1. Chapter 1: Exploring Data
Sec. 1.1 Analyzing Categorical Data

2. Data Analysis: Making Sense of Data
IDENTIFY the individuals and variables in a set of data
CLASSIFY variables as categorical or quantitative
DISPLAY categorical data with a bar graph
IDENTIFY what makes some graphs of categorical data deceptive
CALCULATE and DISPLAY the marginal distribution of a categorical variable from a two-way table
CALCULATE and DISPLAY the conditional distribution of a categorical variable for a particular value of the other categorical variable in a two-way table
DESCRIBE the association between two categorical variables

3. Data Analysis
Statistics is the science of data. Data Analysis is the process of organizing, displaying, summarizing, and asking questions about data.

4. Data Analysis Individuals
objects described by a set of data (don’t necessarily have to be people)
Variable
any characteristic of an individual
Categorical Variable
places an individual into one of several groups or categories.
Quantitative Variable
takes numerical values for which it makes sense to find an average.
Some number values can be considered categorical – for instance, area codes, floors of a building
Some quantitative variables can be converted to categorical – Giving letter grades instead of numerical grades, meeting standards

5. Example, p. 3
CensusAtSchool is an international project that collects data about primary and secondary school students using surveys. Hundreds of thousands of students from Australia, Canada, New Zealand, South Africa, and the United Kingdom have taken part in the project since Data from the surveys are available at the project’s Web site ( We used the site’s “Random Data Selector” to choose 10 Canadian students who completed the survey in a recent year. The table displays the data.

6. Example, p. 3 Who are the individuals in this data set?
What variables were measured? Identify each as categorical or quantitative.
Describe the individual in the highlighted row.
10 randomly selected Canadian students who participated in the CensusAtSchool survey.
7 variables: Province, Gender, Dominant Hand, and Preferred Communication are Categorical, and Language spoken, Height, and Wrist Circumference are Quantitative.
Student lives in Ontario and is male. He speaks 4 languages, is left-handed, is cm tall with a wrist circumference of 147 mm and prefers text messaging.
Notice that the female who is 166 cm tall has a wrist circumference of 65 mm. This is probably wrong. Always check to see if your data makes sense!

7. Data Analysis Distribution
tells us what values a variable takes and how often it takes those values.
How to Explore Data
Start by determining the variable of interest.
Display the data in a graph.
Add Numerical Summaries.

8. Categorical Variables
Categorical variables place individuals into one of several groups or categories.
Frequency Table
Format
Count of Stations
Adult Contemporary
1556
Adult Standards
1196
Contemporary Hit
569
Country
2066
News/Talk
2179
Oldies
1060
Religious
2014
Rock
869
Spanish Language
750
Other Formats
1579
Total
13838
Relative Frequency Table
Format
Percent of Stations
Adult Contemporary
11.2
Adult Standards
8.6
Contemporary Hit
4.1
Country
14.9
News/Talk
15.7
Oldies
7.7
Religious
14.6
Rock
6.3
Spanish Language
5.4
Other Formats
11.4
Total
99.9
Variable
Count
Percent
“Values” do not necessarily mean numbers. A categorical variable for instance can have a value of “red” if you are talking about colors.
Values

9. Count vs. Percent Frequency Table – displays counts in each category
Relative Frequency Table – displays the percent in each category (% found by count in category / total count)
Usually percents are rounded. If the rounded percents do not add up to exactly 100%, it may be due to round-off error.

10. Displaying Categorical Data – Bar Graph
Bars don’t touch and must be equally wide
Can rearrange categories
Scale may be in counts or percents
Bar graphs compare several quantities by comparing the heights of bars that represent those quantities. Our eyes, however, react to the area of the bars as well as to their height. Must make bars equally wide because of this.

11. Displaying Categorical Data – Pie Chart
Must be Out of a Whole (100%)
Read more about Pie Charts on p. 8.
You may get data where they give the % out of that category. This data cannot be displayed in a pie chart.

12. Graphs: Good and Bad There are two important lessons to keep in mind:
Who Buys iMacs?
There are two important lessons to keep in mind:
beware the pictograph, and
watch those scales.
It is tempting to replace the bars with pictures for greater eye appeal. Don’t do it!

13. Two-Way Tables and Marginal Distributions
A two-way table describes two categorical variables, organizing counts according to a row variable and a column variable.
Typically, we put the explanatory variables in the columns and the response variables in the rows.

14. Example, p. 12 What are the variables described by this two-way table?
I’m Gonna Be Rich! A survey of 4826 randomly selected young adults (aged 19 to 25) asked, “What do you think the chances are you will have much more than a middle-class income at age 30?” The table below shows the responses.
What are the variables described by this
two-way table?
How many young adults were surveyed?
Describes two categorical variables: Gender and opinion about becoming rich young adults were surveyed.

15. Two-Way Tables and Marginal Distributions
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: Percents are often more informative than counts, especially when comparing groups of different sizes.
How to examine a marginal distribution:
Use the data in the table to calculate the marginal distribution (in percents) of the row or column totals. (row or column total / table total)
Make a graph to display the marginal distribution.

16. Example, p. 13
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%

17. Example, p. 13 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%

18. Relationships Between Categorical Variables
A conditional distribution of a variable describes the values of that variable among individuals who have a specific value of another variable.
How to examine or compare conditional distributions:
Select the row(s) or column(s) of interest.
Use the data in the table to calculate the conditional distribution (in percents) of the row(s) or column(s).
Make a graph to display the conditional distribution.
Use a bar graph to compare distributions.

19. Example, p. 15
Calculate the conditional distribution of opinion among males. Examine the relationship between gender and opinion.
Response
Male
Almost no chance
98/2459 = 4.0%
Some chance
286/2459 = 11.6%
A chance
720/2459 = 29.3%
A good chance
758/2459 = 30.8%
Almost certain
597/2459 = 24.3%
Female
96/2367 = 4.1%
426/2367 = 18.0%
696/2367 = 29.4%
663/2367 = 28.0%
486/2367 = 20.5%
Segmented bar graph

20. We can also compare categories with a side-by-side bar graph.

21. Example, p. 15
Can we say there is an association between gender and opinion in the population of young adults? Making this determination requires formal inference, which will have to wait a few chapters.
Caution!
Even a strong association between two categorical variables can be influenced by other variables lurking in the background.

22. Data Analysis: Making Sense of Data
DISPLAY categorical data with a bar graph
IDENTIFY what makes some graphs of categorical data deceptive
CALCULATE and DISPLAY the marginal distribution of a categorical variable from a two-way table
CALCULATE and DISPLAY the conditional distribution of a categorical variable for a particular value of the other categorical variable in a two-way table
DESCRIBE the association between two categorical variables
A dataset contains information on individuals.
For each individual, data give values for one or more variables.
Variables can be categorical or quantitative.
The distribution of a variable describes what values it takes and how often it takes them.
Inference is the process of making a conclusion about a population based on a sample set of data.

23. Homework – Due Block Day
P. 6 # 3 & 6
P. 20 # 9, 16, 19, 21