1. Chapter 1 Exploring Data
Introduction: Data Analysis: Making Sense of Data
1.1 Analyzing Categorical Data
1.2 Displaying Quantitative Data with Graphs
1.3 Describing Quantitative Data with Numbers

2. Introduction Data Analysis: Making Sense of Data
Learning Objectives
After this section, you should be able to…
DEFINE “Individuals” and “Variables”
DISTINGUISH between “Categorical” and “Quantitative” variables
DEFINE “Distribution”
DESCRIBE the idea behind “Inference”

3. Data Analysis Statistics is the science of data.
Data Analysis is the process of organizing, displaying, summarizing, and asking questions about data.
Data Analysis
Definitions:
Individuals – objects (people, animals, things) described by a set of data
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.

4. Now do Exercise 3 on P- 7 Answers:
Read Example : Census at school ( P-3)
Now do Exercise 3 on P- 7
Answers:
a) Individuals: AP Stat students who completed a questionnaire on the first day of class.
b) Categorical: gender, handedness, favorite type of music
Quantitative: height( inches), amount of time the students
expects to spend on HW (mins) , the total value of coin(cents)
c) Female, right-handed, 58 inches tall, spends 60
mins on HW, prefers alternative music, had 76
cents in her pocket.

5. Do Activity : Hiring Discrimination – It just won’t fly ……
Follow the directions on Page 5
Perform 5 repetitions of your simulation.
Turn in your results to your teacher.
(Make Groups of 4)

6. Dotplot of MPG Distribution
A variable generally takes on many different values. In data analysis, we are interested in how often a variable takes on each value.
Definition:
Distribution – tells us what values a variable takes and how often it takes those values
Example
Dotplot of MPG Distribution
Variable of Interest:
MPG

7. Data Analysis How to Explore Data Examine each variable by itself.
Then study relationships among the variables.
Start with a graph or graphs
Add numerical summaries

8. Data Analysis From Data Analysis to Inference Population Sample
Collect data from a representative Sample...
Make an Inference about the Population.
Perform Data Analysis, keeping probability in mind…

9. Inference:
Inference is the process of making a conclusion about a population based on a sample set of data.

10. Introduction Data Analysis: Making Sense of Data
Summary
In this section, we learned that…
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.

11. Do Notes worksheet : First page

12. Looking Ahead… In the next Section…
We’ll learn how to analyze categorical data.
Bar Graphs
Pie Charts
Two-Way Tables
Conditional Distributions
We’ll also learn how to organize a statistical problem.

13. Section 1.1 Analyzing Categorical Data
Learning Objectives
After this section, you should be able to…
CONSTRUCT and INTERPRET bar graphs and pie charts
RECOGNIZE “good” and “bad” graphs
CONSTRUCT and INTERPRET two-way tables
DESCRIBE relationships between two categorical variables
ORGANIZE statistical problems

14. Analyzing Categorical Data
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.
Analyzing Categorical Data
Example, page 8
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

15. Analyzing Categorical Data
Displaying categorical data
Frequency tables can be difficult to read. Sometimes is is easier to analyze a distribution by displaying it with a bar graph or pie chart.
Analyzing Categorical Data
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

16. Analyzing Categorical Data
Graphs: Good and Bad
Bar graphs compare several quantities by comparing the heights of bars that represent those quantities.
Our eyes react to the area of the bars as well as height. Be sure to make your bars equally wide.
Avoid the temptation to replace the bars with pictures for greater appeal…this can be misleading!
Analyzing Categorical Data
Alternate Example: The following ad for DIRECTV has multiple problems. See how many your students can point out. First, the heights of the bars are not accurate. According to the graph, the difference between 81 and 95 is much greater than the difference between 56 and 81. Also, the extra width for the DIRECTV bar is deceptive since our eyes respond to the area, not just the height.
Alternate Example
This ad for DIRECTV has multiple problems. How many can you point out?

17. What personal media do you own?
Example: Bar graph
What personal media do you own?
Here are the percents of year olds who own the following personal media devices.
a) Make a well-labeled bar graph. What do you see?
b) Would it be appropriate to make a pie chart for these data? Why or why not?
Device
% who won
Cell Phone 85
MP3 player 83
Handheld Video game player 41
Laptop 38
Portable CD/tape player 20

18. Analyzing Categorical Data
Two-Way Tables and Marginal Distributions
When a dataset involves two categorical variables, we begin by examining the counts or percents in various categories for one of the variables.
Analyzing Categorical Data
Definition:
Two-way Table – describes two categorical variables, organizing counts according to a row variable and a column variable.
Example, p. 12
What are the variables described by this two-way table?
How many young adults were surveyed?
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
Alternate Example: Super Powers
A sample of 200 children from the United Kingdom ages 9-17 was selected from the CensusAtSchool website ( The gender of each student was recorded along with which super power they would most like to have: invisibility, super strength, telepathy (ability to read minds), ability to fly, or ability to freeze time. Here are the results:

19. Analyzing Categorical Data
Two-Way Tables and Marginal Distributions
Analyzing Categorical Data
Definition:
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.
To examine a marginal distribution,
Use the data in the table to calculate the marginal distribution (in percents) of the row or column totals.
Make a graph to display the marginal distribution.

20. Analyzing Categorical Data
Two-Way Tables and Marginal Distributions
Analyzing Categorical Data
Example, p. 13
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%

21. Relationships Between Categorical Variables
Marginal distributions tell us nothing about the relationship between two variables.
Definition:
A Conditional Distribution of a variable describes the values of that variable among individuals who have a specific value of another variable.
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 side-by-side bar graph or segmented bar graph to compare distributions.

22. Young adults by gender and chance of getting rich
Two-Way Tables and Conditional Distributions
Example, p. 15
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
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%

23. Analyzing Categorical Data
Organizing a Statistical Problem
As you learn more about statistics, you will be asked to solve more complex problems.
Here is a four-step process you can follow.
Analyzing Categorical Data
How to Organize a Statistical Problem: A Four-Step Process
State: What’s the question that you’re trying to answer?
Plan: How will you go about answering the question? What statistical techniques does this problem call for?
Do: Make graphs and carry out needed calculations.
Conclude: Give your practical conclusion in the setting of the real-world problem.

24. Do : P- 25 # 26. State: Do the data support the idea that people who get angry easily tend to have more heart disease? Plan: We suspect that people with different anger levels will have different rates of CHD, so we will compare the conditional distributions of CHD for each anger level. Do: We will display the conditional distributions in a table to compare the rate of CHD occurrence for each of the 3 anger levels Conclude: The conditional distributions show that while CHD occurrence is quite small overall, the percent of the population with CHD does increase as eth anger level increases.

25. Section 1.1 Analyzing Categorical Data
Summary
In this section, we learned that…
The distribution of a categorical variable lists the categories and gives the count or percent of individuals that fall into each category.
Pie charts and bar graphs display the distribution of a categorical variable.
A two-way table of counts organizes data about two categorical variables.
The row-totals and column-totals in a two-way table give the marginal distributions of the two individual variables.
There are two sets of conditional distributions for a two-way table.

26. Section 1.1 Analyzing Categorical Data
Summary, continued
In this section, we learned that…
We can use a side-by-side bar graph or a segmented bar graph to display conditional distributions.
To describe the association between the row and column variables, compare an appropriate set of conditional distributions.
Even a strong association between two categorical variables can be influenced by other variables lurking in the background.
You can organize many problems using the four steps state, plan, do, and conclude.

27. Looking Ahead… In the next Section…
We’ll learn how to display quantitative data.
Dotplots
Stemplots
Histograms
We’ll also learn how to describe and compare distributions of quantitative data.
In the next Section…

28. Section 1.2 Displaying Quantitative Data with Graphs
Learning Objectives
After this section, you should be able to…
CONSTRUCT and INTERPRET dotplots, stemplots, and histograms
DESCRIBE the shape of a distribution
COMPARE distributions
USE histograms wisely

29. Displaying Quantitative Data
Dotplots
One of the simplest graphs to construct and interpret is a dotplot. Each data value is shown as a dot above its location on a number line.
Displaying Quantitative Data
How to Make a Dotplot
Draw a horizontal axis (a number line) and label it with the variable name.
Scale the axis from the minimum to the maximum value.
Mark a dot above the location on the horizontal axis corresponding to each data value.
Number of Goals Scored Per Game by the 2004 US Women’s Soccer Team 3 2 7 8 4 5 1 6

30. Displaying Quantitative Data
Examining the Distribution of a Quantitative Variable
The purpose of a graph is to help us understand the data. After you make a graph, always ask, “What do I see?”
Displaying Quantitative Data
How to Examine the Distribution of a Quantitative Variable
In any graph, look for the overall pattern and for striking departures from that pattern.
Describe the overall pattern of a distribution by its:
Shape
Center
Spread
Outliers
Note individual values that fall outside the overall pattern. These departures are called outliers.
Don’t forget your SOCS!

31. Displaying Quantitative Data
Examine this data
The table and dotplot below displays the Environmental Protection Agency’s estimates of highway gas mileage in miles per gallon (MPG) for a sample of 24 model year 2009 midsize cars.
Displaying Quantitative Data
Example, page 28
Describe the shape, center, and spread of the distribution. Are there any outliers?

32. Displaying Quantitative Data
Describing Shape
When you describe a distribution’s shape, concentrate on the main features. Look for rough symmetry or clear skewness.
Displaying Quantitative Data
Definitions:
A distribution is roughly symmetric if the right and left sides of the graph are approximately mirror images of each other.
A distribution is skewed to the right (right-skewed) if the right side of the graph (containing the half of the observations with larger values) is much longer than the left side.
It is skewed to the left (left-skewed) if the left side of the graph is much longer than the right side.
Symmetric
Skewed-left
Skewed-right

33. Battery Life (minutes)
You Do: Smart Phone Battery Life
Here is the estimated battery life for each of 9 different smart phones (in minutes) according to
Make a dot plot.
* Then describe the shape, center, and spread.
of the distribution. Are there any outliers?
Smart Phone
Battery Life (minutes)
Apple iPhone
300
Motorola Droid
385
Palm Pre
Blackberry Bold
360
Blackberry Storm
330
Motorola Cliq
Samsung Moment
Blackberry Tour
HTC Droid
460

34. Solution: Shape: There is a peak at 300 and the distribution has a long tail to the right (skewed to the right). Center: The middle value is 330 minutes. Spread: The range is 460 – 300 = 160 minutes. Outliers: There is one phone with an unusually long battery life, the HTC Droid at 460 minutes.

35. Comparing Distributions
Some of the most interesting statistics questions involve comparing two or more groups.
Always discuss shape, center, spread, and possible outliers whenever you compare distributions of a quantitative variable.
Example, page 32
Compare the distributions of household size for these two countries. Don’t forget your SOCS!
( Compare each characteristic)
U.K South Africa
Place

36. Displaying Quantitative Data
Stemplots (Stem-and-Leaf Plots)
Another simple graphical display for small data sets is a stemplot. Stemplots give us a quick picture of the distribution while including the actual numerical values.
Displaying Quantitative Data
How to Make a Stemplot
Separate each observation into a stem (all but the final digit) and a leaf (the final digit).
Write all possible stems from the smallest to the largest in a vertical column and draw a vertical line to the right of the column.
Write each leaf in the row to the right of its stem.
Arrange the leaves in increasing order out from the stem.
Provide a key that explains in context what the stems and leaves represent.

37. Displaying Quantitative Data
Stemplots (Stem-and-Leaf Plots)
These data represent the responses of 20 female AP Statistics students to the question, “How many pairs of shoes do you have?” Construct a stemplot.
Displaying Quantitative Data 50 26 31 57 19 24 22 23 38 13 34 30 49 15 51
Stems 1 2 3 4 5
Add leaves
4 9
Order leaves
4 9
Add a key
Key: 4|9 represents a female student who reported having 49 pairs of shoes.

38. Splitting Stems and Back-to-Back Stemplots
When data values are “bunched up”, we can get a better picture of the distribution by splitting stems.
Two distributions of the same quantitative variable can be compared using a back-to-back stemplot with common stems.
Females
Males 50 26 31 57 19 24 22 23 38 13 34 30 49 15 51 14 7 6 5 12 38 8 10 11 4 22 35
Females
333 95
4332 66
410 8 9
100 7
Males
0 4 1
2 2 2 3
3 58 4 5 1 2 3 4 5
“split stems”
Key: 4|9 represents a student who reported having 49 pairs of shoes.

39. From both sides, compare and write about the center , spread ,
shape and possible outliers.
Do CYU : P- 34 .
Check Your Understanding, page 34:
1. In general, it appears that females have more pairs of shoes than males. The median report for the males was 9 pairs while the female median was 26 pairs. The females also have a larger range of =44 in comparison to the range of 38-4=34 for the males .
Finally, both males and females have distributions that are skewed to the right, though the distribution for the males is more heavily skewed as evidenced by the three likely outliers at 22, 35 and 38. The females do not have any likely outliers.
2. b
3. b
4. b

40. Histograms
Quantitative variables often take many values. A graph of the distribution may be clearer if nearby values are grouped together.
The most common graph of the distribution of one quantitative variable is a histogram.
How to Make a Histogram
Divide the range of data into classes of equal width.
Find the count (frequency) or percent (relative frequency) of individuals in each class.
Label and scale your axes and draw the histogram. The height of the bar equals its frequency. Adjacent bars should touch, unless a class contains no individuals.

41. Displaying Quantitative Data
Making a Histogram
The table on page 35 presents data on the percent of residents from each state who were born outside of the U.S.
Example, page 35
Displaying Quantitative Data
Frequency Table
Class
Count
0 to <5 20
5 to <10 13
10 to <15 9
15 to <20 5
20 to <25 2
25 to <30 1
Total 50
Percent of foreign-born residents
Number of States

42. Displaying Quantitative Data
Using Histograms Wisely
Here are several cautions based on common mistakes students make when using histograms.
Displaying Quantitative Data
Cautions
Don’t confuse histograms and bar graphs.
Don’t use counts (in a frequency table) or percents (in a relative frequency table) as data. ( P- 40: Read the example)
Use percents instead of counts on the vertical axis when comparing distributions with different numbers of observations.
Just because a graph looks nice, it’s not necessarily a meaningful display of data.

43. Try Check your understanding: P – 39 Many people………
Try Check your understanding: P – 39 Many people……….. (Also put these #s in the calculator.) Solution: 2. The distribution is roughly symmetric and bell-shaped. The median IQ appears to be between 110 and 120 and the IQ’s vary from 80 to 150. There do not appear to be any outliers.

44. Section 1.2 Displaying Quantitative Data with Graphs
Summary
In this section, we learned that…
You can use a dotplot, stemplot, or histogram to show the distribution of a quantitative variable.
When examining any graph, look for an overall pattern and for notable departures from that pattern. Describe the shape, center, spread, and any outliers. Don’t forget your SOCS!
Some distributions have simple shapes, such as symmetric or skewed. The number of modes (major peaks) is another aspect of overall shape.
When comparing distributions, be sure to discuss shape, center, spread, and possible outliers.
Histograms are for quantitative data, bar graphs are for categorical data. Use relative frequency histograms when comparing data sets of different sizes.

45. Do # 56 P- 46

46. Looking Ahead… In the next Section…
We’ll learn how to describe quantitative data with numbers.
Mean and Standard Deviation
Median and Interquartile Range
Five-number Summary and Boxplots
Identifying Outliers
We’ll also learn how to calculate numerical summaries with technology and how to choose appropriate measures of center and spread.