1. Elementary Statistics: Looking at the Big Picture
(C) 2007 Nancy Pfenning
Lecture 5: Chapter 4, Section 1 Single Variables (Focus on Categorical Variables)
Displays and Summaries
Data Production Issues
Looking Ahead to Inference
Details about Displays and Summaries
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture
Elementary Statistics: Looking at the Big Picture

2. Elementary Statistics: Looking at the Big Picture
(C) 2007 Nancy Pfenning
Looking Back: Review
4 Stages of Statistics
Data Production (discussed in Lectures 1-4)
Displaying and Summarizing
Single variables: 1 categorical, 1 quantitative
Relationships between 2 variables
Probability
Statistical Inference
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture
Elementary Statistics: Looking at the Big Picture

3. First Process of Statistics
(C) 2007 Nancy Pfenning
First Process of Statistics
1. Data Production: Take sample data from the population, with sampling and study designs that avoid bias.
POPULATION
SAMPLE
SAMPLE
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture
Elementary Statistics: Looking at the Big Picture

4. Second Process of Statistics
(C) 2007 Nancy Pfenning
Second Process of Statistics
1. Data Production: Take sample data from the population, with sampling and study designs that avoid bias.
2. Displaying and Summarizing: Use appropriate displays and summaries of the sample data, according to variable types and roles.
POPULATION C Q
SAMPLE
CQ
CC
QQ
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture
Elementary Statistics: Looking at the Big Picture

5. Third Process of Statistics:
(C) 2007 Nancy Pfenning
Third Process of Statistics:
1. Data Production: Take sample data from the population, with sampling and study designs that avoid bias.
2. Displaying and Summarizing: Use appropriate displays and summaries of the sample data, according to variable types and roles.
POPULATION C
SAMPLE CQ
QQ
PROBABILITY
3. Probability: Assume we know what’s true for the population; how should random samples behave?
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture
Elementary Statistics: Looking at the Big Picture

6. Fourth Process of Statistics
(C) 2007 Nancy Pfenning
Fourth Process of Statistics
1. Data Production: Take sample data from the population, with sampling and study designs that avoid bias.
2. Displaying and Summarizing: Use appropriate displays and summaries of the sample data, according to variable types and roles.
POPULATION
SAMPLE
PROBABILITY
3. Probability: Assume we know what’s true for the population; how should random samples behave?
INFERENCE C Q
4. Statistical Inference: Assume we only know what’s true about sampled values of a single variable or relationship; what can we infer about the larger population?
CQ
CC
QQ
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture
Elementary Statistics: Looking at the Big Picture

7. Handling Single Categorical Variables
(C) 2007 Nancy Pfenning
Handling Single Categorical Variables
Display:
Pie chart
Bar graph
Summary:
Count
Percent
Proportion
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture
Elementary Statistics: Looking at the Big Picture

8. Definitions and Notation
(C) 2007 Nancy Pfenning
Definitions and Notation
Statistic: number summarizing sample
Parameter: number summarizing population
: sample proportion (a statistic) [“p-hat”]
p: population proportion (a parameter)
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture
Elementary Statistics: Looking at the Big Picture

9. Example: Issues to Consider
(C) 2007 Nancy Pfenning
Example: Issues to Consider
Background: 246 of 446 students at a certain university had eaten breakfast on survey day.
Questions:
Are intro stat students representative of all students at that university?
Would they respond without bias?
Responses:
Yes, if years and class times are representative.
Yes (not sensitive).
Looking Back: These are data production issues.
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture
Practice: 4.4e p.79
Elementary Statistics: Looking at the Big Picture

10. Example: More Issues to Consider
(C) 2007 Nancy Pfenning
Example: More Issues to Consider
Background: 246 of 446 students at a certain university had eaten breakfast on survey day.
Questions:
How do we display and summarize the info?
Can we conclude that a majority of all students at that university eat breakfast?
Responses:
Display: pie chart
Summary: 246/446= 55% or 0.55 ate breakfast.
Can’t yet say if majority eat breakfast overall.
Looking Ahead: This would be statistical inference.
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture
Elementary Statistics: Looking at the Big Picture

11. Example: Statistics vs. Parameters
(C) 2007 Nancy Pfenning
Example: Statistics vs. Parameters
Background: 246 of 446 students at a certain university had eaten breakfast on survey day.
Questions:
Is 246/446=0.55 a statistic or a parameter? How do we denote it?
Is the proportion of all students eating breakfast a statistic or a parameter? How do we denote it?
Responses:
246/446=0.55 is a statistic denoted .
Proportion of all students eating breakfast is a parameter denoted p.
Practice: 4.3c p.79
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture
Elementary Statistics: Looking at the Big Picture

12. Example: Summary Issues
(C) 2007 Nancy Pfenning
Example: Summary Issues
Background: Location (state) for all 1,696 TV series in 2004 with known settings:
601 in California (601/1696=0.35)
412 in New York (412/1696=0.24)
683 in other states (683/1696=0.40)
Questions:
=0.99mistake?
Why is it not appropriate to use this info to draw conclusions about a larger population in 2004?
Responses:
No mistake; roundoff error
There is no larger population (data is for all series in 2004).
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture
Elementary Statistics: Looking at the Big Picture

13. Elementary Statistics: Looking at the Big Picture
(C) 2007 Nancy Pfenning
Example: Notation
Background: In study of 20 antarctic prions (birds), 17 correctly chose the one of two bags that had contained their mate.
Questions: How do we denote sample and population proportions? Are they statistics or parameters?
Responses:
sample proportion is a statistic.
population proportion p is a parameter (not known in this case)
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture
Elementary Statistics: Looking at the Big Picture

14. Elementary Statistics: Looking at the Big Picture
(C) 2007 Nancy Pfenning
Definitions
Mode: most common value
Majority: more common of two possible values (same as mode)
Minority: less common of two possible values
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture
Elementary Statistics: Looking at the Big Picture

15. Example: Role of Sample Size
(C) 2007 Nancy Pfenning
Example: Role of Sample Size
Background: In study of 20 antarctic prions (birds), 17 correctly chose the one of two bags that had contained their mate.
Question: Would we be more convinced that a majority of all prions would choose correctly, if 170 out of 200 were correct?
Response: Yes, as long as the sampling and study design are sound.
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture
Practice: 4.3i p.79
Elementary Statistics: Looking at the Big Picture

16. Example: Sampling Design
(C) 2007 Nancy Pfenning
Example: Sampling Design
Background: In study of 20 antarctic prions (birds), 17 correctly chose the one of two bags that had contained their mate.
Question: Is the sample biased?
Response: No, unless maybe they were too long in captivity.
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture
Elementary Statistics: Looking at the Big Picture

17. Elementary Statistics: Looking at the Big Picture
(C) 2007 Nancy Pfenning
Example: Study Design
Background: Antarctic prions presented with Y-shaped maze, a bag at the end of each arm. One bag had contained mate, the other not.
Question:
What were researchers attempting to show?
Response:
that prions know mate by smell ?
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture
Elementary Statistics: Looking at the Big Picture

18. Elementary Statistics: Looking at the Big Picture
(C) 2007 Nancy Pfenning
Example: Study Design
Background: Antarctic prions presented with Y-shaped maze, a bag at the end of each arm. One bag had contained mate, the other not.
Question:
Why use bags and not birds themselves?
Response:
Didn’t want them to find mate by sight.
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture
Elementary Statistics: Looking at the Big Picture

19. Elementary Statistics: Looking at the Big Picture
(C) 2007 Nancy Pfenning
Example: Study Design
Background: Antarctic prions presented with Y-shaped maze, a bag at the end of each arm. One bag had contained mate, the other not.
Question:
Why “had” contained (bird no longer in bag)?
Response:
Didn’t want them to find mate by sound.
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture
Elementary Statistics: Looking at the Big Picture

20. Elementary Statistics: Looking at the Big Picture
(C) 2007 Nancy Pfenning
Example: Study Design
Background: Antarctic prions presented with Y-shaped maze, a bag at the end of each arm. One bag had contained mate, the other not.
Question:
OK to always place correct bag on right?
Response:
Position of correct bag should be randomized, in case birds have natural preference for left or right.
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture
Elementary Statistics: Looking at the Big Picture

21. Elementary Statistics: Looking at the Big Picture
(C) 2007 Nancy Pfenning
Example: Study Design
Background: Antarctic prions presented with Y-shaped maze, a bag at the end of each arm. One bag had contained mate, the other not.
Question: Should the other be just any empty bag?
Response: It should have formerly contained a non-mate bird.
Looking Ahead: Researchers were careful to avoid bias in their study design. A success rate of 85% is impressive but we need inference methods to quantify claims that prions in general can recognize their mate by smell.
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture
Elementary Statistics: Looking at the Big Picture

22. Two or More Possible Values
(C) 2007 Nancy Pfenning
Two or More Possible Values
Looking Ahead: In Probability and Inference, most categorical variables discussed have just two possibilities.
Still, we often summarize and display categorical data with more than two possibilities.
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture
Elementary Statistics: Looking at the Big Picture

23. Example: Proportions in Three Categories
(C) 2007 Nancy Pfenning
Example: Proportions in Three Categories
Background: Student wondered if she should resist changing answers in multiple choice tests. “Ask Marilyn” replied:
50% of changes go from wrong to right
25% of changes go from right to wrong
25% of changes go from wrong to wrong
Question: How to display information?
Response: Use a pie chart:
Instructor may survey students: To which category do your changes typically belong ?
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture
Elementary Statistics: Looking at the Big Picture

24. Elementary Statistics: Looking at the Big Picture
(C) 2007 Nancy Pfenning
Definition
Bar graph: shows counts, percents, or proportions in various categories (marked on horizontal axis) with bars of corresponding heights
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture
Elementary Statistics: Looking at the Big Picture

25. Elementary Statistics: Looking at the Big Picture
(C) 2007 Nancy Pfenning
Example: Bar Graph
Background: Instructor can survey students to find proportion in each year (1st, 2nd, 3rd, 4th, Other).
Questions:
How can we display the information?
What should we look for in the display?
Responses:
Construct a bar graph:
years 1, 2, 3, 4, Other on horizontal axis
counts or proportions in each year graphed vertically
Look for mode (tallest bar) to tell what year is most common; compare heights of all 5 bars.
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture
Practice: 4.6a p.80
Elementary Statistics: Looking at the Big Picture

26. Elementary Statistics: Looking at the Big Picture
(C) 2007 Nancy Pfenning
Example: Bar Graph
Background: Instructor can survey students to find proportion in each year (1st, 2nd, 3rd, 4th, Other).
Questions:
How can we display the information?
What should we look for in the display?
Responses: For the particular sample whose data accompany the book, mode is 2nd year, next is 3rd, then 1st and 4th, then Other.
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture
Practice: 4.6a p.80
Elementary Statistics: Looking at the Big Picture

27. Overlapping Categories
(C) 2007 Nancy Pfenning
Overlapping Categories
If more than two categorical variables are considered at once, we must note the possibility that categories overlap.
Looking Ahead: In Probability, we will need to distinguish between situations where categories do and do not overlap.
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture
Elementary Statistics: Looking at the Big Picture

28. Example: Overlapping Categories
(C) 2007 Nancy Pfenning
Example: Overlapping Categories
Background: Report by ResumeDoctor.com on over 160,000 resumes:
13% said applicant had “communication skills”
7% said applicant was a “team player”
Question: Can we conclude that 20% claimed communication skills or team player?
Response: No: adding 13%+7% counts those in both categories twice.
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture
Elementary Statistics: Looking at the Big Picture

29. Processing Raw Categorical Data
(C) 2007 Nancy Pfenning
Processing Raw Categorical Data
Small categorical data sets are easily handled without software.
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture
Elementary Statistics: Looking at the Big Picture

30. Example: Proportion from Raw Data
(C) 2007 Nancy Pfenning
Example: Proportion from Raw Data
Background: Harvard study claimed 44% of college students are binge drinkers. Agree on survey design and have students self-report: on one occasion in past month, alcoholic drinks more than 5 (males) or 4 (females)? Or use these data:
Question: Are data consistent with claim of 44%?
Response: % “yes”=28/66=42%, very close to 44%
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture
Practice: 4.7a p.80
Elementary Statistics: Looking at the Big Picture

31. Example: Proportion from Raw Data
(C) 2007 Nancy Pfenning
Example: Proportion from Raw Data
Background: Harvard study claimed 44% of college students are binge drinkers. Agree on survey design and have students self-report: on one occasion in past month, alcoholic drinks more than 5 (males) or 4 (females)? Or use these data:
Looking Ahead: How different would the sample percentage have to be, to convince you that your sample is significantly different from Harvard’s?
This is an inference question.
Question: Are data consistent with claim of 44%?
Response: % “yes”=28/66=42%, very close to 44%
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture
Practice: 4.7a p.80
Elementary Statistics: Looking at the Big Picture

32. Lecture Summary (Categorical Variables)
(C) 2007 Nancy Pfenning
Lecture Summary (Categorical Variables)
Display: pie chart, bar graph
Summarize: count, percent, proportion
Sampling: data unbiased (representative)?
Design: produced unbiased summary of data?
Inference: will we ultimately draw conclusion about population based on sample?
Mode, Majority: most common values
Larger samples: provide more info
Other issues: Two or more possibilities? Categories overlap? How to handle raw data?
©2011 Brooks/Cole, Cengage Learning
Elementary Statistics: Looking at the Big Picture
Elementary Statistics: Looking at the Big Picture