac{132}{400} = 0.33$ sleep less than 7 hours. The marginal relative frequencies for one variable form that variable's distribution (0.40 and 0.60 for job; 0.33, 0.46 and 0.21 for sleep), and they appear as the totals row and column of the joint table above.

Conditional relative frequency: restrict to one group first

A conditional relative frequency is computed inside one row or one column only. You first restrict attention to a single category of one variable (the condition), then ask what share of that group falls in a category of the other:

Both of the opening questions are conditional, but they restrict to different groups:

• "Of the students with a job, what proportion sleep less than 7 hours?" The group is the Yes row, 160 students: $\frac{72}{160} = 0.45$.
• "Of the students who sleep less than 7 hours, what proportion have a job?" The group is the first column, 132 students: $\frac{72}{132} \approx 0.545$.

The numerator is the same 72 students. The denominators differ because the questions are about different groups. The words that set the condition ("of the," "among," "out of those who," "what percent of ... are") name the denominator. Read them, say the group out loud, and divide by its total. Topic 2.6 turns this same idea into conditional probability, $P(A \mid B)$.

Using conditional relative frequencies to judge association

Topic 2.1 compared groups by the proportions within each group. Those are conditional relative frequencies, and the full set for one group is a conditional distribution. For the juniors, condition on having a job:

The two conditional distributions differ: 45% of working juniors sleep less than 7 hours, compared with 25% of non-working juniors, and working juniors are less likely to sleep more than 8 hours (0.15 against 0.25). Having a job and sleep amount are associated in this sample. You can state the size of the difference too: the proportion sleeping less than 7 hours is $0.45 - 0.25 = 0.20$ higher for working students, or $\frac{0.45}{0.25} = 1.8$ times as large.

There is a reason the marginal proportion 0.33 sits between 0.45 and 0.25. The marginal is a weighted average of the conditionals, weighted by group size:

(The two products are the joint relative frequencies 0.18 and 0.15 from the joint table.) This gives a clean description of no association: if every group's conditional proportion is the same, the weighted average equals that common value, so each conditional distribution matches the marginal distribution. When the conditional distributions differ from each other, and so from the marginal, the variables are associated.

This survey shows that job and sleep go together, not why: working may cut into sleep, or students who stay up late anyway may be the ones who take evening shifts.

Worked examples

Where students lose points