TwoWay Tables
Statisticians use twoway tables and segmented bar charts to
examine the relationship between two categorical variables.
Entries in the cells of a twoway table can be displayed as
frequency
counts or as
relative
frequencies (just like a
oneway
table). Or they can be displayed graphically as a segmented
bar chart.
TwoWay Frequency Tables
Below, the twoway table shows the favorite leisure activities
for 50 adults  20 men and 30 women. Because entries in the table
are frequency counts, the table is a
frequency
table.

Dance 
Sports 
TV 
Total 
Men 
2 
10 
8 
20 
Women 
16 
6 
8 
30 
Total 
18 
16 
16 
50 
Entries in the "Total" row and "Total" column are called
marginal frequencies or the
marginal distribution. Entries in the body
of the table are called joint frequencies.
If we looked only at the marginal frequencies in the Total row,
we might conclude that the three activities had roughly equal appeal.
Yet, the joint frequencies show a strong preference for dance among
women; and little interest in dance among men.
TwoWay Relative Frequency Tables
The table above used frequency counts to describe preferences for leisure activities. Alternatively, we could have used relative frequencies, like percentages or proportions,
to describe the same data.
When we use relative frequencies in a twoway table, table entries are are called
conditional frequencies or the conditional distribution. Here is a version of the leisureactivity table with proportions in the table cells.

Dance 
Sports 
TV 
Total 
Men 
0.04 
0.20 
0.16 
0.40 
Women 
0.32 
0.12 
0.16 
0.60 
Total 
0.36 
0.32 
0.32 
1.00 
Relative Frequency for the Whole Table
Twoway tables can show relative frequencies for the
whole table, for rows, or for columns. The table above shows relative frequencies for the whole table.
The following table shows relative frequencies (proportions) for rows.

Dance 
Sports 
TV 
Total 
Men 
0.10 
0.50 
0.40 
1.00 
Women 
0.53 
0.20 
0.27 
1.00 
Total 
0.36 
0.32 
0.32 
1.00 
Relative Frequency for Table Rows
And, the next table show relative frequencies (proportions, again) for columns.

Dance 
Sports 
TV 
Total 
Men 
0.11 
0.62 
0.50 
0.40 
Women 
0.89 
0.38 
0.50 
0.60 
Total 
1.00 
1.00 
1.00 
1.00 
Relative Frequency for Table Columns
Each type of relative frequency table makes a different contribution to
understanding the relationship between gender and preferences for
leisure activities. For example, the "Relative Frequency for Rows"
table most clearly shows the probability that each gender will
prefer a particular leisure activity. It is easy
to see that the probability that a man will prefer dance is
10%; the probability that a woman will prefer dance is 53%; the
probability that a man will prefer sports is 50%; and so on.
Segmented Bar Charts
Sometimes, relationships are easier to detect when they are
displayed graphically in a segmented bar chart.
A segmented bar chart has one bar for each level of a
categorical variable. Each bar is divided into "segments", such
that the length of each segment indicates proportion or percentage
of observations in a second variable.
The segmented bar chart above uses data from the
"Relative Frequency for Rows" table that we discussed earlier. It shows that women
have an strong preference for dance; while men seldom make dance
their first choice. Men are most likely to prefer sports, but
the degree of male preference for sports over TV is not great.
Test Your Understanding
Problem
A public opinion survey explored the relationship between age
and support for increasing the minimum wage. The results are
summarized below in a twoway frequency table.

For 
Against 
No opinion 
Total 
21  40 
25 
20 
5 
50 
41  60 
20 
35 
20 
75 
Over 60 
55 
15 
5 
75 
Total 
100 
70 
30 
200 
In the 21 to 40 age group, what percentage supports increasing
the minimum wage?
(A) 12.5%
(B) 20%
(C) 25%
(D) 50%
(E) 75%
Solution
The correct answer is (D). A total of 50 people in the 21 to 40 age
group were surveyed. Of those, 25 were for increasing the
minimum wage. Thus, half of the respondents in the 21 to
50 age group (50%) supported increasing the minimum wage.