1. Two-Way Tables Normal Distributions
Basic Practice of Statistics - 3rd Edition
Two-Way Tables Normal Distributions
Chapter 5

2. Categorical Variables
Review: Categorical Variables place individuals into one of several groups or categories.
The values of a categorical variable are labels for the different categories.
The distribution of a categorical variable lists the count or percent of individuals who fall into each category.
When a dataset involves two categorical variables, we begin by examining the counts or percents in various categories for one of the variables.
Two-way Table – describes two categorical variables, organizing counts according to a row variable and a column variable.

3. Young adults by gender and chance of getting rich
Two-Way Table
Young adults by gender and chance of getting rich
Female
Male
Total
Almost no chance 96 98
194
Some chance, but probably not
426
286
712
A chance
696
720
1416
A good chance
663
758
1421
Almost certain
486
597
1083
2367
2459
4826
What are the variables described by this two-way table?
How many young adults were surveyed?

4. Practice 1: Complete the following tables
Students are asked
if they prefer to go
swimming or to the gym.
2) Some people are
asked about their favorite outdoor sport.
Swimming
Gym
Total
Boys 25 45
Girls 64
100
Hiking
Canoeing
Climbing
Total
Under 15’s 42
150
15 – 30 30 47
100
Over 30 15 75
300

5. Eat breakfast on a regular basis
Fill in the gaps
Sex
TOTALS
Male
Female
Eat breakfast on a regular basis
Yes
190
300 No
165
320
275
110
130
295
595

6. Marginal Distribution
The Marginal Distribution of one of the categorical variables in a two-way table of counts is the distribution of values of that variable among all individuals described by the table.
Note: Percent's are often more informative than counts, especially when comparing groups of different sizes.
To examine a marginal distribution:
Use the data in the table to calculate the marginal distribution (in percent's) of the row or column totals.
Make a graph to display the marginal distribution.

7. Example: Age and Education
“Age groups” is the categorical explanatory variable “Education level” is the categorical response variable
Variables
Marginal distributions
BPS
Chapter 6

8. Example: Marginal Totals
Variables
Marginal totals
37, , ,008
27,858 58,077 44,465 44,828
BPS
Chapter 6

9. Marginal Distributions
Marginal distributions are used as background information only.
They do not address association
BPS
Chapter 6

10. Marginal Distribution, Row Variable
% not completed HS =
27,859 / 175,230 × 100% = 15.9%
% graduated HS =
58,077 / 175,230 × 100% = 33.1%
% finished 1-3 yrs col. =
44,465 / 175,230 × 100% = 25.4%
% finished ≥4 yrs col. =
44,828 / 175,230 × 100% = 25.6%
BPS
Chapter 6

11. Marginal Distribution, Column Variable
% age 25–34 =
37,786 / 175,230 × 100% = 21.6%
% age 35–54 =
81,435 / 175,230 × 100% = 46.5%
% 55 and over =
56,008 / 175,230 × 100% = 32.0%
BPS
Chapter 6 11

12. Marginal Distribution
Young adults by gender and chance of getting rich
Female
Male
Total
Almost no chance 96 98
194
Some chance, but probably not
426
286
712
A chance
696
720
1416
A good chance
663
758
1421
Almost certain
486
597
1083
2367
2459
4826
Examine the marginal distribution of chance of getting rich.
Response
Percent
Almost no chance
194/4826 = 4.0%
Some chance
712/4826 = 14.8%
A chance
1416/4826 = 29.3%
A good chance
1421/4826 = 29.4%
Almost certain
1083/4826 = 22.4%

13. New Chips Mashed Total Boys 34 100 Girls 25 37 Teachers 12 22 50 104
Practice 2: Copy and complete the following two-way table about ways of eating potato and then answer the questions.
How many boys liked mashed?
How many teachers preferred new potatoes?
How many girls were asked?
Out of the people who liked chips, how many were boys?
New
Chips
Mashed
Total
Boys 34
100
Girls 25 37
Teachers 12 22 50
104
250

14. Practice 3: For the following two-way table answer the questions on probability. A person is picked at random from the sample.
New
Chips
Mash
Total
Boys 15 51 34
100
Girls 25 37 38
Teachers 12 16 22 50 52
104 94
250
What is the probability the person picked is a boy?
What is the probability the person liked mash?
What is the probability the person was a teacher who preferred new potatoes?
What is the probability that, out of the girls, the person liked chips?
Out of the people who liked chips, what is the probability the person was a boy?

15. TAIL
For the following table, copy and complete it, then answer the questions about types of coffee bought.
Coffee is sold in three types and in three weights.
How many people bought 100g of powdered coffee?
How many people bought ground coffee?
Out of the packets weighing 200g, what is the probability the packet bought contained granules?
100g
200g
300g
Total
Ground 50
120
Powder 80 35 26
Granules 40 45
135
400
Are you ready for the answers ?
Next

16. Answers 55 141 54 130 100g 200g 300g Total Ground 15 50 120 Powder 80
100g
200g
300g
Total
Ground
 15 50
55 
120
Powder 80 35 26
141 
Granules 40 45 54
 139
135
130 
400
(a) 80 (b) 120 (c) 45/130

17. Task One – Data Collection
Has an Ipod
TOTALS
Yes No
Has been to Myrtle Beach

18. What is the probability that…
A student has an Ipod?
A student has been to Myrtle Beach?
A student does not have an Ipod?
A student has not been to Myrtle Beach?
A student has an Ipod and has been to the Myrtle Beach?
A student has an Ipod or has been to Myrtle Beach)
A student does not have an Ipod or has not been to the Myrtle Beach?

19. Hours watching TV last night
Example Two
Brown Hair
TOTALS
Yes No
Hours watching TV last night
Less than 2 hours
2 or more hours

20. What is the probability that…
A student has brown hair?
A student spent less than 2 hours watching TV last night?
A brunette student spent less than 2 hours watching TV last night?
A student who was watching TV for more than 2 hours last night is brunette?

21. Has correct supplies for class
Example 3
Did Homework
TOTALS
Male
Female
Has correct supplies for class
Yes No

22. Questions?
Write your own set of at least 5 questions you could answer using the previous two-way table.

23. Homework