A two-way table is a frequency table that displays data collected from one source that belongs to two different categories. One category of data is represented by rows and the other is represented by columns. We want to construct a two-way table for the given data and interpret the relative frequencies by column. To do so, we will follow four steps.

Construct an empty table with the appropriate column and row headers.
Fill in the table with the given values.
Find the rest of values.
Find and interpret the relative frequencies by column.

Let's do these things one at a time.

Constructing the Table

We can start by looking at the given Venn diagram.

In the universal set, which includes sets for jogging and aerobics, 24 students only jog, 13 students only do aerobics, 8 students do both activities, and 5 students do not participate in either jogging or aerobics.

We know that the diagram shows the number of students that exercise in different ways. Notice that some students jog, while others do aerobics. This information is enough to determine the appropriate column and row headers for our table.

	Jogging	No Jogging	Total
Aerobics
No Aerobics
Total

Filling in the Table

To fill in the table, we can look at the diagram once again.

Notice that

24

students jog and do no aerobics,

13

students do aerobics and do not jog, and

8

students both jog and do aerobics. We can also see that

5

students neither jog nor do aerobics. Let's write the given information in our table.

	Jogging	No Jogging
Aerobics	$8;$	$13;$
No Aerobics	$24;$	$5;$
Total

Finding the Missing Values

Let's calculate the sum of each column and each row to find the missing values.

Students Who Do Aerobics : Students Who Do No Aerobics : Students Who Jog : Students Who Do Not Jog : 8 + 13 = 21 24 + 5 = 29 8 + 24 = 32 13 + 5 = 18

We can write this newly-obtained information in our table.

	Jogging	No Jogging	Total
Aerobics	$8;$	$13$	$21;$
No Aerobics	$24;$	$5;$	$29$
Total	$32;$	$18;$

Finally, we have two ways of calculating the grand total. We can add the number of students who do

aerobics

to the number of students who do

no

aerobics,

or we can add the students who

jog

to the students who do

not

jog .

These two numbers must be the same!

\underline{Grand Total} 21 + 29 = 50 32 + 18 = 50 ✓

Now, we can complete our table!

	Jogging	No Jogging	Total
Aerobics	$8;$	$13$	$21;$
No Aerobics	$24;$	$5;$	$29$
Total	$32;$	$18;$	$50$

Finding and Interpreting the Relative Frequencies by Column

To find the relative frequencies by column, we calculate the ratios of each value and the total in that column. We will round the results to the nearest hundredth.

	Jogging	No Jogging	Total
Aerobics	$8; \frac{8}{3 2} = 0.25$	$13; \frac{1 3}{1 8} \approx 0.72$	$21; \frac{2 1}{5 0} = 0.42$
No Aerobics	$24; \frac{2 4}{3 2} = 0.75$	$5; \frac{5}{1 8} \approx 0.28$	$29; \frac{2 9}{5 0} = 0.58$
Total	$32; \frac{3 2}{3 2} = 1.00$	$18; \frac{1 8}{1 8} = 1.00$	$50; \frac{5 0}{5 0} = 1.00$

Looking at the relative frequencies, we can say that most of the students who jog do no aerobics. Most of the students who do not jog do aerobics.

Exercise

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