Statistics: Analyzing 2 Categorical Variables MIDDLE SCHOOL LEVEL  Session #1  Presented by: Dr. Del Ferster

 Why are statistics significant? Why should we have young students be aware of statistics?  What kind of statistics can we consider with elementary students?  Why do many people who have studied statistics have “bad memories” of the subject?

 We’re going to spend time today on QUALITATIVE STATISTICS.  We’ll consider effective ways to summarize qualitative statistics.  We’ll build TWO WAY TABLES.  We’ll do an activity involving qualitative statistics that you might be able to adapt for use with your students.

 Qualitative variables classify the data into categories.  The categories may or may not have a natural ordering to them.  Qualitative variables are also called categorical variables.  EXAMPLES ◦ Eye color ◦ Political party ◦ Gender ◦ Do you smoke?

 Quantitative variables have numerical values that are measurements (length, weight, and so on) or counts (of how many).  Examples: ◦ How many are in your family? ◦ How many cars do you own?

 We further distinguish quantitative variables based on whether or not the values fall on a continuum. ◦ A discrete variable is one for which you can count the number of possible values.  How many siblings a person has ◦ A continuous variable can take on any value within a given interval.  A person’s weight

 We’ll take a closer look at quantitative variables during our next meeting.

A look at ways to represent our data

 The distribution of a categorical variable provides the possible values that a variable can take on and how often these possible values occur.  The distribution of a categorical variable shows the pattern of variation of the variable.

 According to the Bureau of Justice, the following data represent the number of inmates by ethnicity in White338,400 Black301,900 Hispanic125,600

 Often, rather than simply presenting numerical values, we choose to graph our data.  When generating a graph of 1 categorical variable, we might consider the following types of graph. ◦ Pie Chart ◦ Bar Graph

 A pie chart displays the distribution of the qualitative variable by dividing the circle into wedges corresponding to the categories of the variable such that the angle of each wedge is proportional to the percentage of items in that category.  Pie Charts are easy to do in EXCEL.

White338,400 Black301,900 Hispanic125,600

White338,400 Black301,900 Hispanic125,600

 A bar graph displays the distribution of a qualitative variable by listing the categories of the variable along one axis and drawing a bar over each category with a height equal to the percentage of items in that category.  The bars should all be of equal width.  We could also do one using percents.

White338,400 Black301,900 Hispanic125,600

How does one variable compare to another? 2 Way Tables

 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.

 When a dataset involves two categorical variables, we begin by examining the counts or percents in various categories for one of the variables. Two-way Table – describes two categorical variables, organizing counts according to a row variable and a column variable.

 Two-way tables come about when we are interested in the relationship between two categorical variables. ◦ One of the variables is the row variable. ◦ The other is the column variable. ◦ The combination of a row variable and a column variable is a cell.

 Dr. F is hosting 38 of his friends to a cookout. Now, Dr. F. has limited cooking skills, so everyone is having a burger. However, he has bought sufficient tomatoes so that anyone who wants tomato on his or her burger will be happy.  The following slide details the results of his burger and tomato survey. ◦ For the record….a good burger needs only 2 things…CHEESE….and KETCHUP!

 Let’s look at the components of a 2 way table Row variable Column variable Column Totals Row Totals Overall Total Cells

 Dr. F. decided to survey a group of young adults, to determine whether they expected to be rich by the age of 30.  He decided to consider gender as one variable  The other variable indicates each participant’s expected likelihood of being rich (using the following options) ◦ Almost no chance ◦ Some chance, but probably not ◦ A chance ◦ A good chance ◦ Almost certain

 Note: Percents are often more informative than counts, especially when comparing groups of different sizes. 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.

 To examine a marginal distribution: 1. Use the data in the table to calculate the marginal distribution (in percents) of the row or column totals. 2. Make a graph to display the marginal distribution

Examine the marginal distribution of chance of getting rich.

 Marginal distributions tell us nothing about the relationship between two variables. 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: 1. Select the row(s) or column(s) of interest. 2. Use the data in the table to calculate the conditional distribution (in percents) of the row(s) or column(s). 3. Make a graph to display the conditional distribution. 4. Use a side-by-side bar graph or segmented bar graph to compare distributions.

Conditional Distribution Calculate the conditional distribution of opinion among males. Examine the relationship between gender and opinion.

An example concerning marginal and conditional distributions

 Enrollment of recent high school graduates. The table below gives some census data concerning the enrollment status of recent high school graduates aged 16 to 24 years.

 How many male recent high school graduates aged 16 to 24 years were enrolled full-time in two-year colleges?  How many female recent high school graduates aged 16 to 24 years were enrolled in graduate schools?

 See if you can use the skills developed in this presentation to complete the handout that Dr. F. will distribute.  Feel free to consult your notes, work together, or ask me if you get really stuck or frustrated.  RELAX…it’s just for FUN!

 The marginal distribution of gender statusmenwomen 2 year college, full time year college, part time year college, full time year college, part time graduate school total Marginal Distributions of gender46.5%53.5%

 Graph of The marginal distribution of gender

 The marginal distribution of status statusPercent 2 year college, full time17.8% 2 year college, part time7.1% 4 year college, full time59.7% 4 year college, part time6.1% graduate school6.4% vocational school2.9%

 Graph of The marginal distribution of status

 Conditional Distribution of Gender for each status 2 yearcollege,full time 2 yearcollege,part time 4 yearcollege,full time 4 yearcollege,part time graduateschool vocationalschool Men47.9%45.8%46.6%39.4%45.5%53.9% Women52.1%54.2%53.4%60.6%54.5%46.1%

 Graph of Conditional Distribution of Gender for each status

 Conditional Distribution of Status for Each Gender 2 yearcollege,full time 2 yearcollege,part time 4 yearcollege,full time 4 yearcollege,part time graduateschool vocationalschool Men18.4%7.0%59.8%5.1%6.3%3.3% Women17.4%7.2%59.5%6.9%6.6%2.5%

 Graph of Conditional Distribution of Status for Each Gender

 A Different Graph of Conditional Distribution of Status for Each Gender

 Next time we’ll be looking at:  1.Analysis of quantitative statistics  2. We’ll consider linear regression (without having to actually calculate the equation of the regression line.  3.We’ll also look at correlation