Learn how to read and use two-way frequency tables.
Let's jump right in and look at a two-way frequency table that came from asking 100 students whether they prefer cats or dogs.
PreferenceMaleFemale
Prefers dogs3622
Prefers cats826
No preference26
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 two-way frequency table comes from.
The cells tell us the number (or frequency) of students. For example, the 36 is in the male column and the prefers dogs row. This tells us that there are 36 male students who prefer dogs.
How many female students prefer cats?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i

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:
PreferenceMaleFemale
Prefers dogs3622
Prefers catsstart color blueD, 8, end color blueDstart color blueD, 26, end color blueD
No preference26
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
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i

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 two-way 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 to 22, question mark
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i
students
AgeUses FacebookDoes not use Facebook
18 to 22784
23, space, plus7067

To find how many students in the survey were in the age category of 18 to 22, comma we can add to find the total number of students in that row.
AgeUses FacebookDoes not use FacebookTOTAL
18 to 22start color blueD, 78, end color blueDstart color blueD, 4, end color blueDstart color blueD, 82, end color blueD
23, space, plus7067137
TOTAL14871219
There were 82 students in the survey in the age category of 18 to 22.

Problem 2

Researchers once surveyed students on which superpower they would most like to have. The following two-way 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
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i
students
SuperpowerMaleFemale
Fly2612
Invisibility1432
Other108

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".
SuperpowerMaleFemale
Flystart color blueD, 26, end color blueD12
Invisibility1432
Other108
There were 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 two-way 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
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i
graduates
IncomeUniversity AUniversity B
Under dollar sign, 20, comma, 0003540
dollar sign, 20, comma, 000 to dollar sign, 39, comma, 9999063
dollar sign, 40, comma, 000 and over3537

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.
IncomeUniversity AUniversity BTOTAL
Under dollar sign, 20, comma, 000start color blueD, 35, end color blueD4075
dollar sign, 20, comma, 000 to dollar sign, 39, comma, 999start color blueD, 90, end color blueD63153
dollar sign, 40, comma, 000 and overstart color blueD, 35, end color blueD3772
TOTALstart color blueD, 160, end color blueD140300
There were 160 graduates in the sample that came from University A.

Challenge problem

Lena knows the following information about her box of 18 candies:
  • 10 candies contain both chocolate and caramel.
  • 3 candies contain neither chocolate nor caramel.
  • 12 candies in total contain chocolate.
Help Lena organize the results in the following two-way frequency table.
Contain caramelDo not contain caramel
Contain chocolate
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i
Do not contain chocolate
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, p, i or 2, slash, 3, space, p, i

Contain caramelDo not contain caramel
Contain chocolate102
Do not contain chocolate33