Chapter 1: Exploring Data Section 1.1 Analyzing Categorical Data
What is Statistics? Statistics is the science of • Collecting data • Analyzing data • Drawing conclusions from data
Major Branches of Statistics • Descriptive Statistics • Organizing, Summarizing Information • Graphical techniques • Numerical techniques • Inferential Statistics • Estimation • Decision making
Important Terms Variable – A variable is any characteristic whose value may change from one individual to another Examples: • Brand of television • Height of a building • Number of students in a class Examples from Algebra?
Important Terms Data results from making observations either on a single variable or simultaneously on two or more variables. A univariate data setconsists of observations on a single variable made on individuals in a sample or population. A bivariate data set consists of observations on two variables made on individuals in a sample or population. A multivariate data setconsists of observations on two or more variables made on individuals in a sample or population.
Data Sets • A univariate data set is categorical (or qualitative) if the individual observations are categorical responses. • A univariate data set is numerical (or quantitative) if the individual observations are numerical responses where numerical operations generally have meaning.
Categorical Variables place individuals into one of several groups or categories • EXAMPLES?
Analyzing Categorical Data • Thedistributionof a categorical variable lists the count or percent of individuals who fall into each category. Example, page 8 Variable Count Percent
Analyzing Categorical Data • 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.
Types of Numerical Data Numerical data is discrete if the possible values are isolated points on the number line. Numerical data is continuous if the set of possible values form an entire interval on the number line.
Examples of Discrete Data • The number of students served in the MHS lunchroom over a one hour time period for a sample of 7 different days: 230 220 310 280 210 270 320 • The number of textbooks bought by students at a given school during a semester for a sample of 16 students 5 3 6 8 6 1 3 6 12 3 5 7 6 7 5 4
Examples of Continuous Data • The height of students that are taking AP Statistics for a sample of 10 students. 72.1” 64.3” 68.2” 74.1” 66.3” 61.2” 68.3” 71.1” 65.9” 70.8” Note: Even though the heights are only measured accurately to 1 tenth of an inch, the actual height could be any value in some reasonable interval.
Examples of Continuous Data The crushing strength of a sample of four jacks used to support trailers. 7834 lb 8248 lb 9817 lb 8141 lb Gasoline mileage (miles per gallon) for a brand of car is measured by observing how far each of a sample of seven cars of this brand of car travels on ten gallons of gasoline. 23.1 26.4 29.8 25.0 25.9 22.6 24.3
To decide if a given data set represents continuous or discrete numerical data, use the following guidelines: • Discrete data is typically gathered by counting during observation • Continuous data is typically found by a measuring process
Deceptive Graphs • It is important to be an informed consumer by: • Extracting information from charts and graphs • Following numerical arguments • Knowing the basics of how data should be gathered, summarized and analyzed to draw statistical conclusions • Understanding the validity and appropriateness of processes and decisions that affect your life
The NY Times ran an article in the “Week in Review” on the housing bubble on 9-23-2007. Included was a graphic that, with some clever scaling, exaggerates the changes in housing prices over the last 20 years.
The following bar graph gives the percent of owners of three brands of trucks who are satisfied with their truck. Truck Brands From this graph we may legitimately conclude A) there is very little difference in the satisfaction of owners for the three brands. B) Chevrolet owners are substantially more satisfied than Ford or Toyota owners. C) owners of other brands of trucks are less satisfied than the owners of these three brands. D) Chevrolet probably sells more trucks than Ford or Toyota.
Graphs: Good and Bad Analyzing Categorical Data Bar graphs compare several quantities by comparing the heights of bars that represent those quantities. Our eyes react to the area of the bars as well as height. Be sure to make your bars equally wide. Avoid the temptation to replace the bars with pictures for greater appeal…this can be misleading! Alternate Example This ad for DIRECTV has multiple problems. How many can you point out?
Analyzing Categorical Data • 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. Definition: Two-way Table – describes two categorical variables, organizing counts according to a row variable and a column variable. Example, p. 12 What are the variables described by this two-way table? How many young adults were surveyed?
Analyzing Categorical Data • Two-Way Tables and Marginal Distributions Definition: 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: Are percents or counts more informative when comparing groups of different sizes? • 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.
Analyzing Categorical Data • Two-Way Tables and Marginal Distributions Example, p. 13 Examine the marginal distribution of chance of getting rich.
Analyzing Categorical Data • Relationships between Categorical Variables • Marginal distributions tell us nothing about the relationship between two variables. Definition: 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, • 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.
Analyzing Categorical Data • Two-Way Tables and Conditional Distributions Example, p. 15 Calculate the conditional distribution of opinion among males. Examine the relationship between gender and opinion.
Analyzing Categorical Data • Organizing a Statistical Problem • As you learn more about statistics, you will be asked to solve more complex problems. • Here is a four-step process you can follow. How to Organize a Statistical Problem: A 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.