1. CHAPTER 1 Exploring Data
Introduction
Data Analysis:
Making Sense of Data

2. Data Analysis: Making Sense of Data
IDENTIFY the individuals and variables in a set of data
CLASSIFY variables as categorical or quantitative

3. What is Statistics?

4. What is going on here!
Take your piece of the graph and try to make one complete graph by finding the other pieces that go with your piece.

5. A) The months all OMHS students were born
Puzzle Exercise
Once you have completed your graph, discuss with the other members in your group what data your graph represents. Your options are as follows:
A) The months all OMHS students were born
B) The distribution of GPA’s at Harvard
C) The distribution of SAT scores in Maryland
D) The distribution of salaries for students right out of college

7. Is the data convincing?
One of the main things we will be looking at this year is how to determine whether or not we have enough data to make a conclusion about some hypothesis.

8. Stats Starts Here
According to 100% of people surveyed, this is the greatest class ever offered in college.
List of people surveyed: ME

9. The Exam (FICTIONAL DATA)
The class average last semester was a 94 on the final exam.
It was out of 500 points!!!!
A group of individuals averaged a 78% on an algebra exam……
The group of individuals were 7 year olds…
Or…the group of individuals were Algebra teachers!

10. Dream Job
The average salary at a company that has 25 employees is $8,500,000 per year.
Would you like to be hired by this company?
The CEO makes
$212,400,000 per year…
The other 24 employees
Average $4, per/year

11. The “W’s” To provide context we need the W’s Who
What (and in what units)
When
Where
Why (if possible)
and How
of the data.
Note: the answers to “who” and “what” are essential.

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

13. WEBQUEST Types of Variables

14. Dotplot of MPG Distribution
Data Analysis
A variable generally takes on many different values.
We are interested in how often a variable takes on each value.
Distribution
tells us what values a variable takes and how often it takes those values.
Dotplot of MPG Distribution
Variable of Interest:
MPG

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

16. 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…

17. Data Analysis: Making Sense of Data
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.

18. CHAPTER 1 Exploring Data
1.1 Analyzing Categorical Data

19. Analyzing Categorical 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

20. 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

21. 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.
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

22. Graphs: Good and Bad
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.
When you draw a bar graph, make the bars equally wide.
It is tempting to replace the bars with pictures for greater eye appeal.
Don’t do it!
There are two important lessons to keep in mind:
beware the pictograph, and
watch those scales.

23. 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.
A two-way table describes two categorical variables, organizing counts according to a row variable and a column variable.
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?

24. 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.
Make a graph to display the marginal distribution.

25. Two-Way Tables and Marginal Distributions
Examine the marginal distribution of chance of getting rich.
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
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%

26. 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 side-by-side bar graph or segmented bar graph to compare distributions.

27. Relationships Between Categorical Variables
Calculate the conditional distribution of opinion among males. Examine the relationship between gender and opinion.
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
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%

28. Relationships Between Categorical Variables
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.

29. 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

30. CHAPTER 1 Exploring Data
1.2 Displaying Quantitative Data with Graphs

31. Displaying Quantitative Data with Graphs
MAKE and INTERPRET dotplots and stemplots of quantitative data
DESCRIBE the overall pattern of a distribution and IDENTIFY any outliers
IDENTIFY the shape of a distribution
MAKE and INTERPRET histograms of quantitative data
COMPARE distributions of quantitative data

32. Number of Goals Scored Per Game by the 2012 US Women’s Soccer Team
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.
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 2012 US Women’s Soccer Team 2 1 5 3 4 13 14

33. 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?”
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
Note individual values that fall outside the overall pattern. These departures are called outliers.
Don’t forget your SOCS!

34. Describing Shape
When you describe a distribution’s shape, concentrate on the main features. Look for rough symmetry or clear skewness.
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

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.
Compare the distributions of household size for these two countries.
Don’t forget your SOCS!

36. Stemplots
Another simple graphical display for small data sets is a stemplot. (Also called a stem-and-leaf plot.)
Stemplots give us a quick picture of the distribution while including the actual numerical values.
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. Stemplots
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. 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. 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. 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.

40. Histograms
This table presents data on the percent of residents from each state who were born outside of the U.S.
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

41. Using Histograms Wisely
Here are several cautions based on common mistakes students make when using histograms.
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.
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.

42. Data Analysis: Making Sense of Data
MAKE and INTERPRET dotplots and stemplots of quantitative data
DESCRIBE the overall pattern of a distribution
IDENTIFY the shape of a distribution
MAKE and INTERPRET histograms of quantitative data
COMPARE distributions of quantitative data