A conditional relative frequency is the ratio of a joint relative frequency to the marginal relative frequency. Consider the joint and marginal frequency table.

	In Favor of Planting a Community Garden
Gender	Yes	No	Total
Girls	$0.386$	$0.092$	$0.478$
Boys	$0.332$	$0.190$	$0.522$
Total	$0.717$	$0.283$	$1$

We want to construct a two-way table that shows the conditional relative frequencies based on the totals of the rows. To do so, we will divide each joint frequency by its corresponding marginal row frequency. Let's do it!

	In Favor of Planting a Community Garden
Gender	Yes	No
Girls	$\frac{0 . 3 8 6}{0 . 4 7 8} \approx 0.807$	$\frac{0 . 0 9 2}{0 . 4 7 8} \approx 0.193$
Boys	$\frac{0 . 3 3 2}{0 . 5 2 2} \approx 0.635$	$\frac{0 . 1 9 0}{0 . 5 2 2} \approx 0.365$

In the context of the exercise, we can say that

80.7 %

of the girls are in favor of planting a community garden and

19.3 %

are against. Additionally,

63.5 %

of the surveyed boys are in favor and

36.5 %

are against.

Alternative Solution

A shorter method

It is also possible to find conditional frequencies using only the original two-way frequency table.

	In Favor of Planting a Community Garden
Gender	Yes	No	Total
Girls	$71$	$17$	$88$
Boys	$61$	$35$	$96$
Total	$132$	$52$	$184$

We can divide the number of responses in each cell by the conditional total — either the column total or the row total depending on which condition is being considered. In this case, we want to know the conditional frequencies by row.

	In Favor of Planting a Community Garden
Gender	Yes	No
Girls	$\frac{7 1}{8 8} \approx 0.807$	$\frac{1 7}{8 8} \approx 0.193$
Boys	$\frac{6 1}{9 6} \approx 0.635$	$\frac{3 5}{9 6} \approx 0.365$

To see why this works, let's consider just the first cell from the original solution.

\frac{0 . 3 8 6}{0 . 4 7 8} \approx 0.807

Where did the values on the left-hand side of this approximation come from? Recall that they are the joint relative frequency and the marginal relative frequency.

\frac{Joint relative frequency}{Marginal relative frequency} \to \frac{0 . 3 8 6}{0 . 4 7 8}

The joint relative frequencies are found by dividing the individual cell values by the grand total and the marginal relative frequencies are found by dividing the row and column totals by the grand total. For this one cell, we had the following numbers.

\frac{7 1}{1 8 4} \approx 0.368 and \frac{8 8}{1 8 4} \approx 0.478

Let's now see what happens when we find the conditional relative frequency using the fractions rather than using the rounded decimal numbers.

\frac{Joint relative frequency}{Marginal relative frequency}

SubstituteValues

Substitute values

\frac{( \frac{7 1}{1 8 4} )}{( \frac{8 8}{1 8 4} )}

DivFracByFracSameDenom

$\frac{a / 1 8 4}{b / 1 8 4} = \frac{a}{b}$

\frac{7 1}{8 8}

We end up with the same fraction that was used to calculate the conditional frequency in the alternative solution method!

Exercise