California Math 1 - 2020 Edition
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6.06 Two-way frequency tables

Two-way frequency tables

Like Venn diagrams, two-way frequency tables are a visual way of representing information.

Two-way tables allow us to display and examine the relationship between two sets of categorical data. The categories are labeled at the top and the left side of the table, and the frequency of the different characteristics appear in the interior of the table. Often the totals of each row and column also appear.

The following two-way table was made by surveying $100$100 students who were $9$9th and $10$10th graders. They were asked two questions, if they are right or left-handed and if they were in the $9$9th or $10$10th grade, the results are as follows:

  Right-handed Left-handed Total
9th Grade 43 9 52
10th Grade 44 4 48
Total: 87 13 100

It's called a two-way table because we can read information from it in two directions. Here we have information about the two categories grade and handedness. If read across each row, we can tell how many students in each grade surveyed are right or left-handed. If we read down each column, we can tell how many of the right or left-handed people surveyed were in $9$9th grade and how many were in $10$10th grade. 

Notice the last column contains the totals for each row and the last row contains the totals for each column. They both add up to $100$100 which is the total number of students surveyed. These are all called marginal frequencies. Think about how they are located on the edge of the table similar to the margins of a page.

Where a particular row and column overlap shows are how many people satisfy both categories. For example, there were $4$4 left-handed $10$10th graders surveyed. These are called joint frequencies because they represent where two categories join together.

Knowing there are $4$4 left-handed $10$10th graders we can find the proportion of $10$10th graders that are left-handed by dividing $4$4 by the total number of $10$10th graders $48$48. We get $\frac{4}{48}=\frac{1}{12}$448=112. So we can conclude that $\frac{1}{12}$112 or about $0.083$0.083 of the $10$10th graders surveyed were left-handed. This is called a conditional frequency because it represents what proportion of a group satisfies a given condition or qualification.


Worked example

Question 1

The following are the statistics of the passengers and crew who sailed on the Titanic on its fateful maiden voyage in $1912$1912

  First Class Second Class Third Class Crew Total
Survived $202$202 $118$118 $178$178 $212$212 $710$710
Died $123$123 $167$167 $\editable{}$ $696$696 $1514$1514
Total: $325$325 $285$285 $\editable{}$ $908$908 $\editable{}$

a) According to the table, what is the total number of passengers (first, second, and third class) and crew on-board the ship?

Think: Adding the totals in the final column will give the total number of people on-board the ship.


Total passengers and crew $=$= $710+1514$710+1514
  $=$= $2224$2224

b) Find the missing values in the "Third Class" column.

Think: We now know the total number of passengers and crew. This number will also be the sum of the values in the "Total" row. We can use this to find the missing value of the total number of third class passengers. Once we have this we can use it to find the number of third class passengers who died.


Total third class passengers $=$= $2224-908-285-325$2224908285325
  $=$= $706$706
Third class passengers who died $=$= $\text{Total}-\text{survivors}$Totalsurvivors
  $=$= $706-178$706178
  $=$= $528$528

c) What proportion of first class passengers survived?

Think: We need the fraction of first class survivors out of total number of first class passengers written as a decimal.


Proportion of first class passengers that survived $=$= $\frac{\text{first class survivors}}{\text{total number first class passengers}}$first class survivorstotal number first class passengers
  $=$= $\frac{202}{325}$202325
  $\approx$ $0.62$0.62

d) What percentage of total passengers and crew survived?

Think: We need the fraction of total survivors out of total number of passengers and crew written as a percentage.


Percentage of passengers and crew that survived $=$= $\frac{\text{total survivors}}{\text{total number passengers and crew}}$total survivorstotal number passengers and crew
  $=$= $\frac{710}{2224}\times100%$7102224×100%
  $\approx$ $31.9%$31.9%
Reflect: This type of table allows us to examine if relationships exist between the categories. For example, for the given statistics above can you use supporting calculations to show if the survival rate is dependent on the class of the passenger?


Practice questions


Yuri surveyed a group of people about the type of jobs they had. He recorded the data in the following graph.

  1. Complete the two way table with the information.

      No Job Casual Part time Full time
    Men $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
    Women $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$

Question 3

$150$150 tennis players were asked whether they would support equal prize money for the women’s and men’s draw.

  1.   Support Do not support
    Males $\editable{}$ $35$35
    Females $66$66 $12$12

    Find the missing value in the table.

  2. How many more players are there in support of equal prize money than those against it?

  3. What percentage of the male tennis players support equal prize money?

    Give your answer to one decimal place if necessary.


$36$36 students were asked whether or not they were allergic to nuts and dairy. The two way table is provided below.

  Allergic to nuts Not allergic to nuts
Allergic to dairy $10$10 $6$6
Not allergic to dairy $6$6 $14$14
  1. How many students are allergic to nuts?

  2. How many students are allergic to nuts or dairy, or both?

  3. How many students are allergic to at most one of the two things?


In a biology study, the appearance of an animal when carrying a certain gene is recorded. Assume that all animals must cary Gene $A$A or Gene $B$B but not both.

  Gene $A$A Gene $B$B
Light fur $36$36 $27$27
Dark fur $28$28 $x$x
  1. If, out of all the animals, the proportion carrying Gene A and having light fur is $\frac{4}{19}$419, what is the value of $x$x?




Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. '[Linear focus; discuss general principle.]

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