Name AP Statistics UNIT 1 Summer Work Section II: Notes Analyzing Categorical Data

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1 Name AP Statistics UNIT 1 Summer Work Date Section II: Notes Analyzing Categorical Data Essential Understanding: How can I represent the data when it is treated as a categorical variable? I. Distribution of a Categorical Variable Values of a Categorical Variable: the labels for the categories Ex) Distribution of a Categorical Variable: Describes the possible results of a random event where the results are distributed into categories. It lists the categories and gives either the count or percent of individuals who fall in each category. Exs) Categorical Variable Distributions as a Table: Frequency Table: Relative Frequency Table: shows the percents (relative frequencies) displays the counts (frequencies) in each format category FT: Counts should add up to: RFT: Percents should add up to: Individuals for this ex): Variable measured: Round- off Error: When exact percents don t add up to exactly 100% (effect of rounding off results) Ex from reading) II. Visual Displays of Categorical Distributions: Bar Graphs, Segmented and Side- by- Side Bar Graphs, Pie Charts and Pictographs Used to display the distribution of a categorical variable Always think about whether the graph you chose will help you answer your questions about the data Pie Charts: When you use it: Slices are sized by the counts or percents for the categories Must include all categories Can be generated with technology - StatCrunch

2 Bar Graphs: Represent each category as a bar Bar heights show the category s counts or percents More flexible than pie charts because it can also compare any set of quantities that are measured in the same units. Bars should be the same width Example from Radio Data: Bar Charts: the term for a display of the counts of a categorical with bars Difference between histogram and bar graph: : OK to make a bar graph and can rearrange the order of bars : OK to make a histogram and cannot change order of bars Side- by- Side and Segmented Bar Graphs: Used to compare distributions when given categorical variable. Example of Women s and Men s Opinions Type of Graph: Type of Graph: Individuals: Variable(s): Segmented bar graphs display the conditional distribution(part V in these notes) of a categorical variable within each category of another variable III. Graphs: What Can Go Wrong? Area Principle: - In a statistical display, each data value should be represented by the same amount of area - It says that the area occupied by part of the graph should correspond to the magnitude of the value it represents.

3 Bar Graphs: Make the bars equally wide Choose bar graph over a pictograph- eyes respond to the area of the picture instead Watch the scales on the bar graph Examine the bar graphs on the right and comment: Pictographs: Beware/Don t use. How are these two graphs misleading? IV. Two - Way Tables and Marginal Distribution Two- way table: Describes two categorical variables Ex from reading) Row variable: variable in each row Ex above) row variable = Column variable: variable in each column Column variable = Look at the distribution of each variable separately: Distribution of a categorical variable says how often each outcome occurred. Calculate totals if they are missing from a two- way table (last row for column total, last column for row Type of Car by Gender total) Marginal Distribution of one of the categorical variables: In a two- way table of counts, it is the distribution of values of that variable among all individuals described by the table. Percents are more often informative for this. found in the last column and last row (margins) on the two- way table tell us nothing about the relationship between two variables Marginal Distribution calculations only use over Ex from reading) Ex from Car and Gender)

4 V. Conditional Distributions Relationships between categorical variables (Often used when finding conditional probabilities!) Conditional Distribution of a Variable: Describes the values of that variable among individuals who have a specific value of another variable. There is a separate conditional distribution for each value of the other variable. Conditional distribution calculations use over Ex from reading: Ex from Car and Gender: Software will calculate conditional distributions for you. You can choose which you would like to calculate. There are of conditional distributions for any two- way table. Ex from reading: Ex from Car and Gender: To Display Conditional Distributions: (other than with the two- way table) Segmented Bar Graph: Sometimes used to compare conditional distributions. Each bar has a total of 100% OR Side- by Side Bar Graph: Easier to make Comparing Conditional Distributions: Think about what information the distributions give you, and what you are trying to analyze. Which conditional distributions should we find and compare? Ex) VI. Organizing a Statistical Problem - General Framework for Organizing Your Thinking What information would provide a context? A context ideally tells Who was measured, What was measured (and units), how the data were collected, Where the data were collected, and When and Why the study was performed. Ex) In December 2011, Consumer Reports published an evaluation of 25 tablets from a variety of manufacturers. Describe the population of interest, the sample, and the Who of the study. Population of interest: Sample: Who:

5 General Framework for Organizing Statistical Problems throughout the Entire Course The 4 Step Process THINK think about where the data come from, then: #1 STATE the problem #2 PLAN by identifying the procedure & variables, by checking the appropriate conditions, and by reporting the W s (Who, What, When, Where, how, Why) SHOW - calculation and graph (most of the math) #3 DO the calculations (ex- marginal distributions for categorical data)before looking at two variables together Make appropriate displays (ex- pie chart or bar chart for categorical variables) TELL- you aren t done!! ½ of the work is left CONCLUDE and EXPLAIN #4 CONCLUDE by interpreting the patterns in the table and displays using the data Be careful not to overstate For quantitative variables, describe the shape, center and spread Example of a Statistical Problem with Categorical Variables: (use p.17 for help) Based on the survey data, can we conclude that young men and women differ in their opinions about the likelihood of future wealth? Give appropriate evidence to support your answer. Follow the four- step process. 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 DO: We ll make a side- by- side bar graph to compare the opinions of males and females: CONCLUDE: Based on the sample data,

6 Association: There is an association between two variables if specific values of one variable tend to occur in common with specific values of the other. The last example s graph provides That is, the value of one variable ( ) tend to occur more or less frequently in combination with specific values of the other variable ( ). Men more often rated their chances of becoming rich in the two highest categories Women said some chance but probably not much more frequently. What about an association between gender and opinion in the POPULATION of young adults? This requires formal inference later on in the course. Since this was a random sample, an inference method allows us to predict a population DATA EXPLORATION A Titanic disaster (p.19 in book) In 1912 the luxury liner Titanic, on its first voyage across the Atlantic, struck an iceberg and sank. Some passengers got off the ship in lifeboats, but many died. The two- way table below gives information about adult passengers who lived and who died, by class of travel. Answer the questions from the book: For #1, use the 4- step process: STATE, PLAN, DO, CONCLUDE. Make sure to state the problem, give a plan, calculate conditional distributions, make two side- by- side bar graphs(try this kind)- LABEL-, and include a conclusion on what you see. For #2, use your statistics found (conditional probabilities), and be sure to comment about class AND gender. Ex) % of first class that are female NOW: Complete the book assignment from Section II