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In probability, tables are used to display a data set. For example, frequency tables show how often an outcome appears in a category. To represent a data set that includes two categories, another type of table is needed. This lesson will discuss how to create and interpret these tables.
### Catch-Up and Review

**Here are a few recommended readings before getting started with this lesson.**

Zosia attends North High School in Honolulu. She asked 50 students whether they prefer a chocolate bar or a piece of fruit as a lunchtime snack and whether or not they surf. She obtained the following information.

Letting A be the event that a student surfs and B be the event that a student prefers fruit as a lunchtime snack, Zosia wants to calculate the following probabilities.{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"mlmath-simple\"><\/span>"},"formTextBefore":"<span class=\"mlmath-simple\"><span class=\"variable\">P<\/span>(<span class=\"variable\">A<\/span>)<span class=\"space big-before big-after\">=<\/span><\/span>","formTextAfter":null,"answer":{"text":["0.84","\\dfrac{42}{50}","\\dfrac{21}{25}"]}}

{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"mlmath-simple\"><\/span>"},"formTextBefore":"<span class=\"mlmath-simple\"><span class=\"variable\">P<\/span>(<span class=\"variable\">B<\/span>)<span class=\"space big-before big-after\">=<\/span><\/span>","formTextAfter":null,"answer":{"text":["0.16","\\dfrac{8}{50}","\\dfrac{4}{25}"]}}

{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"mlmath-simple\"><\/span>"},"formTextBefore":"<span class=\"mlmath-simple\"><span class=\"variable\">P<\/span>(<span class=\"variable\">A<\/span>\u2223<span class=\"variable\">B<\/span>)<span class=\"space big-before big-after\">=<\/span><\/span>","formTextAfter":null,"answer":{"text":["0.96","\\dfrac{27}{28}","0.964"]}}

{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"mlmath-simple\"><\/span>"},"formTextBefore":"<span class=\"mlmath-simple\"><span class=\"variable\">P<\/span>(<span class=\"variable\">B<\/span>\u2223<span class=\"variable\">A<\/span>)<span class=\"space big-before big-after\">=<\/span><\/span>","formTextAfter":null,"answer":{"text":["0.6","\\dfrac{27}{42}","\\dfrac{9}{14}","0.64","0.642","0.643"]}}

A two-way frequency table, also known as a **two-way table**, is a table that displays categorical data that can be grouped into two categories. One of the categories is represented by the rows of the table, the other by the columns. For example, the table below shows the results of a survey in which 100 participants were asked if they have a driver's license and if they own a car.

Here, the two categories are car

and driver's license,

both with possible answers of yes

and no.

The entries in the table are called joint frequencies. Two-way frequency tables often include the total of the rows and columns. These totals are called marginal frequencies.

Totalrow and the

Totalcolumn is equal to the sum of all joint frequencies and is called the

Organizing data in a two-way frequency table can help with visualization, which in turn makes it easier to analyze and present the data. To draw a two-way frequency table, three steps must be followed.

- Determine the Categories
- Fill the Table With Given Data
- Find Any Missing Frequencies

Suppose that 53 people took part in an online survey, where they were asked whether they prefer top hats or berets. Out of the 18 males that participated, 12 of them prefer berets. Also, 15 of the females chose top hats as their preference. The steps listed above will be developed for this example.

Determine the Categories

First, the two categories of the table must be determined, after which the table can be drawn without frequencies. Here, the participants gave their hat preference and their gender, which are the two categories. Hat preference can be further divided into top hat and beret, and gender into female and male.

The total row and total column are included to write the marginal frequencies.

Fill the Table With Given Data

The given joint and marginal frequencies can now be added to the table.

Find Any Missing Frequencies

As part of a school project, Zain asked 50 people whether they are a night owl or an early bird. Zain decided to categorize the participants by their answer and age — younger than 30, or aged 30 or older. They obtained the following information.

- 11 people aged 30 or older said they are early birds.
- 23 people younger than 30 participated in the survey.
- 28 people said they are night owls.

Zain made a two-way frequency table and added this information.

Find the missing joint and marginal frequencies to help Zain complete the table!{"type":"pair","form":{"alts":[[{"id":0,"text":"<span class=\"mlmath-simple\"><span class=\"text bold\">A)<\/span><\/span>"},{"id":1,"text":"<span class=\"mlmath-simple\"><span class=\"text bold\">B)<\/span><\/span>"},{"id":2,"text":"<span class=\"mlmath-simple\"><span class=\"text bold\">C)<\/span><\/span>"},{"id":3,"text":"<span class=\"mlmath-simple\"><span class=\"text bold\">D)<\/span><\/span>"},{"id":4,"text":"<span class=\"mlmath-simple\"><span class=\"text bold\">E)<\/span><\/span>"}],[{"id":0,"text":"<span class=\"mlmath-simple\">27<\/span>"},{"id":1,"text":"<span class=\"mlmath-simple\">16<\/span>"},{"id":2,"text":"<span class=\"mlmath-simple\">22<\/span>"},{"id":3,"text":"<span class=\"mlmath-simple\">11<\/span>"},{"id":4,"text":"<span class=\"mlmath-simple\">12<\/span>"}]],"lockLeft":true,"lockRight":false},"formTextBefore":"","formTextAfter":"","answer":[[0,1,2,3,4],[0,1,2,3,4]]}

Start by finding the number of people aged 30 or older who participated in the survey. To do that, calculate the difference between the grand total 50 and 23, the number of participants younger than 30.

Start by finding the missing marginal frequency of the last column of the table, labeled **A.** Note that 50 people participated in the survey and 23 of them are younger than 30. Therefore, the number of participants aged 30 or older can be found by calculating the difference between these two values.

$A)50−23=27 $

This information can be added to the table.
With this information, the joint frequency that represents the number of night owls aged 30 or older can be calculated. Of the 27 participants aged 30 or older, 11 are early birds. Therefore, the number of night owls aged 30 or older is the difference between these two values.
$B)27−11=16 $

This information can also be added to the table.
The missing marginal frequency in the last row will now be calculated. Of the 50 participants, 28 said they are night owls. To find the number of early birds, the difference between these two values must be calculated.
$C)50−28=22 $

One more cell can be filled in!
Finally, the missing joint frequencies in the first row can be completed.
$D)E) 22−11=1128−16=12 $

The table can be completed with this information! Click on each cell to see its interpretation.
In a two-way frequency table, a joint relative frequency is the ratio of a joint frequency to the grand total. Similarly, a marginal relative frequency is the ratio of a marginal frequency to the grand total. Consider an example two-way table.

Here, the grand total is 100. The joint and marginal frequencies can now be divided by 100 to obtain the $joint$ and $marginal$ *relative* frequencies. Clicking in each cell will display its interpretation.

Zain made a two-way frequency table as a part of a school project. The table categorizes the 50 participants as night owls

or early birds

and as younger than 30 years or aged 30 or older.

To get a deeper understanding of the preferences of the participants, Zain wants to calculate the joint and marginal relative frequencies.

Help Zain complete the table!{"type":"pair","form":{"alts":[[{"id":0,"text":"<span class=\"mlmath-simple\"><span class=\"text bold\">A)<\/span><\/span>"},{"id":1,"text":"<span class=\"mlmath-simple\"><span class=\"text bold\">B)<\/span><\/span>"},{"id":2,"text":"<span class=\"mlmath-simple\"><span class=\"text bold\">C)<\/span><\/span>"},{"id":3,"text":"<span class=\"mlmath-simple\"><span class=\"text bold\">D)<\/span><\/span>"},{"id":4,"text":"<span class=\"mlmath-simple\"><span class=\"text bold\">E)<\/span><\/span>"},{"id":5,"text":"<span class=\"mlmath-simple\"><span class=\"text bold\">F)<\/span><\/span>"},{"id":6,"text":"<span class=\"mlmath-simple\"><span class=\"text bold\">G)<\/span><\/span>"},{"id":7,"text":"<span class=\"mlmath-simple\"><span class=\"text bold\">H)<\/span><\/span>"}],[{"id":0,"text":"<span class=\"mlmath-simple\">0.24<\/span>"},{"id":1,"text":"<span class=\"mlmath-simple\">0.22<\/span>"},{"id":2,"text":"<span class=\"mlmath-simple\">0.46<\/span>"},{"id":3,"text":"<span class=\"mlmath-simple\">0.32<\/span>"},{"id":4,"text":"<span class=\"mlmath-simple\">0.22<\/span>"},{"id":5,"text":"<span class=\"mlmath-simple\">0.54<\/span>"},{"id":6,"text":"<span class=\"mlmath-simple\">0.56<\/span>"},{"id":7,"text":"<span class=\"mlmath-simple\">0.44<\/span>"}]],"lockLeft":true,"lockRight":false},"formTextBefore":"","formTextAfter":"","answer":[[0,1,2,3,4,5,6,7],[0,1,2,3,4,5,6,7]]}

To obtain the joint and marginal relative frequencies, the joint and marginal frequencies must be divided by the grand total, 50.

The table below shows the joint and marginal relative frequencies.

A conditional relative frequency is the ratio of a joint frequency to either of its corresponding two marginal frequencies. Alternatively, it can be calculated using joint and marginal relative frequencies. As an example, the following data will be used.

Using the column totals, the left column of joint frequencies should be divided by 67, and the right column by 33. Since the column totals are used, the sum of the conditional relative frequencies of each column is 1.

The resulting two-way frequency table can be interpreted to obtain the following information.

- Out of all the participants with a driver's license, about $64%$ of them own a car.
- Out of all the participants with a driver's license, about $36%$ of them do not own a car.
- Out of all the participants without a driver's license, about $12%$ of them own a car.
- Out of all the participants without a driver's license, about $88%$ of them do not own a car.

Using their two-way frequency table, Zain now wants to find the conditional relative frequencies.

To do so, Zain will use the row totals.

Help Zain to complete this table!{"type":"pair","form":{"alts":[[{"id":0,"text":"<span class=\"mlmath-simple\"><span class=\"text bold\">A)<\/span><\/span>"},{"id":1,"text":"<span class=\"mlmath-simple\"><span class=\"text bold\">B)<\/span><\/span>"},{"id":2,"text":"<span class=\"mlmath-simple\"><span class=\"text bold\">C)<\/span><\/span>"},{"id":3,"text":"<span class=\"mlmath-simple\"><span class=\"text bold\">D)<\/span><\/span>"}],[{"id":0,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.48312em;vertical-align:0em;\"><\/span><span class=\"mrel\">\u2248<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">0<\/span><span class=\"mord\">.<\/span><span class=\"mord\">5<\/span><span class=\"mord\">2<\/span><\/span><\/span><\/span>"},{"id":1,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.48312em;vertical-align:0em;\"><\/span><span class=\"mrel\">\u2248<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">0<\/span><span class=\"mord\">.<\/span><span class=\"mord\">4<\/span><span class=\"mord\">8<\/span><\/span><\/span><\/span>"},{"id":2,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.48312em;vertical-align:0em;\"><\/span><span class=\"mrel\">\u2248<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">0<\/span><span class=\"mord\">.<\/span><span class=\"mord\">5<\/span><span class=\"mord\">9<\/span><\/span><\/span><\/span>"},{"id":3,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.48312em;vertical-align:0em;\"><\/span><span class=\"mrel\">\u2248<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:0.64444em;vertical-align:0em;\"><\/span><span class=\"mord\">0<\/span><span class=\"mord\">.<\/span><span class=\"mord\">4<\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span>"}]],"lockLeft":true,"lockRight":false},"formTextBefore":"","formTextAfter":"","answer":[[0,1,2,3],[0,1,2,3]]}

Since Zain uses the row totals, the joint frequencies in the first row must be divided by 23 and the joint frequencies in the second row must be divided by 27.

Zain uses the row totals. Therefore, the joint frequencies in the first row must be divided by 23 and the joint frequencies in the second row must be divided by 27.

The table below shows the conditional relative frequencies.

Zain will now consider the two-way table that shows conditional relative frequencies obtained using *row* totals.

They want to calculate some conditional probabilities by using the table. Help Zain find these probabilities!

a Knowing that a person is aged 30 or older, find the probability that they are a night owl.

b Knowing that a person is younger than 30, find the probability that they are an early bird.

c Knowing that a person is younger than 30, find the probability that they are a night owl.

d Knowing that a person is aged 30 or older, find the probability that they are an early bird.

{"type":"pair","form":{"alts":[[{"id":0,"text":"<span class=\"mlmath-simple\"><span class=\"text bold\">A<\/span><\/span>"},{"id":1,"text":"<span class=\"mlmath-simple\"><span class=\"text bold\">B<\/span><\/span>"},{"id":2,"text":"<span class=\"mlmath-simple\"><span class=\"text bold\">C<\/span><\/span>"},{"id":3,"text":"<span class=\"mlmath-simple\"><span class=\"text bold\">D<\/span><\/span>"}],[{"id":0,"text":"<span class=\"mlmath-simple\">0.59<\/span>"},{"id":1,"text":"<span class=\"mlmath-simple\">0.48<\/span>"},{"id":2,"text":"<span class=\"mlmath-simple\">0.52<\/span>"},{"id":3,"text":"<span class=\"mlmath-simple\">0.41<\/span>"}]],"lockLeft":true,"lockRight":false},"formTextBefore":"","formTextAfter":"","answer":[[0,1,2,3],[0,1,2,3]]}

Consider the fact that the conditional relative frequencies were found using *row* totals.

The table was created using *row* totals. Therefore, the first cell of the first row shows the probability of a person being a night owl given that they are younger than 30. Similarly, the second cell of the first row shows the probability of a person being an early bird given that they are younger than 30.

Likewise, the first cell of the second row shows the probability of a person being a night owl given that they aged 30 or older. Similarly, the second cell of the second row shows the probability of a person being an early bird given that they are aged 30 or older.

Paulina conducted a survey at Washington High. She asked 170 students whether they have cable TV and whether they took a vacation last summer. She displays the results in a two-way frequency table.

Using the table, Paulina wants to find out whether or nottaking a vacationand

having cable TVare independent events for this population of 170 students.

{"type":"choice","form":{"alts":["The events are independent.","The events are not independent."],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":1}

What is the probability that a student chosen at random took a vacation last summer? What is the probability that a random student who has cable TV took a vacation last summer?

Let A be the event that a student took a vacation last summer and B be the event that a student has cable TV. The table shows that from a total of 170 participants, 56 students took a vacation last summer.

With this information, the probability of randomly choosing a student who took a vacation can be found.$P(A)=17056 $

$P(A)≈0.33$

$P(A∣B)=7241 $

$P(A∣B)≈0.57$

At the beginning of the lesson, Zosia asked 50 students of North High School in Honolulu whether they prefer a chocolate bar or a piece of fruit as a lunchtime snack and whether they surf or not.

Letting A be the event that a student surfs and B the event that a student prefers a piece of fruit as a lunchtime snack, Zosia wants to calculate the following probabilities.{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"mlmath-simple\"><\/span>"},"formTextBefore":"<span class=\"mlmath-simple\"><span class=\"variable\">P<\/span>(<span class=\"variable\">B<\/span>)<span class=\"space big-before big-after\">=<\/span><\/span>","formTextAfter":null,"answer":{"text":["0.56","\\dfrac{28}{50}","\\dfrac{14}{25}"]}}

Make a two-way frequency table to display the obtained information.

A two-way frequency table can be made to organize the obtained information.

Next, the missing marginal frequencies can be calculated.

Now, two of the three missing joint frequencies can be calculated.

Finally, the last empty cell can be filled.

Now that the two-way table is complete, the desired probabilities can be found. Out of a total of 50 students, 42 surf and 28 prefer fruit as a lunch snack.

With this information, P(A) and P(B) can be calculated. {{ 'mldesktop-placeholder-grade' | message }} {{ article.displayTitle }}!

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