Comparitive Graphs

Two Way Table Describes two categorical variables. One variable is shown in the rows and the other is in the columns.

Example of Two Way Table Young adults by gender & chance of getting rich   Gender Opinion 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 godd chance 663 758 1421 Almost certain 486 597 1083 2367 2459 4826

Reading a Two-Way Table Look at the distribution of each variable separately. The totals on the right are strictly the values for the distribution of opinions about becoming rich for all. The totals at the bottom are for gender

Marginal Distribution 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. It’s the distribution of each category alone.

Percentages Often are more informative Used when comparing groups of different sizes.

Young adults by gender & chance of getting rich Find the percent of young adults who they there is a good chance they will be rich. Young adults by gender & chance of getting rich   Gender Opinion 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 godd chance 663 758 1421 Almost certain 486 597 1083 2367 2459 4826

Young adults by gender & chance of getting rich Find the marginal distribution (in %) of opinions. Make a graph to display the marginal distribution. Young adults by gender & chance of getting rich   Gender Opinion 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 godd chance 663 758 1421 Almost certain 486 597 1083 2367 2459 4826

Response Percent Almost no chance 4.0% Some chance but probably not 14.8% A 50-50 chance 29.3% A good chance 29.4% Almost certain 22.4%

Young adults by gender & chance of getting rich Find the marginal distribution (in %) of gender. Make a graph to display the marginal distribution. Young adults by gender & chance of getting rich   Gender Opinion 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 godd chance 663 758 1421 Almost certain 486 597 1083 2367 2459 4826

Response Percent Male 51% Female 49%

Conditional Distribution It describes the values of that variable among individuals who have a specific value of another variable. To describe the relationship between the two categorical variables

Conditional Distribution of young women and men and their opinion. Young adults by gender & chance of getting rich   Gender Opinion Female Male Almost no chance 96 98 Some chance but probably not 426 286 A 50-50 chance 696 720 A godd chance 663 758 Almost certain 486 597 Total 2367 2459

Segmented Bar Graph Response Women Men Almost no chance 4.1% 4% Some chance but probably not 18.0% 11.6% A 50-50 chance 29.4% 29.3% A good chance 28% 30.8% Almost certain 20.5% 24.3%

Did we look at the right conditional distribution? Our goal was to analyze the relationship between gender and opinion about chances of becoming rich for these young adults.

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.

State What is the relationship between gender and responses to the question “What do you think are the chances you will have much more than a middle-class income at age 30?”

Plan We suspect that gender might influence a young adult’s opinion about the chance of getting rich. So we’ll compare the conditional distributions of response for men alone and for women alone. Response Women Men Almost no chance 4.1% 4% Some chance but probably not 18.0% 11.6% A 50-50 chance 29.4% 29.3% A good chance 28% 30.8% Almost certain 20.5% 24.3%

Do We’ll make a side-by side bar graph to compare the opinions of males and females. I could have used a segmented as well!

Side-by Side Comparative Bar Graph Response Women Men Almost no chance 4.1% 4% Some chance but probably not 18.0% 11.6% A 50-50 chance 29.4% 29.3% A good chance 28% 30.8% Almost certain 20.5% 24.3%

Conclude Based on the sample data, men seem somewhat more optimistic about their future income than women. Men were less likely to say that they have “some chance but probably no” than women (11.6% vs 18.0%). Men were more likely to say that they have a “good chance” (30.8% vs 28.0%) aor alre “almost certain” (24.3% vs 20.5%) to have much more than a middle-class income by age 30 than women were.

Association We say there is an association between two variables if specific values of one variable tend to occur in common with specific values of the other. Be careful though….even a strong association between two categorical variables can be influenced by other variables lurking in the background.

Simpson’s Paradox An association between two variables that holds for each individual value of a thrid variable can be changed or even reversed when the data for all values of the third variable are combined. This reversal is called Simpson’s paradox.

Accident victims are sometimes taken by helicopter from the accident scene to a hospital. Helicopters save taim. Do they also save lives?   Helicopter Road Victim Died 64 260 Victim survived 136 840 Total 200 1100 32% of helicopter patients died, but only 24% of the others did. This seems discouraging!

Helicopter is sent mostly to serious accidents.   Helicopter Road Died 48 60 Survived 52 40 Total 100 Less Serious Accident   Helicopter Road Died 16 200 Survived 84 800 Total 100 1000

Titanic Disaster

Homework Page 24 (19, 21, 23, 24, 25, 27-32, 33, 35, 36)