Two-Way Tables: Finding Patterns in Categorical Data

Sorting by two categories at once

A two-way frequency table counts how many fall into each combination of two categories.

Students asked whether they play a sport and whether they play an instrument:

	Instrument	No instrument	Total
Sport	$30$	$50$	$80$
No sport	$45$	$25$	$70$
Total	$75$	$75$	$150$

Each inner cell is one combination. The edges hold the marginal totals, and the corner holds the grand total.

Two checks worth running before anything else:

• each row’s cells add to its row total: $30 + 50 = 80$ ✓
• the row totals and column totals both add to $150$ ✓

If either fails, the table is wrong and nothing computed from it will be right.

Reading a cell

Every inner cell answers a “both” question.

Question	Cell	Answer
Play sport and instrument?	top left	$30$
Play sport, no instrument?	top right	$50$
Play neither?	bottom right	$25$
Play sport at all?	row total	$80$

Why raw counts mislead

Do sporty students play instruments less often?

Comparing the two “instrument” counts directly: $30$ against $45$. It looks like sporty students play instruments less.

But the two groups are not the same size — $80$ against $70$ — so the counts are not comparable. Convert each row to percentages of its own total:

	Instrument	No instrument
Sport	$30/80 = 37.5\%$	$62.5\%$
No sport	$45/70 = 64.3\%$	$35.7\%$

Now the comparison is fair, and the gap is larger than the raw counts suggested: $37.5\%$ against $64.3\%$.

Dividing each row by its own total is what puts unequal groups on equal footing. It is the same reasoning as a unit rate: reduce both to a common basis, then compare.

Deciding whether there is an association

Two categories are associated when knowing one changes what you would expect of the other.

The test is whether the row percentages differ:

• Similar rows — no association. Knowing whether a student plays sport tells you nothing about the instrument.
• Different rows — association. Here $37.5\%$ against $64.3\%$ is a substantial gap, so the two are associated.

Compare against a table with no association:

	Instrument	No instrument
Sport	$50\%$	$50\%$
No sport	$52\%$	$48\%$

Near-identical rows mean the categories are close to independent.

Row or column percentages

Which you compute depends on the question.

Question	Divide by
Of sporty students, how many play an instrument?	the row total
Of instrument players, how many play sport?	the column total

These give different numbers and answer different questions:

$$\frac{30}{80} = 37.5\% \qquad \frac{30}{75} = 40\%$$

Same cell, two denominators. Read the question to see which group is being asked about, and divide by that group’s total.

The three kinds of relative frequency

Every relative frequency in a two-way table is one count divided by another. Which two counts decides its name.

Kind	Divide	Answers
joint	a cell by the grand total	what share of everyone is in both groups
marginal	a row or column total by the grand total	what share of everyone is in one group
conditional	a cell by its row or column total	within one group, what share is in the other

Using the sport and instrument table, with $150$ students in all:

$$\text{joint: } \frac{30}{150} = 20\% \qquad \text{marginal: } \frac{80}{150} \approx 53.3\% \qquad \text{conditional: } \frac{30}{80} = 37.5\%$$

The denominator names the kind. Divide by the grand total and you get a joint or marginal frequency. Divide by a row or column total and you get a conditional one.

Dividing every cell by the grand total gives a whole table of joint relative frequencies, with the marginal ones along its edges:

	Instrument	No instrument	Total
Sport	$20\%$	$33.3\%$	$53.3\%$
No sport	$30\%$	$16.7\%$	$46.7\%$
Total	$50\%$	$50\%$	$100\%$

The whole table adds to $100\%$, which is the same check as before in a new unit.

Association is judged with conditional frequencies. Joint frequencies mix up two things at once — how common a combination is and how big each group is — so comparing them repeats the raw-count problem from earlier.

Why association is not cause

A table can show two categories occurring together more often than chance would suggest. It cannot show that one produces the other.

Sport and instrument playing are associated here, and nothing in the table says which way round — or whether some third thing, such as how much free time a student has, drives both.

That limit is not a flaw in the method. It is what the method measures: a two-way table records how often things co-occur, and co-occurrence is not causation.

Worked examples

Common mistakes

Practice problems

Use this table for questions 1 to 6.

	Bus	Walk	Total
Grade 7	$40$	$60$	$100$
Grade 8	$75$	$25$	$100$
Total	$115$	$85$	$200$