Twoway frequency tables
Learn how to read and use twoway frequency tables.
Let's jump right in and look at a twoway frequency table that came from asking $100$100 students whether they prefer cats or dogs.
Preference  Male  Female 

Prefers dogs  $36$36  $22$22 
Prefers cats  $8$8  $26$26 
No preference  $2$2  $6$6 
The columns of the table tell us whether the student is a male or a female. The rows of the table tell us whether the student prefers dogs, cats, or doesn't have a preference.
Notice that there are two variables gender and preference this is where the two in twoway frequency table comes from.
The cells tell us the number (or frequency) of students. For example, the $36$36 is in the male column and the prefers dogs row. This tells us that there are $36$36 male students who prefer dogs.
How many female students prefer cats?
 Your answer should be
 an integer, like $6$6
 a simplified proper fraction, like $3/5$3, slash, 5
 a simplified improper fraction, like $7/4$7, slash, 4
 a mixed number, like $1\ 3/4$1, space, 3, slash, 4
 an exact decimal, like $0.75$0, point, 75
 a multiple of pi, like $12\ \text{pi}$12, space, p, i or $2/3\ \text{pi}$2, slash, 3, space, p, i
$26$26 female students prefer cats
Row and column totals
To find the number of students that prefer cats, we just add up the number of students in the prefers cats row:
Preference  Male  Female 

Prefers dogs  $36$36  $22$22 
Prefers cats  $\blueD{8}$start color blueD, 8, end color blueD  $\blueD{26}$start color blueD, 26, end color blueD 
No preference  $2$2  $6$6 
$\text{Students who prefer cats} = \blueD{8} + \blueD{26} = 34$S, t, u, d, e, n, t, s, space, w, h, o, space, p, r, e, f, e, r, space, c, a, t, s, equals, start color blueD, 8, end color blueD, plus, start color blueD, 26, end color blueD, equals, 34
How many of the students are male?
 Your answer should be
 an integer, like $6$6
 a simplified proper fraction, like $3/5$3, slash, 5
 a simplified improper fraction, like $7/4$7, slash, 4
 a mixed number, like $1\ 3/4$1, space, 3, slash, 4
 an exact decimal, like $0.75$0, point, 75
 a multiple of pi, like $12\ \text{pi}$12, space, p, i or $2/3\ \text{pi}$2, slash, 3, space, p, i
$36 + 8 + 2 = 46$36, plus, 8, plus, 2, equals, 46 male students
Let's practice!
Problem 1
Researchers once surveyed college students on their Facebook use. The following twoway table displays data for the sample of students who responded to the survey.
How many students in the survey were in the age category of $18$18 to $22?$22, question mark
 Your answer should be
 an integer, like $6$6
 a simplified proper fraction, like $3/5$3, slash, 5
 a simplified improper fraction, like $7/4$7, slash, 4
 a mixed number, like $1\ 3/4$1, space, 3, slash, 4
 an exact decimal, like $0.75$0, point, 75
 a multiple of pi, like $12\ \text{pi}$12, space, p, i or $2/3\ \text{pi}$2, slash, 3, space, p, i
Age  Uses Facebook  Does not use Facebook 

$18$18 to $22$22  $78$78  $4$4 
$23\,\text{+}$23, space, plus  $70$70  $67$67 
To find how many students in the survey were in the age category of $18$18 to $22,$22, comma we can add to find the total number of students in that row.
Age  Uses Facebook  Does not use Facebook  TOTAL 

$18$18 to $22$22  $\blueD{78}$start color blueD, 78, end color blueD  $\blueD{4}$start color blueD, 4, end color blueD  $\blueD{82}$start color blueD, 82, end color blueD 
$23\,\text{+}$23, space, plus  $70$70  $67$67  $137$137 
TOTAL  $148$148  $71$71  $219$219 
There were $82$82 students in the survey in the age category of $18$18 to $22$22.
Problem 2
Researchers once surveyed students on which superpower they would most like to have. The following twoway table displays data for the sample of students who responded to the survey.
How many males in the survey chose to fly as their superpower?
 Your answer should be
 an integer, like $6$6
 a simplified proper fraction, like $3/5$3, slash, 5
 a simplified improper fraction, like $7/4$7, slash, 4
 a mixed number, like $1\ 3/4$1, space, 3, slash, 4
 an exact decimal, like $0.75$0, point, 75
 a multiple of pi, like $12\ \text{pi}$12, space, p, i or $2/3\ \text{pi}$2, slash, 3, space, p, i
Superpower  Male  Female 

Fly  $26$26  $12$12 
Invisibility  $14$14  $32$32 
Other  $10$10  $8$8 
To find how many males in the survey chose to fly as their superpower, we look for the count in the table that corresponds to both "Male" and "Fly".
Superpower  Male  Female 

Fly  $\blueD{26}$start color blueD, 26, end color blueD  $12$12 
Invisibility  $14$14  $32$32 
Other  $10$10  $8$8 
There were $26$26 males in the survey who chose to fly as their superpower.
Problem 3
Researchers surveyed recent graduates of two different universities about their income. The following twoway table displays data for the sample of graduates who responded to the survey.
How many graduates in the sample came from University A?
 Your answer should be
 an integer, like $6$6
 a simplified proper fraction, like $3/5$3, slash, 5
 a simplified improper fraction, like $7/4$7, slash, 4
 a mixed number, like $1\ 3/4$1, space, 3, slash, 4
 an exact decimal, like $0.75$0, point, 75
 a multiple of pi, like $12\ \text{pi}$12, space, p, i or $2/3\ \text{pi}$2, slash, 3, space, p, i
Income  University A  University B 

Under $\$20\text{,}000$dollar sign, 20, comma, 000  $35$35  $40$40 
$\$20\text{,}000$dollar sign, 20, comma, 000 to $\$39\text{,}999$dollar sign, 39, comma, 999  $90$90  $63$63 
$\$40\text{,}000$dollar sign, 40, comma, 000 and over  $35$35  $37$37 
To find how many graduates in the sample came from University A, we can add to find the total number of students in that column.
Income  University A  University B  TOTAL 

Under $\$20\text{,}000$dollar sign, 20, comma, 000  $\blueD{35}$start color blueD, 35, end color blueD  $40$40  $75$75 
$\$20\text{,}000$dollar sign, 20, comma, 000 to $\$39\text{,}999$dollar sign, 39, comma, 999  $\blueD{90}$start color blueD, 90, end color blueD  $63$63  $153$153 
$\$40\text{,}000$dollar sign, 40, comma, 000 and over  $\blueD{35}$start color blueD, 35, end color blueD  $37$37  $72$72 
TOTAL  $\blueD{160}$start color blueD, 160, end color blueD  $140$140  $300$300 
There were $160$160 graduates in the sample that came from University A.
Challenge problem
Lena knows the following information about her box of $18$18 candies:
 $10$10 candies contain both chocolate and caramel.
 $3$3 candies contain neither chocolate nor caramel.
 $12$12 candies in total contain chocolate.
Help Lena organize the results in the following twoway frequency table.
Contain caramel  Do not contain caramel  

Contain chocolate 


Do not contain chocolate 


Contain caramel  Do not contain caramel  

Contain chocolate  $10$10  $2$2 
Do not contain chocolate  $3$3  $3$3 