Elementary Statistics: Looking at the Big Picture (C) 2007 Nancy Pfenning Lecture 5: Chapter 4, Section 1 Single Variables (Focus on Categorical Variables) Displays and Summaries Data Production Issues Looking Ahead to Inference Details about Displays and Summaries ©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture Elementary Statistics: Looking at the Big Picture

Elementary Statistics: Looking at the Big Picture (C) 2007 Nancy Pfenning Looking Back: Review 4 Stages of Statistics Data Production (discussed in Lectures 1-4) Displaying and Summarizing Single variables: 1 categorical, 1 quantitative Relationships between 2 variables Probability Statistical Inference ©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture Elementary Statistics: Looking at the Big Picture

First Process of Statistics (C) 2007 Nancy Pfenning First Process of Statistics 1. Data Production: Take sample data from the population, with sampling and study designs that avoid bias. POPULATION SAMPLE SAMPLE ©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture Elementary Statistics: Looking at the Big Picture

Second Process of Statistics (C) 2007 Nancy Pfenning Second Process of Statistics 1. Data Production: Take sample data from the population, with sampling and study designs that avoid bias. 2. Displaying and Summarizing: Use appropriate displays and summaries of the sample data, according to variable types and roles. POPULATION C Q SAMPLE CQ CC QQ ©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture Elementary Statistics: Looking at the Big Picture

Third Process of Statistics: (C) 2007 Nancy Pfenning Third Process of Statistics: 1. Data Production: Take sample data from the population, with sampling and study designs that avoid bias. 2. Displaying and Summarizing: Use appropriate displays and summaries of the sample data, according to variable types and roles. POPULATION C SAMPLE CQ QQ PROBABILITY 3. Probability: Assume we know what’s true for the population; how should random samples behave? ©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture Elementary Statistics: Looking at the Big Picture

Fourth Process of Statistics (C) 2007 Nancy Pfenning Fourth Process of Statistics 1. Data Production: Take sample data from the population, with sampling and study designs that avoid bias. 2. Displaying and Summarizing: Use appropriate displays and summaries of the sample data, according to variable types and roles. POPULATION SAMPLE PROBABILITY 3. Probability: Assume we know what’s true for the population; how should random samples behave? INFERENCE C Q 4. Statistical Inference: Assume we only know what’s true about sampled values of a single variable or relationship; what can we infer about the larger population? CQ CC QQ ©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture Elementary Statistics: Looking at the Big Picture

Handling Single Categorical Variables (C) 2007 Nancy Pfenning Handling Single Categorical Variables Display: Pie chart Bar graph Summary: Count Percent Proportion ©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture Elementary Statistics: Looking at the Big Picture

Definitions and Notation (C) 2007 Nancy Pfenning Definitions and Notation Statistic: number summarizing sample Parameter: number summarizing population : sample proportion (a statistic) [“p-hat”] p: population proportion (a parameter) ©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture Elementary Statistics: Looking at the Big Picture

Example: Issues to Consider (C) 2007 Nancy Pfenning Example: Issues to Consider Background: 246 of 446 students at a certain university had eaten breakfast on survey day. Questions: Are intro stat students representative of all students at that university? Would they respond without bias? Responses: Yes, if years and class times are representative. Yes (not sensitive). Looking Back: These are data production issues. ©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture Practice: 4.4e p.79 Elementary Statistics: Looking at the Big Picture

Example: More Issues to Consider (C) 2007 Nancy Pfenning Example: More Issues to Consider Background: 246 of 446 students at a certain university had eaten breakfast on survey day. Questions: How do we display and summarize the info? Can we conclude that a majority of all students at that university eat breakfast? Responses: Display: pie chart Summary: 246/446= 55% or 0.55 ate breakfast. Can’t yet say if majority eat breakfast overall. Looking Ahead: This would be statistical inference. ©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture Elementary Statistics: Looking at the Big Picture

Example: Statistics vs. Parameters (C) 2007 Nancy Pfenning Example: Statistics vs. Parameters Background: 246 of 446 students at a certain university had eaten breakfast on survey day. Questions: Is 246/446=0.55 a statistic or a parameter? How do we denote it? Is the proportion of all students eating breakfast a statistic or a parameter? How do we denote it? Responses: 246/446=0.55 is a statistic denoted . Proportion of all students eating breakfast is a parameter denoted p. Practice: 4.3c p.79 ©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture Elementary Statistics: Looking at the Big Picture

Example: Summary Issues (C) 2007 Nancy Pfenning Example: Summary Issues Background: Location (state) for all 1,696 TV series in 2004 with known settings: 601 in California (601/1696=0.35) 412 in New York (412/1696=0.24) 683 in other states (683/1696=0.40) Questions: 0.35+0.24+0.40=0.99mistake? Why is it not appropriate to use this info to draw conclusions about a larger population in 2004? Responses: No mistake; roundoff error There is no larger population (data is for all series in 2004). ©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture Elementary Statistics: Looking at the Big Picture

Elementary Statistics: Looking at the Big Picture (C) 2007 Nancy Pfenning Example: Notation Background: In study of 20 antarctic prions (birds), 17 correctly chose the one of two bags that had contained their mate. Questions: How do we denote sample and population proportions? Are they statistics or parameters? Responses: sample proportion . is a statistic. population proportion p is a parameter (not known in this case) ©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture Elementary Statistics: Looking at the Big Picture

Elementary Statistics: Looking at the Big Picture (C) 2007 Nancy Pfenning Definitions Mode: most common value Majority: more common of two possible values (same as mode) Minority: less common of two possible values ©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture Elementary Statistics: Looking at the Big Picture

Example: Role of Sample Size (C) 2007 Nancy Pfenning Example: Role of Sample Size Background: In study of 20 antarctic prions (birds), 17 correctly chose the one of two bags that had contained their mate. Question: Would we be more convinced that a majority of all prions would choose correctly, if 170 out of 200 were correct? Response: Yes, as long as the sampling and study design are sound. ©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture Practice: 4.3i p.79 Elementary Statistics: Looking at the Big Picture

Example: Sampling Design (C) 2007 Nancy Pfenning Example: Sampling Design Background: In study of 20 antarctic prions (birds), 17 correctly chose the one of two bags that had contained their mate. Question: Is the sample biased? Response: No, unless maybe they were too long in captivity. ©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture Elementary Statistics: Looking at the Big Picture

Elementary Statistics: Looking at the Big Picture (C) 2007 Nancy Pfenning Example: Study Design Background: Antarctic prions presented with Y-shaped maze, a bag at the end of each arm. One bag had contained mate, the other not. Question: What were researchers attempting to show? Response: that prions know mate by smell ? ©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture Elementary Statistics: Looking at the Big Picture

Elementary Statistics: Looking at the Big Picture (C) 2007 Nancy Pfenning Example: Study Design Background: Antarctic prions presented with Y-shaped maze, a bag at the end of each arm. One bag had contained mate, the other not. Question: Why use bags and not birds themselves? Response: Didn’t want them to find mate by sight. ©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture Elementary Statistics: Looking at the Big Picture

Elementary Statistics: Looking at the Big Picture (C) 2007 Nancy Pfenning Example: Study Design Background: Antarctic prions presented with Y-shaped maze, a bag at the end of each arm. One bag had contained mate, the other not. Question: Why “had” contained (bird no longer in bag)? Response: Didn’t want them to find mate by sound. ©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture Elementary Statistics: Looking at the Big Picture

Elementary Statistics: Looking at the Big Picture (C) 2007 Nancy Pfenning Example: Study Design Background: Antarctic prions presented with Y-shaped maze, a bag at the end of each arm. One bag had contained mate, the other not. Question: OK to always place correct bag on right? Response: Position of correct bag should be randomized, in case birds have natural preference for left or right. ©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture Elementary Statistics: Looking at the Big Picture

Elementary Statistics: Looking at the Big Picture (C) 2007 Nancy Pfenning Example: Study Design Background: Antarctic prions presented with Y-shaped maze, a bag at the end of each arm. One bag had contained mate, the other not. Question: Should the other be just any empty bag? Response: It should have formerly contained a non-mate bird. Looking Ahead: Researchers were careful to avoid bias in their study design. A success rate of 85% is impressive but we need inference methods to quantify claims that prions in general can recognize their mate by smell. ©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture Elementary Statistics: Looking at the Big Picture

Two or More Possible Values (C) 2007 Nancy Pfenning Two or More Possible Values Looking Ahead: In Probability and Inference, most categorical variables discussed have just two possibilities. Still, we often summarize and display categorical data with more than two possibilities. ©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture Elementary Statistics: Looking at the Big Picture

Example: Proportions in Three Categories (C) 2007 Nancy Pfenning Example: Proportions in Three Categories Background: Student wondered if she should resist changing answers in multiple choice tests. “Ask Marilyn” replied: 50% of changes go from wrong to right 25% of changes go from right to wrong 25% of changes go from wrong to wrong Question: How to display information? Response: Use a pie chart: Instructor may survey students: To which category do your changes typically belong ? ©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture Elementary Statistics: Looking at the Big Picture

Elementary Statistics: Looking at the Big Picture (C) 2007 Nancy Pfenning Definition Bar graph: shows counts, percents, or proportions in various categories (marked on horizontal axis) with bars of corresponding heights ©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture Elementary Statistics: Looking at the Big Picture

Elementary Statistics: Looking at the Big Picture (C) 2007 Nancy Pfenning Example: Bar Graph Background: Instructor can survey students to find proportion in each year (1st, 2nd, 3rd, 4th, Other). Questions: How can we display the information? What should we look for in the display? Responses: Construct a bar graph: years 1, 2, 3, 4, Other on horizontal axis counts or proportions in each year graphed vertically Look for mode (tallest bar) to tell what year is most common; compare heights of all 5 bars. ©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture Practice: 4.6a p.80 Elementary Statistics: Looking at the Big Picture

Elementary Statistics: Looking at the Big Picture (C) 2007 Nancy Pfenning Example: Bar Graph Background: Instructor can survey students to find proportion in each year (1st, 2nd, 3rd, 4th, Other). Questions: How can we display the information? What should we look for in the display? Responses: For the particular sample whose data accompany the book, mode is 2nd year, next is 3rd, then 1st and 4th, then Other. ©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture Practice: 4.6a p.80 Elementary Statistics: Looking at the Big Picture

Overlapping Categories (C) 2007 Nancy Pfenning Overlapping Categories If more than two categorical variables are considered at once, we must note the possibility that categories overlap. Looking Ahead: In Probability, we will need to distinguish between situations where categories do and do not overlap. ©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture Elementary Statistics: Looking at the Big Picture

Example: Overlapping Categories (C) 2007 Nancy Pfenning Example: Overlapping Categories Background: Report by ResumeDoctor.com on over 160,000 resumes: 13% said applicant had “communication skills” 7% said applicant was a “team player” Question: Can we conclude that 20% claimed communication skills or team player? Response: No: adding 13%+7% counts those in both categories twice. ©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture Elementary Statistics: Looking at the Big Picture

Processing Raw Categorical Data (C) 2007 Nancy Pfenning Processing Raw Categorical Data Small categorical data sets are easily handled without software. ©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture Elementary Statistics: Looking at the Big Picture

Example: Proportion from Raw Data (C) 2007 Nancy Pfenning Example: Proportion from Raw Data Background: Harvard study claimed 44% of college students are binge drinkers. Agree on survey design and have students self-report: on one occasion in past month, alcoholic drinks more than 5 (males) or 4 (females)? Or use these data: Question: Are data consistent with claim of 44%? Response: % “yes”=28/66=42%, very close to 44% ©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture Practice: 4.7a p.80 Elementary Statistics: Looking at the Big Picture

Example: Proportion from Raw Data (C) 2007 Nancy Pfenning Example: Proportion from Raw Data Background: Harvard study claimed 44% of college students are binge drinkers. Agree on survey design and have students self-report: on one occasion in past month, alcoholic drinks more than 5 (males) or 4 (females)? Or use these data: Looking Ahead: How different would the sample percentage have to be, to convince you that your sample is significantly different from Harvard’s? This is an inference question. Question: Are data consistent with claim of 44%? Response: % “yes”=28/66=42%, very close to 44% ©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture Practice: 4.7a p.80 Elementary Statistics: Looking at the Big Picture

Lecture Summary (Categorical Variables) (C) 2007 Nancy Pfenning Lecture Summary (Categorical Variables) Display: pie chart, bar graph Summarize: count, percent, proportion Sampling: data unbiased (representative)? Design: produced unbiased summary of data? Inference: will we ultimately draw conclusion about population based on sample? Mode, Majority: most common values Larger samples: provide more info Other issues: Two or more possibilities? Categories overlap? How to handle raw data? ©2011 Brooks/Cole, Cengage Learning Elementary Statistics: Looking at the Big Picture Elementary Statistics: Looking at the Big Picture