Notice that the ratios have different denominators between the relative frequencies of the columns and rows.
See solution.
Practice makes perfect
Let's consider an example two-way frequency table that shows the relationship between favorite type of coffee and gender.
Favorite Type of Coffee
Gender
Cappuccino
Latte
Espresso
Total
Male
23
36
41
100
Female
25
38
12
75
Total
48
74
53
175
Now, let's assume that we want to determine what percent of men like espresso the most compared to the percent of women who prefer espresso. This means that we need to create a relative frequency table by rows. To do this, let's divide each data value by the row total and then multiply it by 100 %.
Favorite Type of Coffee
Gender
Cappuccino
Latte
Espresso
Total
Male
23/100* 100 %=23 %
36/100* 100 %=36 %
41/100* 100 %=41 %
100/100* 100 %=100 %
Female
25/75* 100 %≈ 33 %
38/75* 100 %≈ 51 %
12/75* 100 %≈ 16 %
75/75* 100 %=100 %
Total
48/175* 100 %≈ 28 %
74/175* 100 %≈ 42 %
53/175* 100 %≈ 30 %
175/175* 100 %=100 %
We can see that the percentages in columns do not add up to 100 %. This is because we used row totals as the denominator of the relative frequency. Now let's assume that we want to determine whether women or men like lattes more. To do this, we will divide each data value by the column total and multiply it by 100 %.
Favorite Type of Coffee
Gender
Cappuccino
Latte
Espresso
Total
Male
23/48* 100 %≈ 48 %
36/74* 100 %≈ 49 %
41/53* 100 %≈ 77 %
100/175* 100 %≈ 57 %
Female
25/48* 100 %≈ 52 %
38/74* 100 %≈ 51 %
12/53* 100 %≈ 23 %
75/175* 100 %≈ 43 %
Total
48/48* 100 %=100 %
74/74* 100 %=100 %
53/53* 100 %=100 %
175/175* 100 %=100 %
This time we created a relative frequency by columns. We can see that the percentages in rows do not add up to 100 % because we used column totals as the denominator of the ratio.
