Introduction to Statistics

Introduction to Statistics Online Edition

Primary author and editor: David M. Lane1

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Other authors: David Scott1, Mikki Hebl1, Rudy Guerra1, Dan Osherson1, and Heidi Ziemer2

1Rice University; 2University of Houston, Downtown Campus

Section authors specified on each section.

…………………………………………………………………………….1. Introduction 10

………………………………………………………………………………….What Are Statistics 11

…………………………………………………………………………..Importance of Statistics 13

……………………………………………………………………………….Descriptive Statistics 15

…………………………………………………………………………………Inferential Statistics 20

………………………………………………………………………………………………..Variables 26

……………………………………………………………………………………………..Percentiles 29

……………………………………………………………………………Levels of Measurement 34

…………………………………………………………………………………………..Distributions 40

………………………………………………………………………………Summation Notation 52

…………………………………………………………………………..Linear Transformations 55

…………………………………………………………………………………………….Logarithms 58

 

 

…………………………………………………………………………………..Statistical Literacy 61

………………………………………………………………………………………………..Exercises 62

……………………………………………………………..2. Graphing Distributions 65

………………………………………………………………..Graphing Qualitative Variables 67

………………………………………………………………Graphing Quantitative Variables 76

…………………………………………………………………………..Stem and Leaf Displays 77

…………………………………………………………………………………………….Histograms 83

………………………………………………………………………………..Frequency Polygons 87

……………………………………………………………………………………………….Box Plots 93

……………………………………………………………………………………………Bar Charts 102

…………………………………………………………………………………………Line Graphs 106

……………………………………………………………………………………………..Dot Plots 110

…………………………………………………………………………………Statistical Literacy 114

……………………………………………………………………………………………References 116

………………………………………………………………………………………………Exercises 117

………………………………………………………3. Summarizing Distributions 124

……………………………………………………………………What is Central Tendency? 125

……………………………………………………………….Measures of Central Tendency 132

………………………………………………………………………………..Median and Mean 135

……………………………………………….Additional Measures of Central Tendency 137

……………………………………………..Comparing Measures of Central Tendency 141

………………………………………………………………………….Measures of Variability 145

………………………………………………………………………….Shapes of Distributions 153

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…………………………………………………………..Effects of Linear Transformations 155

……………………………………………………………………………..Variance Sum Law I 157

…………………………………………………………………………………Statistical Literacy 159

………………………………………………………………………………………………Exercises 160

………………………………………………………..4. Describing Bivariate Data 165

………………………………………………………………..Introduction to Bivariate Data 166

…………………………………………………………..Values of the Pearson Correlation 171

…………………………………………………………………………Properties of Pearson’s r 176

…………………………………………………………………………..Computing Pearson’s r 177

…………………………………………………………………………….Variance Sum Law II 179

…………………………………………………………………………………Statistical Literacy 181

………………………………………………………………………………………………Exercises 182

……………………………………………………………………………..5. Probability 186

………………………………………………..Remarks on the Concept of “Probability” 187

……………………………………………………………………………………..Basic Concepts 190

…………………………………………………………….Permutations and Combinations 199

…………………………………………………………………………….Binomial Distribution 204

………………………………………………………………………………Poisson Distribution 208

……………………………………………………………………….Multinomial Distribution 209

………………………………………………………………….Hypergeometric Distribution 211

……………………………………………………………………………………………Base Rates 213

…………………………………………………………………………………Statistical Literacy 216

………………………………………………………………………………………………Exercises 217

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……………………………………………………………………..6. Research Design 223

………………………………………………………………………………….Scientific Method 224

……………………………………………………………………………………….Measurement 226

……………………………………………………………………….Basics of Data Collection 232

………………………………………………………………………………………Sampling Bias 236

……………………………………………………………………………Experimental Designs 239

…………………………………………………………………………………………….Causation 244

…………………………………………………………………………………Statistical Literacy 247

……………………………………………………………………………………………References 248

………………………………………………………………………………………………Exercises 249

………………………………………………………………7. Normal Distributions 250

……………………………………………………..Introduction to Normal Distributions 251

…………………………………………………………History of the Normal Distribution 254

…………………………………………………………Areas Under Normal Distributions 258

………………………………………………………………..Standard Normal Distribution 261

………………………………………………….Normal Approximation to the Binomial 265

…………………………………………………………………………………Statistical Literacy 268

………………………………………………………………………………………………Exercises 269

……………………………………………………………………8. Advanced Graphs 274

………………………………………………………………….Quantile-Quantile (q-q) Plots 275

……………………………………………………………………………………….Contour Plots 291

……………………………………………………………………………………………….3D Plots 294

…………………………………………………………………………………Statistical Literacy 299

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………………………………………………………………………………………………Exercises 300

…………………………………………………………….9. Sampling Distributions 301

……………………………………………………Introduction to Sampling Distributions 302

…………………………………………………………Sampling Distribution of the Mean 309

………………………………..Sampling Distribution of Difference Between Means 313

……………………………………………………….Sampling Distribution of Pearson’s r 318

……………..Figure 2. The sampling distribution of r for N = 12 and ρ = 0.90. 320

…………………………………………………………………….Sampling Distribution of p 321

…………………………………………………………………………………Statistical Literacy 324

………………………………………………………………………………………………Exercises 325

……………………………………………………………………………10. Estimation 330

……………………………………………………………………..Introduction to Estimation 331

………………………………………………………………………………Degrees of Freedom 332

………………………………………………………………….Characteristics of Estimators 335

……………………………………………………………………………..Confidence Intervals 338

……………………………………………………….Introduction to Confidence Intervals 339

………………………………………………………………………………………..t Distribution 341

…………………………………………………………..Confidence Interval for the Mean 345

……………………………………………………………………..Difference between Means 351

…………………………………………………………………………………………..Correlation 358

……………………………………………………………………………………………Proportion 360

…………………………………………………………………………………Statistical Literacy 362

………………………………………………………………………………………………Exercises 364

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…………………………………………………..11. Logic of Hypothesis Testing 371

…………………………………………………………………………………………Introduction 372

………………………………………………………………………………Significance Testing 377

………………………………………………………………………………Type I and II Errors 379

……………………………………………………………………One- and Two-Tailed Tests 381

………………………………………………………………Interpreting Significant Results 385

……………………………………………………….Interpreting Non-Significant Results 387

……………………………………………………………………Steps in Hypothesis Testing 390

………………………………………….Significance Testing and Confidence Intervals 391

…………………………………………………………………………………….Misconceptions 393

…………………………………………………………………………………Statistical Literacy 394

……………………………………………………………………………………………References 395

………………………………………………………………………………………………Exercises 396

………………………………………………………………………12. Testing Means 400

……………………………………………………………………………Testing a Single Mean 401

…………………………….Differences between Two Means (Independent Groups) 408

………………………………………………..All Pairwise Comparisons Among Means 414

……………………………………………Specific Comparisons (Independent Groups) 420

……………………………………Difference Between Two Means (Correlated Pairs) 430

………………………………………Specific Comparisons (Correlated Observations) 434

……………………………………..Pairwise Comparisons (Correlated Observations) 438

…………………………………………………………………………………Statistical Literacy 440

……………………………………………………………………………………………References 441

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………………………………………………………………………………………………Exercises 442

…………………………………………………………………………………..13. Power 449

……………………………………………………………………………Introduction to Power 450

……………………………………………………………………………Example Calculations 452

…………………………………………………………………………Factors Affecting Power 456

…………………………………………………………………………………Statistical Literacy 460

………………………………………………………………………………………………Exercises 461

……………………………………………………………………………14. Regression 463

…………………………………………………………..Introduction to Linear Regression 464

……………………………………………………………Partitioning the Sums of Squares 470

………………………………………………………………Standard Error of the Estimate 475

……………………………………………………………….Inferential Statistics for b and r 478

…………………………………………………………………………Influential Observations 484

…………………………………………………………………Regression Toward the Mean 489

………………………………………………………..Introduction to Multiple Regression 498

…………………………………………………………………………………Statistical Literacy 510

……………………………………………………………………………………………References 511

………………………………………………………………………………………………Exercises 512

………………………………………………………………15. Analysis of Variance 518

…………………………………………………………………………………………Introduction 519

………………………………………………………………….Analysis of Variance Designs 521

…………………………………………………….Between- and Within-Subjects Factors 522

………………………………………………….One-Factor ANOVA (Between Subjects) 524

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…………………………………………………..Multi-Factor Between-Subjects Designs 535

…………………………………………………………………………..Unequal Sample Sizes 547

………………………………………………………………..Tests Supplementing ANOVA 556

……………………………………………………………………….Within-Subjects ANOVA 565

…………………………………………………………………………………Statistical Literacy 572

………………………………………………………………………………………………Exercises 573

……………………………………………………………………16. Transformations 579

…………………………………………………………………………….Log Transformations 580

………………………………………………………………………..Tukey Ladder of Powers 583

……………………………………………………………………..Box-Cox Transformations 591

…………………………………………………………………………………Statistical Literacy 597

……………………………………………………………………………………………References 598

………………………………………………………………………………………………Exercises 599

…………………………………………………………………………..17. Chi Square 600

…………………………………………………………………………Chi Square Distribution 601

………………………………………………One-Way Tables (Testing Goodness of Fit) 604

………………………………………………………………………………Contingency Tables 608

…………………………………………………………………………………Statistical Literacy 611

……………………………………………………………………………………………References 612

………………………………………………………………………………………………Exercises 613

…………………………………………………………..18. Distribution-Free Tests 619

………………………………………………………………………………………………..Benefits 620

……………………………………………………Randomization Tests: Two Conditions 621

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……………………………………….Randomization Tests: Two or More Conditions 623

…………………………………………Randomization Tests: Association (Pearson’s r) 625

…………………Randomization Tests: Contingency Tables: (Fisher’s Exact Test) 627

Rank Randomization: Two Conditions (Mann-Whitney U, Wilcoxon Rank ……………………………………………………………………………………………………Sum) 629

……………….Rank Randomization: Two or More Conditions (Kruskal-Wallis) 634

………………………………..Rank Randomization for Association (Spearman’s ρ) 636

…………………………………………………………………………………Statistical Literacy 639

………………………………………………………………………………………………Exercises 640

…………………………………………………………………………….19. Effect Size 642

…………………………………………………………………………………………..Proportions 643

……………………………………………………………..Difference Between Two Means 646

…………………………………………………………..Proportion of Variance Explained 650

……………………………………………………………………………………………References 656

…………………………………………………………………………………Statistical Literacy 657

………………………………………………………………………………………………Exercises 658

…………………………………………………………………………20. Case Studies 660

……………………………………………………………………………….21. Glossary 662

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1. Introduction This chapter begins by discussing what statistics are and why the study of statistics is important. Subsequent sections cover a variety of topics all basic to the study of statistics. The only theme common to all of these sections is that they cover concepts and ideas important for other chapters in the book.

A. What are Statistics? B. Importance of Statistics C. Descriptive Statistics D. Inferential Statistics E. Variables F. Percentiles G. Measurement H. Levels of Measurement I. Distributions J. Summation Notation K. Linear Transformations L. Logarithms M. Exercises

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What Are Statistics by Mikki Hebl

Learning Objectives 1. Describe the range of applications of statistics 2. Identify situations in which statistics can be misleading 3. Define “Statistics”

Statistics include numerical facts and figures. For instance:

• The largest earthquake measured 9.2 on the Richter scale.

• Men are at least 10 times more likely than women to commit murder.

• One in every 8 South Africans is HIV positive.

• By the year 2020, there will be 15 people aged 65 and over for every new baby born.

The study of statistics involves math and relies upon calculations of numbers. But it also relies heavily on how the numbers are chosen and how the statistics are interpreted. For example, consider the following three scenarios and the interpretations based upon the presented statistics. You will find that the numbers may be right, but the interpretation may be wrong. Try to identify a major flaw with each interpretation before we describe it.

1) A new advertisement for Ben and Jerry’s ice cream introduced in late May of last year resulted in a 30% increase in ice cream sales for the following three months. Thus, the advertisement was effective.

A major flaw is that ice cream consumption generally increases in the months of June, July, and August regardless of advertisements. This effect is called a history effect and leads people to interpret outcomes as the result of one variable when another variable (in this case, one having to do with the passage of time) is actually responsible.

2) The more churches in a city, the more crime there is. Thus, churches lead to crime.

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A major flaw is that both increased churches and increased crime rates can be explained by larger populations. In bigger cities, there are both more churches and more crime. This problem, which we will discuss in more detail in Chapter 6, refers to the third-variable problem. Namely, a third variable can cause both situations; however, people erroneously believe that there is a causal relationship between the two primary variables rather than recognize that a third variable can cause both.

3) 75% more interracial marriages are occurring this year than 25 years ago. Thus, our society accepts interracial marriages.

A major flaw is that we don’t have the information that we need. What is the rate at which marriages are occurring? Suppose only 1% of marriages 25 years ago were interracial and so now 1.75% of marriages are interracial (1.75 is 75% higher than 1). But this latter number is hardly evidence suggesting the acceptability of interracial marriages. In addition, the statistic provided does not rule out the possibility that the number of interracial marriages has seen dramatic fluctuations over the years and this year is not the highest. Again, there is simply not enough information to understand fully the impact of the statistics.

As a whole, these examples show that statistics are not only facts and figures; they are something more than that. In the broadest sense, “statistics” refers to a range of techniques and procedures for analyzing, interpreting, displaying, and making decisions based on data.

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Importance of Statistics by Mikki Hebl

Learning Objectives 1. Give examples of statistics encountered in everyday life 2. Give examples of how statistics can lend credibility to an argument Like most people, you probably feel that it is important to “take control of your life.” But what does this mean? Partly, it means being able to properly evaluate the data and claims that bombard you every day. If you cannot distinguish good from faulty reasoning, then you are vulnerable to manipulation and to decisions that are not in your best interest. Statistics provides tools that you need in order to react intelligently to information you hear or read. In this sense, statistics is one of the most important things that you can study.

To be more specific, here are some claims that we have heard on several occasions. (We are not saying that each one of these claims is true!) • 4 out of 5 dentists recommend Dentine. • Almost 85% of lung cancers in men and 45% in women are tobacco-related. • Condoms are effective 94% of the time. • Native Americans are significantly more likely to be hit crossing the street than

are people of other ethnicities. • People tend to be more persuasive when they look others directly in the eye and

speak loudly and quickly. • Women make 75 cents to every dollar a man makes when they work the same

job. • A surprising new study shows that eating egg whites can increase one’s life span. • People predict that it is very unlikely there will ever be another baseball player

with a batting average over 400. • There is an 80% chance that in a room full of 30 people that at least two people

will share the same birthday. • 79.48% of all statistics are made up on the spot. All of these claims are statistical in character. We suspect that some of them sound familiar; if not, we bet that you have heard other claims like them. Notice how diverse the examples are. They come from psychology, health, law, sports, business, etc. Indeed, data and data interpretation show up in discourse from virtually every facet of contemporary life.

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Statistics are often presented in an effort to add credibility to an argument or advice. You can see this by paying attention to television advertisements. Many of the numbers thrown about in this way do not represent careful statistical analysis. They can be misleading and push you into decisions that you might find cause to regret. For these reasons, learning about statistics is a long step towards taking control of your life. (It is not, of course, the only step needed for this purpose.) The present electronic textbook is designed to help you learn statistical essentials. It will make you into an intelligent consumer of statistical claims.

You can take the first step right away. To be an intelligent consumer of statistics, your first reflex must be to question the statistics that you encounter. The British Prime Minister Benjamin Disraeli is quoted by Mark Twain as having said, “There are three kinds of lies — lies, damned lies, and statistics.” This quote reminds us why it is so important to understand statistics. So let us invite you to reform your statistical habits from now on. No longer will you blindly accept numbers or findings. Instead, you will begin to think about the numbers, their sources, and most importantly, the procedures used to generate them.

We have put the emphasis on defending ourselves against fraudulent claims wrapped up as statistics. We close this section on a more positive note. Just as important as detecting the deceptive use of statistics is the appreciation of the proper use of statistics. You must also learn to recognize statistical evidence that supports a stated conclusion. Statistics are all around you, sometimes used well, sometimes not. We must learn how to distinguish the two cases.   Now let us get to work!

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Descriptive Statistics by Mikki Hebl

Prerequisites • none

Learning Objectives 1. Define “descriptive statistics” 2. Distinguish between descriptive statistics and inferential statistics Descriptive statistics are numbers that are used to summarize and describe data. The word “data” refers to the information that has been collected from an experiment, a survey, an historical record, etc. (By the way, “data” is plural. One piece of information is called a “datum.”) If we are analyzing birth certificates, for example, a descriptive statistic might be the percentage of certificates issued in New York State, or the average age of the mother. Any other number we choose to compute also counts as a descriptive statistic for the data from which the statistic is computed. Several descriptive statistics are often used at one time to give a full picture of the data.

Descriptive statistics are just descriptive. They do not involve generalizing beyond the data at hand. Generalizing from our data to another set of cases is the business of inferential statistics, which you’ll be studying in another section. Here we focus on (mere) descriptive statistics.

Some descriptive statistics are shown in Table 1. The table shows the average salaries for various occupations in the United States in 1999.

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Table 1. Average salaries for various occupations in 1999.

$112,760 pediatricians

$106,130 dentists

$100,090 podiatrists

$76,140 physicists

$53,410 architects,

$49,720 school, clinical, and counseling psychologists

$47,910 flight attendants

$39,560 elementary school teachers

$38,710 police officers

$18,980 floral designers

Descriptive statistics like these offer insight into American society. It is interesting to note, for example, that we pay the people who educate our children and who protect our citizens a great deal less than we pay people who take care of our feet or our teeth.

For more descriptive statistics, consider Table 2. It shows the number of unmarried men per 100 unmarried women in U.S. Metro Areas in 1990. From this table we see that men outnumber women most in Jacksonville, NC, and women outnumber men most in Sarasota, FL. You can see that descriptive statistics can be useful if we are looking for an opposite-sex partner! (These data come from the Information Please Almanac.)

Table 2. Number of unmarried men per 100 unmarried women in U.S. Metro Areas in 1990.

Cities with mostly men

Men per 100 Women

Cities with mostly women

Men per 100 Women

1. Jacksonville, NC 224 1. Sarasota, FL 66

2. Killeen-Temple, TX 123 2. Bradenton, FL 68

3. Fayetteville, NC 118 3. Altoona, PA 69

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4. Brazoria, TX 117 4. Springfield, IL 70

5. Lawton, OK 116 5. Jacksonville, TN 70

6. State College, PA 113 6. Gadsden, AL 70

7. Clarksville- Hopkinsville, TN-KY

113 7. Wheeling, WV 70

8. Anchorage, Alaska 112 8. Charleston, WV 71

9. Salinas-Seaside- Monterey, CA

112 9. St. Joseph, MO 71

10. Bryan-College Station, TX

111 10. Lynchburg, VA 71

NOTE: Unmarried includes never-married, widowed, and divorced persons, 15 years or older.

These descriptive statistics may make us ponder why the numbers are so disparate in these cities. One potential explanation, for instance, as to why there are more women in Florida than men may involve the fact that elderly individuals tend to move down to the Sarasota region and that women tend to outlive men. Thus, more women might live in Sarasota than men. However, in the absence of proper data, this is only speculation.

You probably know that descriptive statistics are central to the world of sports. Every sporting event produces numerous statistics such as the shooting percentage of players on a basketball team. For the Olympic marathon (a foot race of 26.2 miles), we possess data that cover more than a century of competition. (The first modern Olympics took place in 1896.) The following table shows the winning times for both men and women (the latter have only been allowed to compete since 1984).

Table 3. Winning Olympic marathon times.

WomenWomenWomenWomen

Year Winner Country Time

1984 Joan Benoit USA 2:24:52

1988 Rosa Mota POR 2:25:40

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1992 Valentina Yegorova UT 2:32:41

1996 Fatuma Roba ETH 2:26:05

2000 Naoko Takahashi JPN 2:23:14

2004 Mizuki Noguchi JPN 2:26:20

MenMenMenMen

Year Winner Country Time

1896 Spiridon Louis GRE 2:58:50

1900 Michel Theato FRA 2:59:45

1904 Thomas Hicks USA 3:28:53

1906 Billy Sherring CAN 2:51:23

1908 Johnny Hayes USA 2:55:18

1912 Kenneth McArthur S. Afr. 2:36:54

1920 Hannes Kolehmainen FIN 2:32:35

1924 Albin Stenroos FIN 2:41:22

1928 Boughra El Ouafi FRA 2:32:57

1932 Juan Carlos Zabala ARG 2:31:36

1936 Sohn Kee-Chung JPN 2:29:19

1948 Delfo Cabrera ARG 2:34:51

1952 Emil Ztopek CZE 2:23:03

1956 Alain Mimoun FRA 2:25:00

1960 Abebe Bikila ETH 2:15:16

1964 Abebe Bikila ETH 2:12:11

1968 Mamo Wolde ETH 2:20:26

1972 Frank Shorter USA 2:12:19

1976 Waldemar Cierpinski E.Ger 2:09:55

1980 Waldemar Cierpinski E.Ger 2:11:03

1984 Carlos Lopes POR 2:09:21

1988 Gelindo Bordin ITA 2:10:32

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1992 Hwang Young-Cho S. Kor 2:13:23

1996 Josia Thugwane S. Afr. 2:12:36

2000 Gezahenge Abera ETH 2:10.10

2004 Stefano Baldini ITA 2:10:55

There are many descriptive statistics that we can compute from the data in the table. To gain insight into the improvement in speed over the years, let us divide the men’s times into two pieces, namely, the first 13 races (up to 1952) and the second 13 (starting from 1956). The mean winning time for the first 13 races is 2 hours, 44 minutes, and 22 seconds (written 2:44:22). The mean winning time for the second 13 races is 2:13:18. This is quite a difference (over half an hour). Does this prove that the fastest men are running faster? Or is the difference just due to chance, no more than what often emerges from chance differences in performance from year to year? We can’t answer this question with descriptive statistics alone. All we can affirm is that the two means are “suggestive.”

Examining Table 3 leads to many other questions. We note that Takahashi (the lead female runner in 2000) would have beaten the male runner in 1956 and all male runners in the first 12 marathons. This fact leads us to ask whether the gender gap will close or remain constant. When we look at the times within each gender, we also wonder how far they will decrease (if at all) in the next century of the Olympics. Might we one day witness a sub-2 hour marathon? The study of statistics can help you make reasonable guesses about the answers to these questions.

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Inferential Statistics by Mikki Hebl

Prerequisites • Chapter 1: Descriptive Statistics

Learning Objectives 1. Distinguish between a sample and a population 2. Define inferential statistics 3. Identify biased samples 4. Distinguish between simple random sampling and stratified sampling 5. Distinguish between random sampling and random assignment

Populations and samples In statistics, we often rely on a sample — that is, a small subset of a larger set of data — to draw inferences about the larger set. The larger set is known as the population from which the sample is drawn.

Example #1: You have been hired by the National Election Commission to examine how the American people feel about the fairness of the voting procedures in the U.S. Who will you ask?

It is not practical to ask every single American how he or she feels about the fairness of the voting procedures. Instead, we query a relatively small number of Americans, and draw inferences about the entire country from their responses. The Americans actually queried constitute our sample of the larger population of all Americans. The mathematical procedures whereby we convert information about the sample into intelligent guesses about the population fall under the rubric of inferential statistics.

A sample is typically a small subset of the population. In the case of voting attitudes, we would sample a few thousand Americans drawn from the hundreds of millions that make up the country. In choosing a sample, it is therefore crucial that it not over-represent one kind of citizen at the expense of others. For example, something would be wrong with our sample if it happened to be made up entirely of Florida residents. If the sample held only Floridians, it could not be used to infer

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the attitudes of other Americans. The same problem would arise if the sample were comprised only of Republicans. Inferential statistics are based on the assumption that sampling is random. We trust a random sample to represent different segments of society in close to the appropriate proportions (provided the sample is large enough; see below).

Example #2: We are interested in examining how many math classes have been taken on average by current graduating seniors at American colleges and universities during their four years in school. Whereas our population in the last example included all US citizens, now it involves just the graduating seniors throughout the country. This is still a large set since there are thousands of colleges and universities, each enrolling many students. (New York University, for example, enrolls 48,000 students.) It would be prohibitively costly to examine the transcript of every college senior. We therefore take a sample of college seniors and then make inferences to the entire population based on what we find. To make the sample, we might first choose some public and private colleges and universities across the United States. Then we might sample 50 students from each of these institutions. Suppose that the average number of math classes taken by the people in our sample were 3.2. Then we might speculate that 3.2 approximates the number we would find if we had the resources to examine every senior in the entire population. But we must be careful about the possibility that our sample is non-representative of the population. Perhaps we chose an overabundance of math majors, or chose too many technical institutions that have heavy math requirements. Such bad sampling makes our sample unrepresentative of the population of all seniors.

To solidify your understanding of sampling bias, consider the following example. Try to identify the population and the sample, and then reflect on whether the sample is likely to yield the information desired.

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Example #3: A substitute teacher wants to know how students in the class did on their last test. The teacher asks the 10 students sitting in the front row to state their latest test score. He concludes from their report that the class did extremely well. What is the sample? What is the population? Can you identify any problems with choosing the sample in the way that the teacher did?

In Example #3, the population consists of all students in the class. The sample is made up of just the 10 students sitting in the front row. The sample is not likely to be representative of the population. Those who sit in the front row tend to be more interested in the class and tend to perform higher on tests. Hence, the sample may perform at a higher level than the population.

Example #4: A coach is interested in how many cartwheels the average college freshmen at his university can do. Eight volunteers from the freshman class step forward. After observing their performance, the coach concludes that college freshmen can do an average of 16 cartwheels in a row without stopping.

In Example #4, the population is the class of all freshmen at the coach’s university. The sample is composed of the 8 volunteers. The sample is poorly chosen because volunteers are more likely to be able to do cartwheels than the average freshman; people who can’t do cartwheels probably did not volunteer! In the example, we are also not told of the gender of the volunteers. Were they all women, for example? That might affect the outcome, contributing to the non-representative nature of the sample (if the school is co-ed).

Simple Random Sampling Researchers adopt a variety of sampling strategies. The most straightforward is simple random sampling. Such sampling requires every member of the population to have an equal chance of being selected into the sample. In addition, the selection of one member must be independent of the selection of every other member. That is, picking one member from the population must not increase or decrease the probability of picking any other member (relative to the others). In this sense, we can say that simple random sampling chooses a sample by pure chance. To check

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your understanding of simple random sampling, consider the following example. What is the population? What is the sample? Was the sample picked by simple random sampling? Is it biased?

Example #5: A research scientist is interested in studying the experiences of twins raised together versus those raised apart. She obtains a list of twins from the National Twin Registry, and selects two subsets of individuals for her study. First, she chooses all those in the registry whose last name begins with Z. Then she turns to all those whose last name begins with B. Because there are so many names that start with B, however, our researcher decides to incorporate only every other name into her sample. Finally, she mails out a survey and compares characteristics of twins raised apart versus together.

In Example #5, the population consists of all twins recorded in the National Twin Registry. It is important that the researcher only make statistical generalizations to the twins on this list, not to all twins in the nation or world. That is, the National Twin Registry may not be representative of all twins. Even if inferences are limited to the Registry, a number of problems affect the sampling procedure we described. For instance, choosing only twins whose last names begin with Z does not give every individual an equal chance of being selected into the sample. Moreover, such a procedure risks over-representing ethnic groups with many surnames that begin with Z. There are other reasons why choosing just the Z’s may bias the sample. Perhaps such people are more patient than average because they often find themselves at the end of the line! The same problem occurs with choosing twins whose last name begins with B. An additional problem for the B’s is that the “every-other-one” procedure disallowed adjacent names on the B part of the list from being both selected. Just this defect alone means the sample was not formed through simple random sampling.

Sample size matters Recall that the definition of a random sample is a sample in which every member of the population has an equal chance of being selected. This means that the sampling procedure rather than the results of the procedure define what it means for a sample to be random. Random samples, especially if the sample size is small,

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are not necessarily representative of the entire population. For example, if a random sample of 20 subjects were taken from a population with an equal number of males and females, there would be a nontrivial probability (0.06) that 70% or more of the sample would be female. (To see how to obtain this probability, see the section on the binomial distribution in Chapter 5.) Such a sample would not be representative, although it would be drawn randomly. Only a large sample size makes it likely that our sample is close to representative of the population. For this reason, inferential statistics take into account the sample size when generalizing results from samples to populations. In later chapters, you’ll see what kinds of mathematical techniques ensure this sensitivity to sample size.

More complex sampling Sometimes it is not feasible to build a sample using simple random sampling. To see the problem, consider the fact that both Dallas and Houston are competing to be hosts of the 2012 Olympics. Imagine that you are hired to assess whether most Texans prefer Houston to Dallas as the host, or the reverse. Given the impracticality of obtaining the opinion of every single Texan, you must construct a sample of the Texas population. But now notice how difficult it would be to proceed by simple random sampling. For example, how will you contact those individuals who don’t vote and don’t have a phone? Even among people you find in the telephone book, how can you identify those who have just relocated to California (and had no reason to inform you of their move)? What do you do about the fact that since the beginning of the study, an additional 4,212 people took up residence in the state of Texas? As you can see, it is sometimes very difficult to develop a truly random procedure. For this reason, other kinds of sampling techniques have been devised. We now discuss two of them.

Random assignment In experimental research, populations are often hypothetical. For example, in an experiment comparing the effectiveness of a new anti-depressant drug with a placebo, there is no actual population of individuals taking the drug. In this case, a specified population of people with some degree of depression is defined and a random sample is taken from this population. The sample is then randomly divided into two groups; one group is assigned to the treatment condition (drug) and the other group is assigned to the control condition (placebo). This random division of

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the sample into two groups is called random assignment. Random assignment is critical for the validity of an experiment. For example, consider the bias that could be introduced if the first 20 subjects to show up at the experiment were assigned to the experimental group and the second 20 subjects were assigned to the control group. It is possible that subjects who show up late tend to be more depressed than those who show up early, thus making the experimental group less depressed than the control group even before the treatment was administered.

In experimental research of this kind, failure to assign subjects randomly to groups is generally more serious than having a non-random sample. Failure to randomize (the former error) invalidates the experimental findings. A non-random sample (the latter error) simply restricts the generalizability of the results.

Stratified Sampling Since simple random sampling often does not ensure a representative sample, a sampling method called stratified random sampling is sometimes used to make the sample more representative of the population. This method can be used if the population has a number of distinct “strata” or groups. In stratified sampling, you first identify members of your sample who belong to each group. Then you randomly sample from each of those subgroups in such a way that the sizes of the subgroups in the sample are proportional to their sizes in the population.

Let’s take an example: Suppose you were interested in views of capital punishment at an urban university. You have the time and resources to interview 200 students. The student body is diverse with respect to age; many older people work during the day and enroll in night courses (average age is 39), while younger students generally enroll in day classes (average age of 19). It is possible that night students have different views about capital punishment than day students. If 70% of the students were day students, it makes sense to ensure that 70% of the sample consisted of day students. Thus, your sample of 200 students would consist of 140 day students and 60 night students. The proportion of day students in the sample and in the population (the entire university) would be the same. Inferences to the entire population of students at the university would therefore be more secure.

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Variables by Heidi Ziemer

Prerequisites •none

Learning Objectives 1. Define and distinguish between independent and dependent variables 2. Define and distinguish between discrete and continuous variables 3. Define and distinguish between qualitative and quantitative variables

Independent and dependent variables Variables are properties or characteristics of some event, object, or person that can take on different values or amounts (as opposed to constants such as π that do not vary). When conducting research, experimenters often manipulate variables. For example, an experimenter might compare the effectiveness of four types of antidepressants. In this case, the variable is “type of antidepressant.” When a variable is manipulated by an experimenter, it is called an independent variable. The experiment seeks to determine the effect of the independent variable on relief from depression. In this example, relief from depression is called a dependent variable. In general, the independent variable is manipulated by the experimenter and its effects on the dependent variable are measured.

Example #1: Can blueberries slow down aging? A study indicates that antioxidants found in blueberries may slow down the process of aging. In this study, 19-month-old rats (equivalent to 60-year-old humans) were fed either their standard diet or a diet supplemented by either blueberry, strawberry, or spinach powder. After eight weeks, the rats were given memory and motor skills tests. Although all supplemented rats showed improvement, those supplemented with blueberry powder showed the most notable improvement.

1. What is the independent variable? (dietary supplement: none, blueberry, strawberry, and spinach)

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2. What are the dependent variables? (memory test and motor skills test)

Example #2: Does beta-carotene protect against cancer? Beta-carotene supplements have been thought to protect against cancer. However, a study published in the Journal of the National Cancer Institute suggests this is false. The study was conducted with 39,000 women aged 45 and up. These women were randomly assigned to receive a beta-carotene supplement or a placebo, and their health was studied over their lifetime. Cancer rates for women taking the beta-carotene supplement did not differ systematically from the cancer rates of those women taking the placebo.

1. What is the independent variable? (supplements: beta-carotene or placebo)

2. What is the dependent variable? (occurrence of cancer)

Example #3: How bright is right? An automobile manufacturer wants to know how bright brake lights should be in order to minimize the time required for the driver of a following car to realize that the car in front is stopping and to hit the brakes.

1. What is the independent variable? (brightness of brake lights)

2. What is the dependent variable? (time to hit brakes)

Levels of an Independent Variable If an experiment compares an experimental treatment with a control treatment, then the independent variable (type of treatment) has two levels: experimental and control. If an experiment were comparing five types of diets, then the independent variable (type of diet) would have 5 levels. In general, the number of levels of an independent variable is the number of experimental conditions.

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Qualitative and Quantitative Variables An important distinction between variables is between qualitative variables and quantitative variables. Qualitative variables are those that express a qualitative attribute such as hair color, eye color, religion, favorite movie, gender, and so on. The values of a qualitative variable do not imply a numerical ordering. Values of the variable “religion” differ qualitatively; no ordering of religions is implied. Qualitative variables are sometimes referred to as categorical variables. Quantitative variables are those variables that are measured in terms of numbers. Some examples of quantitative variables are height, weight, and shoe size.

In the study on the effect of diet discussed previously, the independent variable was type of supplement: none, strawberry, blueberry, and spinach. The variable “type of supplement” is a qualitative variable; there is nothing quantitative about it. In contrast, the dependent variable “memory test” is a quantitative variable since memory performance was measured on a quantitative scale (number correct).

Discrete and Continuous Variables Variables such as number of children in a household are called discrete variables since the possible scores are discrete points on the scale. For example, a household could have three children or six children, but not 4.53 children. Other variables such as “time to respond to a question” are continuous variables since the scale is continuous and not made up of discrete steps. The response time could be 1.64 seconds, or it could be 1.64237123922121 seconds. Of course, the practicalities of measurement preclude most measured variables from being truly continuous.

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Percentiles by David Lane

Prerequisites •none

Learning Objectives 1. Define percentiles 2. Use three formulas for computing percentiles A test score in and of itself is usually difficult to interpret. For example, if you learned that your score on a measure of shyness was 35 out of a possible 50, you would have little idea how shy you are compared to other people. More relevant is the percentage of people with lower shyness scores than yours. This percentage is called a percentile. If 65% of the scores were below yours, then your score would be the 65th percentile.

Two Simple Definitions of Percentile There is no universally accepted definition of a percentile. Using the 65th percentile as an example, the 65th percentile can be defined as the lowest score that is greater than 65% of the scores. This is the way we defined it above and we will call this “Definition 1.” The 65th percentile can also be defined as the smallest score that is greater than or equal to 65% of the scores. This we will call “Definition 2.” Unfortunately, these two definitions can lead to dramatically different results, especially when there is relatively little data. Moreover, neither of these definitions is explicit about how to handle rounding. For instance, what rank is required to be higher than 65% of the scores when the total number of scores is 50? This is tricky because 65% of 50 is 32.5. How do we find the lowest number that is higher than 32.5% of the scores? A third way to compute percentiles (presented below) is a weighted average of the percentiles computed according to the first two definitions. This third definition handles rounding more gracefully than the other two and has the advantage that it allows the median to be defined conveniently as the 50th percentile.

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A Third Definition Unless otherwise specified, when we refer to “percentile,” we will be referring to this third definition of percentiles. Let’s begin with an example. Consider the 25th percentile for the 8 numbers in Table 1. Notice the numbers are given ranks ranging from 1 for the lowest number to 8 for the highest number.

Table 1. Test Scores.

Number Rank

3 5 7 8 9

11 13 15

1 2 3 4 5 6 7 8

The first step is to compute the rank (R) of the 25th percentile. This is done using the following formula:

Percentiles

 

= 100 × ( + 1)

 

= 25 100 ×

(8 + 1) = 9 4 = 2.25

 

= 25 100 ×

(20 + 1) = 21 4 = 5.25

 

= 85 100 ×

(20 + 1) = 17.85

 

= 50 100 ×

(4 + 1) = 2.5

 

= 50 100 ×

(5 + 1) = 3

 

Summation Notation

 

where P is the desired percentile (25 in this case) and N is the number of numbers (8 in this case). Therefore,

Percentiles

 

= 100 × ( + 1)

 

= 25 100 ×

(8 + 1) = 9 4 = 2.25

 

= 25 100 ×

(20 + 1) = 21 4 = 5.25

 

= 85 100 ×

(20 + 1) = 17.85

 

= 50 100 ×

(4 + 1) = 2.5

 

= 50 100 ×

(5 + 1) = 3

 

Summation Notation

 

If R is an integer, the Pth percentile is be the number with rank R. When R is not an integer, we compute the Pth percentile by interpolation as follows: 1. Define IR as the integer portion of R (the number to the left of the decimal

point). For this example, IR = 2. 2. Define FR as the fractional portion of R. For this example, FR = 0.25.

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3. Find the scores with Rank IR and with Rank IR + 1. For this example, this means the score with Rank 2 and the score with Rank 3. The scores are 5 and 7.

4. Interpolate by multiplying the difference between the scores by FR and add the result to the lower score. For these data, this is (0.25)(7 – 5) + 5 = 5.5.

Therefore, the 25th percentile is 5.5. If we had used the first definition (the smallest score greater than 25% of the scores), the 25th percentile would have been 7. If we had used the second definition (the smallest score greater than or equal to 25% of the scores), the 25th percentile would have been 5.

For a second example, consider the 20 quiz scores shown in Table 2.

Table 2. 20 Quiz Scores.

Score Rank

4 4 5 5 5 5 6 6 6 7 7 7 8 8 9 9 9

10 10 10

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

We will compute the 25th and the 85th percentiles. For the 25th,

Percentiles

 

= 100 × ( + 1)

 

= 25 100 ×

(8 + 1) = 9 4 = 2.25

 

= 25 100 ×

(20 + 1) = 21 4 = 5.25

 

= 85 100 ×

(20 + 1) = 17.85

 

= 50 100 ×

(4 + 1) = 2.5

 

= 50 100 ×

(5 + 1) = 3

 

Summation Notation

 

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IR = 5 and FR = 0.25.

Since the score with a rank of IR (which is 5) and the score with a rank of IR + 1 (which is 6) are both equal to 5, the 25th percentile is 5. In terms of the formula:

25th percentile = (.25) x (5 – 5) + 5 = 5.

For the 85th percentile,

Percentiles

 

= 100 × ( + 1)