Investigation: Analyzing Two-Way Frequency Tables
How to display and analyze data depends on the type of data collected. Quantitative data refers to things that can be
measured such as height of a student in inches or average price of a concert ticket in dollars. Qualitative data refers to
a count of how many items are grouped into a common property such as eye color or type of pet owned. As you work on
the problems in this investigation, look for answers to this question:
How can you analyze qualitative data to recognize possible associations and trends?
Throughout schools across the country, students engage in a variety of
extracurricular activities. Some enjoy athletics such as football, tennis, cross
country, and volleyball. Others spend their time within the arts such as band,
theatre, and chorus. There are also clubs that serve the community by providing
recycling, working at food shelters, tutoring kids and cleaning up the roadways.
Students may participate in one or more of these types of activities.
1. Work together with the whole class to find out about the types of
extracurricular activities enjoyed by students in your class.
a. Make a list of the top activities and have each student determine their
favorite (only one).
b. Organize the data by gender into a two-way frequency table like the one below.
Athletics
Arts
Club
Other
Male
Female
2. Use the information provided in your table to answer the following questions. For each question, explain how you
used the table.
a. How many students are in the class?
b. Which activity was favored the most overall?
c. What percent of the class favored the arts?
d. What percent of males favored athletics?
2012 Olympic Medals
Number of Medals
3. Bar charts and circle graphs are ways to
display qualitative data. The bar chart on
the right shows the 2012 Olympic Medals
won by the top five countries.
a. Create a two-way frequency table
from the bar graph.
b. Compare with another group and
discuss the similarities and
differences.
c. The countries are ranked by the total
number of medals won. Add a row
and a column to the two-way
frequency table and label them
“Total”. Complete the new cells in
the table. The totals are called the
marginal frequencies.
50
45
40
35
30
25
20
15
10
5
0
Gold
Silver
Bronze
USA
China
Great
Britain
Russia
Top 5 Countries
1
South
Korea
Eye Color
4. The table below shows eye color and hair color of
Hair Color
students enrolled in a statistics course at a local
Blond
Black Brown
Red
Total
university. Calculate the missing entries and then
Brown
68
119
26
7
220
use the data to answer the following questions. For
Blue
20
17
94
215
each question, explain how you used the table.
Hazel
15
54
14
10
a. What percentage of the students has hazel
Green
29
14
16
64
eyes?
Total
108
71
127
b. What percentage of the students has blond
hair?
c. A relative marginal frequency is when you are comparing a row total or column total to the table total like in
parts (a) and (b). These can be used to describe the distribution of a category. Determine the relative marginal
frequency for eye color and describe the distribution.
5. Another important type of frequency compares the data in respect to one specific condition and is called relative
conditional frequency.
a. What percentage of students with red hair has brown eyes?
b. What percentage of the students with brown eyes has red hair?
c. Compare the answers for part a and b. Are they the same? Explain.
d. Calculate the relative conditional frequency of blue eyes for each of the different hair colors. Describe the
distribution of hair color for students with blue eyes.
Eye
Color
6. The statistics class at Wade High School collected data of the freshmen students.
a. Using marginal and relative marginal frequencies, describe
the freshmen students.
Blond
b. Compare the relative marginal frequency of the freshmen
Brown
24
students with blond hair to the relative marginal
Blue
3
frequency of university students with blond hair.
Hazel
0
c. Compare the percentage of freshmen students with brown
Green
8
hair and green eyes to the percentage of university
students with brown hair and green eyes.
Hair Color
Black Brown
16
24
24
8
16
6
8
5
Red
8
1
32
12
Eye Color
7. Since there are different sample sizes for the two groups of students, it is easier to compare the relative joint
frequencies instead of the single counts. In activity 6c, you compared the percentage of students with brown hair
and green eyes to the percentage of university students with brown hair and green eyes. In other words, you
compared the percentage of students who have the same two conditions, brown hair and green eyes. A relative joint
frequency is the ratio/percentage based on two conditions. A relative joint frequency table for the university students
has been started below.
a. Complete the table and create one for the freshmen students.
Relative Joint
Hair Color
Frequency of
the University
Blond
Black
Brown
Red
Total
Students
119
26
7
220
68
≈ 11.5%
≈ 20.1%
≈ 4.4%
≈ 37%
≈ 1.2%
Brown
592
592
592
592
592
Blue
Hazel
Green
Total
b. How does the distribution of brown eyes for the freshmen students compare to the distribution of brown eyes for
the university students? How does the relative marginal frequency compare?
c. Compare the two groups. Use the following questions to guide your comparisons.
! Which characteristics or combinations occur more frequently than others?
! Is it the same for both groups?
! What differences are there between the two groups?
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Summarize the Mathematics
a) Describe a two-way frequency table and how it is used to organize data.
b) Explain the three types of relative frequencies: relative marginal frequency, relative conditional frequency, and
relative joint frequency.
c) Describe ways to use the three types of relative frequencies to describe or compare data.
Check Your Understanding
S
u
Children at a YMCA afterschool program were asked “If you were a superhero, what special
power would you most want to have?” The results of the survey are listed below.
Gender
Male
Female
Fly
30
11
Freeze time
19
17
Invisibility
13
11
Super Strength
4
14
Telepathy
26
5
a) Extend the table provided to include the marginal frequencies and explain what they mean in terms of the
context.
b) Compute and record the relative marginal frequencies. Describe the distribution of the superpowers.
c) Describe the distribution by gender of those students who would want to fly.
d) Margo says that 21.4% of those who want super strength are female. Is she correct? If so, explain why. If not,
explain what error she made and how to correct it.
e) Calculate the relative joint frequencies for telepathy and explain what they mean in terms of the context.
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