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

Data Analysis: Making Sense of Data IDENTIFY the individuals and variables in a set of data CLASSIFY variables as categorical or quantitative DEFINE “Distribution” DESCRIBE the idea behind “Inference”

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.

Example: Quantitative or Categorical? Class (fr, soph, jun, sen) Grade point average Email address Name Bus route Phone number Days absent

Example: Quantitative or Categorical? Address Number of Credits earned Allergies Exterior color Mileage Total car length Number of cylinders

Example: Quantitative or Categorical? Cost Model VIN Type of sound system Size of fuel tank Zip code Shoe Size

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

Example

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

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…

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.

CHAPTER 1 Exploring Data 1.1 Analyzing Categorical Data

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

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

Displaying Categorical Data Frequency tables can be difficult to read. Sometimes it 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

Bar Graphs and Pie Charts Each bar must be same width. Can display as count or percentage Scale should start at 0 Pie Charts Need to know the percentages to create. Must add up to 100% (must be parts of a whole) Can add another category when appropriate so that percentages total 100%. Use when you want to emphasize each category’s relation to the whole.

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.

Graphs: Good and Bad

Graphs: Good and Bad This ad for DIRECTV has multiple problems. How many can you point out?

Graphs: Good and Bad

Careful with Scales!

These graphs show the same information: Google: 30% Yahoo: 35% MSN: 35%

Most Deceptive Graph Award

Homework pg. 6-7 # 1, 3, 5, 7, 8 pg. 20-22 #11, 13, 15, 17

Chapter 1: Exploring Data Section 1.1: Analyzing Categorical Data

Recap

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 50-50 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?

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.

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 50-50 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 50-50 chance 1416/4826 = 29.3% A good chance 1421/4826 = 29.4% Almost certain 1083/4826 = 22.4%

Homework Pg. 22 #19, 20

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.

Relationships Between Categorical Variables Calculate the conditional distribution of opinion among males and then females. 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 50-50 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 50-50 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%

Conclusion What does this tell us about the relationship between gender and opinion about future wealth for this sample of young adults? -There does seem to be an association between the two variables for this sample. -Men more often rated their chances in the two highest categories, while women said “some chance” much more frequently than men.

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.

Superpowers A sample of 200 children from the United Kingdom aged 9-17 was selected from the CensusAtSchool Web site. The gender of each student was recorded along with which superpower they would most like to have: invisibility, superstrength, telepathy, ability to fly, ability to freeze time. Here are the results:

Superpowers 1. Use the data in the two-way table to calculate the marginal distribution (in percents) of superpower preferences. 2. Make a graph to display the marginal distribution. Describe what you see. 3. Do these data suggest that there is an association between gender and superpower preference? Give appropriate evidence to support your answer.

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

Homework pg. 22-24 #21, 23, 26, 27-34