The world of statistics at your fingertips
OpenStax
Rice University
6100 Main Street MS-375
Houston, Texas 77005
To learn more about OpenStax, visit https://openstax.org.
Individual print copies and bulk orders can be purchased through our website.
©2018 Rice University. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0). Under this license, any user of this textbook or the textbook contents herein must provide proper attribution as follows:
If you redistribute this textbook in a digital format (including but not limited to PDF and HTML), then you must retain on every page the following attribution: “Download for free at https://openstax.org/details/books/introductory-business-statistics.”
If you redistribute this textbook in a print format, then you must include on every physical page the following attribution: “Download for free at https://openstax.org/details/books/introductory-business-statistics.”
If you redistribute part of this textbook, then you must retain in every digital format page view (including but not limited to PDF and HTML) and on every physical printed page the following attribution: “Download for free at https://openstax.org/details/books/introductory-business-statistics.”
If you use this textbook as a bibliographic reference, please include https://openstax.org/details/books/introductory-business-statistics in your citation.
For questions regarding this licensing, please contact<EMAIL_ADDRESS>The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, OpenStax CNX logo, OpenStax Tutor name, Openstax Tutor logo, Connexions name, Connexions logo, Rice University name, and Rice University logo are not subject to the license and may not be reproduced without the prior and express written consent of Rice University.
PRINT BOOK ISBN-10 1-947172-46-8
PRINT BOOK ISBN-13 978-1-947172-46-3
PDF VERSION ISBN-10 1-947172-47-6
PDF VERSION ISBN-13 978-1-947172-47-0
Revision Number IBS-2017-001(03/18)-LC
Original Publication Year 2017
Welcome to Introductory Business Statistics, an OpenStax resource. This textbook was written to increase student access to high-quality learning materials, maintaining highest standards of academic rigor at little to no cost.
OpenStax is a nonprofit based at Rice University, and it’s our mission to improve student access to education. Our first openly licensed college textbook was published in 2012, and our library has since scaled to over 25 books for college and AP® courses used by hundreds of thousands of students. OpenStax Tutor, our low-cost personalized learning tool, is being used in college courses throughout the country. Through our partnerships with philanthropic foundations and our alliance with other educational resource organizations, OpenStax is breaking down the most common barriers to learning and empowering students and instructors to succeed.
Introductory Business Statistics is licensed under a Creative Commons Attribution 4.0 International (CC BY) license, which means that you can distribute, remix, and build upon the content, as long as you provide attribution to OpenStax and its content contributors.
Because the source book is openly licensed, you are free to use the entire book or pick and choose the sections that are most relevant to the needs of your course. Feel free to remix the content by assigning your students certain chapters and sections in your syllabus, in the order that you prefer. You can even provide a direct link in your syllabus to the sections in the web view of your book.
Instructors also have the option of creating a customized version of their OpenStax book. The custom version can be made available to students in low-cost print or digital form through their campus bookstore. Visit the Instructor Resources section of your book page on OpenStax.org for more information.
This book is built from the OpenStax Introductory Business Statistics content, with minimal added extracts from Wikipedia to represent all types of content for an accurate design sample. You can distribute, remix, and build upon the content of this sample book, as long as you provide attribution to OpenStax and the other sources for the added extracts (quoted next to or below the relevant added content).
All OpenStax textbooks undergo a rigorous review process. However, like any professional-grade textbook, errors sometimes occur. Since our books are web based, we can make updates periodically when deemed pedagogically necessary. If you have a correction to suggest, submit it through the link on your book page on OpenStax.org. Subject matter experts review all errata suggestions. OpenStax is committed to remaining transparent about all updates, so you will also find a list of past errata changes on your book page on OpenStax.org.
You can access this textbook for free in web view or PDF through OpenStax.org, and for a low cost in print.
Introductory Business Statistics is designed to meet the scope and sequence requirements of the one-semester statistics course for business, economics, and related majors. Core statistical concepts and skills have been augmented with practical business examples, scenarios, and exercises. The result is a meaningful understanding of the discipline which will serve students in their business careers and real-world experiences.
Introductory Business Statistics began as a customized version of OpenStax Introductory Statistics by Barbara Illowsky and Susan Dean. Statistics faculty at The University of Oklahoma have used the business statistics adaptation for several years, and the author has continually refined it based on student success and faculty feedback.
The book is structured in a similar manner to most traditional statistics textbooks. The most significant topical changes occur in the latter chapters on regression analysis. Discrete probability density functions have been reordered to provide a logical progression from simple counting formulas to more complex continuous distributions. Many additional homework assignments have been added, as well as new, more mathematical examples.
Introductory Business Statistics places a significant emphasis on the development and practical application of formulas so that students have a deeper understanding of their interpretation and application of data. To achieve this unique approach, the author included a wealth of additional material and purposely de-emphasized the use of the scientific calculator. Specific changes regarding formula use include:
Expanded discussions of the combinatorial formulas, factorials, and sigma notation
Adjustments to explanations of the acceptance/rejection rule for hypothesis testing, as well as a focus on terminology regarding confidence intervals
Deep reliance on statistical tables for the process of finding probabilities (which would not be required if probabilities relied on scientific calculators)
Continual and emphasized links to the Central Limit Theorem throughout the book; Introductory Business Statistics consistently links each test statistic back to this fundamental theorem in inferential statistics
Another fundamental focus of the book is the link between statistical inference and the scientific method. Business and economics models are fundamentally grounded in assumed relationships of cause and effect. They are developed to both test hypotheses and to predict from such models. This comes from the belief that statistics is the gatekeeper that allows some theories to remain and others to be cast aside for a new perspective of the world around us. This philosophical view is presented in detail throughout and informs the method of presenting the regression model, in particular.
The correlation and regression chapter includes confidence intervals for predictions, alternative mathematical forms to allow for testing categorical variables, and the presentation of the multiple regression model.
Examples are placed strategically throughout the text to show students the step-by-step process of interpreting and solving statistical problems. To keep the text relevant for students, the examples are drawn from a broad spectrum of practical topics; these include examples about college life and learning, health and medicine, retail and business, and sports and entertainment.
Practice, Homework, and Bringing It Together give the students problems at various degrees of difficulty while also including real-world scenarios to engage students.
We’ve compiled additional resources for both students and instructors, including Getting Started Guides, an instructor solution manual, and PowerPoint slides. Instructor resources require a verified instructor account, which you can apply for when you log in or create your account on OpenStax.org. Take advantage of these resources to supplement your OpenStax book.
OpenStax partners with the Institute for the Study of Knowledge Management in Education (ISKME) to offer Community Hubs on OER Commons – a platform for instructors to share community-created resources that support OpenStax books, free of charge. Through our Community Hubs, instructors can upload their own materials or download resources to use in their own courses, including additional ancillaries, teaching material, multimedia, and relevant course content. We encourage instructors to join the hubs for the subjects most relevant to your teaching and research as an opportunity both to enrich your courses and to engage with other faculty.
To reach the Community Hubs, visit www.oercommons.org/hubs/OpenStax.
As allies in making high-quality learning materials accessible, our technology partners offer optional low-cost tools that are integrated with OpenStax books. To access the technology options for your text, visit your book page on OpenStax.org.
Alexander Holmes, The University of Oklahoma
Barbara Illowsky, DeAnza College
Susan Dean, DeAnza College
Kevin Hadley, Analyst, Federal Reserve Bank of Kansas City
Birgit Aquilonius, West Valley College
Charles Ashbacher, Upper Iowa University - Cedar Rapids
Abraham Biggs, Broward Community College
Daniel Birmajer, Nazareth College
Roberta Bloom, De Anza College
Bryan Blount, Kentucky Wesleyan College
Ernest Bonat, Portland Community College
Sarah Boslaugh, Kennesaw State University
David Bosworth, Hutchinson Community College
Sheri Boyd, Rollins College
George Bratton, University of Central Arkansas
Franny Chan, Mt. San Antonio College
Jing Chang, College of Saint Mary
Laurel Chiappetta, University of Pittsburgh
Lenore Desilets, De Anza College
Matthew Einsohn, Prescott College
Ann Flanigan, Kapiolani Community College
David French, Tidewater Community College
Mo Geraghty, De Anza College
Larry Green, Lake Tahoe Community College
Michael Greenwich, College of Southern Nevada
Inna Grushko, De Anza College
Valier Hauber, De Anza College
Janice Hector, De Anza College
Jim Helmreich, Marist College
Robert Henderson, Stephen F. Austin State University
Mel Jacobsen, Snow College
Mary Jo Kane, De Anza College
John Kagochi, University of Houston - Victoria
Lynette Kenyon, Collin County Community College
Charles Klein, De Anza College
Alexander Kolovos
Sheldon Lee, Viterbo University
Sara Lenhart, Christopher Newport University
Wendy Lightheart, Lane Community College
Vladimir Logvenenko, De Anza College
Jim Lucas, De Anza College
Suman Majumdar, University of Connecticut
Lisa Markus, De Anza College
Miriam Masullo, SUNY Purchase
Diane Mathios, De Anza College
Robert McDevitt, Germanna Community College
John Migliaccio, Fordham University
Mark Mills, Central College
Cindy Moss, Skyline College
Nydia Nelson, St. Petersburg College
Benjamin Ngwudike, Jackson State University
Jonathan Oaks, Macomb Community College
Carol Olmstead, De Anza College
Barbara A. Osyk, The University of Akron
Adam Pennell, Greensboro College
Kathy Plum, De Anza College
Lisa Rosenberg, Elon University
Sudipta Roy, Kankakee Community College
Javier Rueda, De Anza College
Yvonne Sandoval, Pima Community College
Rupinder Sekhon, De Anza College
Travis Short, St. Petersburg College
Frank Snow, De Anza College
Abdulhamid Sukar, Cameron University
Jeffery Taub, Maine Maritime Academy
Mary Teegarden, San Diego Mesa College
John Thomas, College of Lake County
Philip J. Verrecchia, York College of Pennsylvania
Dennis Walsh, Middle Tennessee State University
Cheryl Wartman, University of Prince Edward Island
Carol Weideman, St. Petersburg College
Kyle S. Wells, Dixie State University
Andrew Wiesner, Pennsylvania State University
Introductory Business Statistics is designed to meet the scope and sequence requirements of the one-semester statistics course for business, economics, and related majors. Core statistical concepts and skills have been augmented with practical business examples, scenarios, and exercises. The result is a meaningful understanding of the discipline, which will serve students in their business careers and real-world experiences.
 If you read any newspaper, watch television, or use the Internet, you will see statistical information. There are statistics about crime, sports, education, politics, and real estate. Typically, when you read a newspaper article or watch a television news program, you are given sample information. With this information, you may make a decision about the correctness of a statement, claim, or "fact." Statistical methods can help you make the "best educated guess."
Since you will undoubtedly be given statistical information at some point in your life, you need to know some techniques for analyzing the information thoughtfully. Think about buying a house or managing a budget. Think about your chosen profession. The fields of economics, business, psychology, education, biology, law, computer science, police science, and early childhood development require at least one course in statistics.
Included in this book are the basic ideas and words of probability and statistics. You will soon understand that statistics and probability work together. You will also learn how data are gathered and what "good" data can be distinguished from "bad".
Once you have collected data, what will you do with it? Data can be described and presented in many different formats. For example, suppose you are interested in buying a house in a particular area. You may have no clue about the house prices, so you might ask your real estate agent to give you a sample data set of prices. Looking at all the prices in the sample often is overwhelming. A better way might be to look at the median price and the variation of prices. The median and variation are just two ways that you will learn to describe data. Your agent might also provide you with a graph of the data.
 This area of statistics is called "Descriptive Statistics". You will learn how to calculate, and even more importantly, how to interpret these measurements and graphs.
A statistical graph is a tool that helps you learn about the shape or distribution of a sample or a population. A graph can be a more effective way of presenting data than a mass of numbers because we can see where data clusters and where there are only a few data values. Newspapers and the Internet use graphs to show trends and to enable readers to compare facts and figures quickly. Statisticians often graph data first to get a picture of the data. Then, more formal tools may be applied.
Some of the types of graphs that are used to summarize and organize data are the dot plot, the bar graph, the histogram, the stem-and-leaf plot, the frequency polygon (a type of broken line graph), the pie chart, and the box plot. In this book, we will look at stem-and-leaf plots, line graphs, and bar graphs, as well as frequency polygons, and time series graphs. Our emphasis will be on histograms and box plots.
It is often necessary to "guess" about the outcome of an event in order to make a decision. Politicians study polls to guess their likelihood of winning an election. Teachers choose a particular course of study based on what they think students can comprehend. Doctors choose the treatments needed for various diseases based on their assessment of likely results. You may have visited a casino where people play games chosen because of the belief that the likelihood of winning is good. You may have chosen your course of study based on the probable availability of jobs.
You have, more than likely, used probability. In fact, you probably have an intuitive sense of probability. Probability deals with the chance of an event occurring. Whenever you weigh the odds of whether or not to do your homework or to study for an exam, you are using probability. In this book, you will learn how to solve probability problems using a systematic approach.
In this part you’ll learn the most important fundamentals of statistics, and how they are used in business. Once you’ve worked through this part, you’ll be ready to tackle more advanced statistics, and to start applying your new skills to the world around you.
How data are gathered and what "good" data can be distinguished from "bad"
After reading this chapter you will be able to:
communicate key statistical concepts to non-statisticians.
identify and articulate strategies for dealing with ethical issues that may arise.
gain proficiency in using statistical terminology in written reports.
calculate key statistics given relevant sample data.
Outline
1.1 Definitions of Statistics, Probability, and Key Terms
1.2 Data, Sampling, and Variation in Data and Sampling
1.3 Levels of Measurement
1.4 Experimental Design and Ethics
You are probably asking yourself the question, "When and where will I use statistics?" If you read any newspaper, watch television, or use the Internet, you will see statistical information. There are statistics about crime, sports, education, politics, and real estate. Typically, when you read a newspaper article or watch a television news program, you are given sample information. With this information, you may make a decision about the correctness of a statement, claim, or "fact." Statistical methods can help you make the "best educated guess."
Since you will undoubtedly be given statistical information at some point in your life, you need to know some techniques for analyzing the information thoughtfully. Think about buying a house or managing a budget. Think about your chosen profession. The fields of economics, business, psychology, education, biology, law, computer science, police science, and early childhood development require at least one course in statistics.
Included in this chapter are the basic ideas and words of probability and statistics. You will soon understand that statistics and probability work together. You will also learn how data are gathered and what "good" data can be distinguished from "bad".
In this chapter we focus on answering the following key questions:
Why is statistics important in daily life?
What is probability?
What is data sampling and what are the red flags and key risks when gathering data samples?
What kinds of measurement can we use for gathering our own samples?
How can we design experiements and what are the ethical considerations we need to take into account?
Statistics reveal the chaotic underbelly, reveal
the tessellating uncertainties, reveal
our deepest truths hiding behind confidence levels
– Asher Smith
The science of statistics deals with the collection, analysis, interpretation, and presentation of data. We see and use data in our everyday lives.
In this course, you will learn how to organize and summarize data. Organizing and summarizing data is called descriptive statistics. Two ways to summarize data are by graphing and by using numbers (for example, finding an average). After you have studied probability and probability distributions, you will use formal methods for drawing conclusions from "good" data. The formal methods are called inferential statistics. Statistical inference uses probability to determine how confident we can be that our conclusions are correct.
Effective interpretation of data (inference) is based on good procedures for producing data and thoughtful examination of the data. You will encounter what will seem to be too many mathematical formulas for interpreting data. The goal of statistics is not to perform numerous calculations using the formulas, but to gain an understanding of your data. The calculations can be done using a calculator or a computer. The understanding must come from you. If you can thoroughly grasp the basics of statistics, you can be more confident in the decisions you make in life.
Probability is a mathematical tool used to study randomness. It deals with the chance (the likelihood) of an event occurring. For example, if you toss a fair coin four times, the outcomes may not be two heads and two tails. However, if you toss the same coin 4,000 times, the outcomes will be close to half heads and half tails. The expected theoretical probability of heads in any one toss is 1212 or 0.5. Even though the outcomes of a few repetitions are uncertain, there is a regular pattern of outcomes when there are many repetitions. After reading about the English statistician Karl Pearson who tossed a coin 24,000 times with a result of 12,012 heads, one of the authors tossed a coin 2,000 times. The results were 996 heads. The fraction is equal to 0.498 which is very close to 0.5, the expected probability.
The theory of probability began with the study of games of chance such as poker. Predictions take the form of probabilities. To predict the likelihood of an earthquake, of rain, or whether you will get an A in this course, we use probabilities. Doctors use probability to determine the chance of a vaccination causing the disease the vaccination is supposed to prevent. A stockbroker uses probability to determine the rate of return on a client's investments. You might use probability to decide to buy a lottery ticket or not. In your study of statistics, you will use the power of mathematics through probability calculations to analyze and interpret your data.
In statistics, we generally want to study a population. You can think of a population as a collection of persons, things, or objects under study. To study the population, we select a sample. The idea of sampling is to select a portion (or subset) of the larger population and study that portion (the sample) to gain information about the population. Data are the result of sampling from a population.
Because it takes a lot of time and money to examine an entire population, sampling is a very practical technique. If you wished to compute the overall grade point average at your school, it would make sense to select a sample of students who attend the school. The data collected from the sample would be the students' grade point averages. In presidential elections, opinion poll samples of 1,000–2,000 people are taken. The opinion poll is supposed to represent the views of the people in the entire country. Manufacturers of canned carbonated drinks take samples to determine if a 16 ounce can contains 16 ounces of carbonated drink.
From the sample data, we can calculate a statistic. A statistic is a number that represents a property of the sample. For example, if we consider one math class to be a sample of the population of all math classes, then the average number of points earned by students in that one math class at the end of the term is an example of a statistic. The statistic is an estimate of a population parameter, in this case the mean. A parameter is a numerical characteristic of the whole population that can be estimated by a statistic. Since we considered all math classes to be the population, then the average number of points earned per student over all the math classes is an example of a parameter.
One of the main concerns in the field of statistics is how accurately a statistic estimates a parameter. The accuracy really depends on how well the sample represents the population. The sample must contain the characteristics of the population in order to be a representative sample. We are interested in both the sample statistic and the population parameter in inferential statistics. In a later chapter, we will use the sample statistic to test the validity of the established population parameter.
A variable, or random variable, usually notated by capital letters such as X and Y, is a characteristic or measurement that can be determined for each member of a population. Variables may be numerical or categorical. Numerical variables take on values with equal units such as weight in pounds and time in hours. Categorical variables place the person or thing into a category. If we let X equal the number of points earned by one math student at the end of a term, then X is a numerical variable. If we let Y be a person's party affiliation, then some examples of Y include Republican, Democrat, and Independent. Y is a categorical variable. We could do some math with values of X (calculate the average number of points earned, for example), but it makes no sense to do math with values of Y (calculating an average party affiliation makes no sense).
Data are the actual values of the variable. They may be numbers or they may be words. Datum is a single value.
Two words that come up often in statistics are mean and proportion. If you were to take three exams in your math classes and obtain scores of 86, 75, and 92, you would calculate your mean score by adding the three exam scores and dividing by three (your mean score would be 84.3 to one decimal place). If, in your math class, there are 40 students and 22 are men and 18 are women, then the proportion of men students is and the proportion of women students is . Mean and proportion are discussed in more detail in later chapters.
Determine what the key terms refer to in the following study. We want to know the mean amount of money spent on school uniforms each year by families with children at Knoll Academy. We randomly survey 100 families with children in the school. Three of the families spent $65, $75, and $95, respectively.
Shall I compare thee to a diff’rent mean?
The control group’s points
art lower than thine
From the preliminary graph I’ve seen.
Do the means differ or in fact align?
First I will find the right parameter.
Data on the sample I will collect.
Which
method to use I can’t yet be sure,
Til the assumptions are thoroughly checked.
The appropriate test statistic found,
Thy humble knave compute the value p
And
conclude whether the diff’rence profound.

Then the confidence interval will tell
The upper bound and the lower as well.
– Cristina Gonzales
Data on the sample I will collect.
Which method to use I can’t yet be
sure,
Til the assumptions are thoroughly checked.
– Cristina Gonzales
Data may come from a population or from a sample. Lowercase letters like x or y generally are used to represent data values. Most data can be put into the following categories:
Qualitative
Quantitative
Qualitative data are the result of categorizing or describing attributes of a population. Qualitative data are also often called categorical data. Hair color, blood type, ethnic group, the car a person drives, and the street a person lives on are examples of qualitative (categorical) data. Qualitative (categorical) data are generally described by words or letters. For instance, hair color might be black, dark brown, light brown, blonde, gray, or red. Blood type might be AB+, O−, or B+. Researchers often prefer to use quantitative data over qualitative (categorical) data because it lends itself more easily to mathematical analysis. For example, it does not make sense to find an average hair color or blood type.
Quantitative data are always numbers. Quantitative data are the result of counting or measuring attributes of a population. Amount of money, pulse rate, weight, number of people living in your town, and number of students who take statistics are examples of quantitative data. Quantitative data may be either discrete or continuous.
All data that are the result of counting are called quantitative discrete data. These data take on only certain numerical values. If you count the number of phone calls you receive for each day of the week, you might get values such as zero, one, two, or three.
Data that are not only made up of counting numbers, but that may include fractions, decimals, or irrational numbers, are called quantitative continuous data. Continuous data are often the results of measurements like lengths, weights, or times. A list of the lengths in minutes for all the phone calls that you make in a week, with numbers like 2.4, 7.5, or 11.0, would be quantitative continuous data.
The data are the number of machines in a gym. You sample five gyms. One gym has 12 machines, one gym has 15 machines, one gym has ten machines, one gym has 22 machines, and the other gym has 20 machines. What type of data is this?
The data are the areas of lawns in square feet. You sample five houses. The areas of the lawns are 144 sq. feet, 160 sq. feet, 190 sq. feet, 180 sq. feet, and 210 sq. feet. What type of data is this?
The registrar at State University keeps records of the number of credit hours students complete each semester. The data he collects are summarized in the histogram. The class boundaries are 10 to less than 13, 13 to less than 16, 16 to less than 19, 19 to less than 22, and 22 to less than 25.
What type of data does this graph show?
Below are tables comparing the number of part-time and full-time students at De Anza College and Foothill College enrolled for the spring 2010 quarter. The tables display counts (frequencies) and percentages or proportions (relative frequencies). The percent columns make comparing the same categories in the colleges easier. Displaying percentages along with the numbers is often helpful, but it is particularly important when comparing sets of data that do not have the same totals, such as the total enrollments for both colleges in this example. Notice how much larger the percentage for part-time students at Foothill College is compared to De Anza College.
|
De Anza College |
Foothill College |
|||||
|
Number |
Percent |
Number |
Percent |
|||
|
Full-time |
9,200 |
40.9% |
Full-time |
4,059 |
28.6% |
|
|
Part-time |
13,296 |
59.1% |
Part-time |
10,124 |
71.4% |
|
Total |
22,496 |
100% |
Total |
14,183 |
100% |
Table 1.2 Fall Term 2007 (Census day)
Tables are a good way of organizing and displaying data. But graphs can be even more helpful in understanding the data. There are no strict rules concerning which graphs to use. Two graphs that are used to display qualitative(categorical) data are pie charts and bar graphs.
In a pie chart, categories of data are represented by wedges in a circle and are proportional in size to the percent of individuals in each category.
In a bar graph, the length of the bar for each category is proportional to the number or percent of individuals in each category. Bars may be vertical or horizontal.
A Pareto chart consists of bars that are sorted into order by category size (largest to smallest).
Look at Figure 1.4 and Figure 1.5 and determine which graph (pie or bar) you think displays the comparisons better.
Sometimes percentages add up to be more than 100% (or less than 100%). In the graph, the percentages add to more than 100% because students can be in more than one category. A bar graph is appropriate to compare the relative size of the categories. A pie chart cannot be used. It also could not be used if the percentages added to less than 100%.
|
Characteristic/category |
Percent |
|
Full-time students |
40.9% |
|
Students who intend to transfer to a 4-year educational institution |
48.6% |
|
Students under age 25 |
61.0% |
|
TOTAL |
150.5% |
Table 1.3 De Anza College Spring 2010
The table displays Ethnicity of Students but is missing the "Other/Unknown" category. This category contains people who did not feel they fit into any of the ethnicity categories or declined to respond. Notice that the frequencies do not add up to the total number of students. In this situation, create a bar graph and not a pie chart.
|
Frequency |
Percent |
|
|
Asian |
8,794 |
36.1% |
|
Black |
1,412 |
5.8% |
|
Filipino |
1,298 |
5.3% |
|
Hispanic |
4,180 |
17.1% |
|
Native American |
146 |
0.6% |
|
Pacific Islander |
236 |
1.0% |
|
White |
5,978 |
24.5% |
|
TOTAL |
22,044 out of 24,382 |
90.4% out of 100% |
Table 1.4 Ethnicity of Students at De Anza College Fall Term 2007 (Census Day)
The following graph is the same as the previous graph but the “Other/Unknown” percent (9.6%) has been included. The “Other/Unknown” category is large compared to some of the other categories (Native American, 0.6%, Pacific Islander 1.0%). This is important to know when we think about what the data are telling us.
This particular bar graph in Figure 1.8 can be difficult to understand visually. The graph in Figure 1.9 is a Pareto chart. The Pareto chart has the bars sorted from largest to smallest and is easier to read and interpret.
The following pie charts have the “Other/Unknown” category included (since the percentages must add to 100%). The chart in Figure 1.10(b) is organized by the size of each wedge, which makes it a more visually informative graph than the unsorted, alphabetical graph in Figure 1.10(a).
Gathering information about an entire population often costs too much or is virtually impossible. Instead, we use a sample of the population. A sample should have the same characteristics as the population it is representing. Most statisticians use various methods of random sampling in an attempt to achieve this goal. This section will describe a few of the most common methods. There are several different methods of random sampling. In each form of random sampling, each member of a population initially has an equal chance of being selected for the sample. Each method has pros and cons. The easiest method to describe is called a simple random sample. Any group of n individuals is equally likely to be chosen as any other group of n individuals if the simple random sampling technique is used. In other words, each sample of the same size has an equal chance of being selected.
Besides simple random sampling, there are other forms of sampling that involve a chance process for getting the sample. Other well-known random sampling methods are the stratified sample, the cluster sample, and the systematic sample.
To choose a stratified sample, divide the population into groups called strata and then take a proportionate number from each stratum. For example, you could stratify (group) your college population by department and then choose a proportionate simple random sample from each stratum (each department) to get a stratified random sample. To choose a simple random sample from each department, number each member of the first department, number each member of the second department, and do the same for the remaining departments. Then use simple random sampling to choose proportionate numbers from the first department and do the same for each of the remaining departments. Those numbers picked from the first department, picked from the second department, and so on represent the members who make up the stratified sample.
To choose a cluster sample, divide the population into clusters (groups) and then randomly select some of the clusters. All the members from these clusters are in the cluster sample. For example, if you randomly sample four departments from your college population, the four departments make up the cluster sample. Divide your college faculty by department. The departments are the clusters. Number each department, and then choose four different numbers using simple random sampling. All members of the four departments with those numbers are the cluster sample.
To choose a systematic sample, randomly select a starting point and take every nth piece of data from a listing of the population. For example, suppose you have to do a phone survey. Your phone book contains 20,000 residence listings. You must choose 400 names for the sample. Number the population 1–20,000 and then use a simple random sample to pick a number that represents the first name in the sample. Then choose every fiftieth name thereafter until you have a total of 400 names (you might have to go back to the beginning of your phone list). Systematic sampling is frequently chosen because it is a simple method.
A type of sampling that is non-random is convenience sampling. Convenience sampling involves using results that are readily available. For example, a computer software store conducts a marketing study by interviewing potential customers who happen to be in the store browsing through the available software. The results of convenience sampling may be very good in some cases and highly biased (favor certain outcomes) in others.
Sampling data should be done very carefully. Collecting data carelessly can have devastating results. Surveys mailed to households and then returned may be very biased (they may favor a certain group). It is better for the person conducting the survey to select the sample respondents.
True random sampling is done with replacement. That is, once a member is picked, that member goes back into the population and thus may be chosen more than once. However for practical reasons, in most populations, simple random sampling is done without replacement. Surveys are typically done without replacement. That is, a member of the population may be chosen only once. Most samples are taken from large populations and the sample tends to be small in comparison to the population. Since this is the case, sampling without replacement is approximately the same as sampling with replacement because the chance of picking the same individual more than once with replacement is very low.
Sampling without replacement instead of sampling with replacement becomes a mathematical issue only when the population is small.
When you analyze data, it is important to be aware of sampling errors and nonsampling errors. The actual process of sampling causes sampling errors. For example, the sample may not be large enough. Factors not related to the sampling process cause nonsampling errors. A defective counting device can cause a nonsampling error.
In reality, a sample will never be exactly representative of the population so there will always be some sampling error. As a rule, the larger the sample, the smaller the sampling error.
In statistics, a sampling bias is created when a sample is collected from a population and some members of the population are not as likely to be chosen as others (remember, each member of the population should have an equally likely chance of being chosen). When a sampling bias happens, there can be incorrect conclusions drawn about the population that is being studied.
We need to evaluate the statistical studies we read about critically and analyze them before accepting the results of the studies. Common problems to be aware of include
Problems with samples: A sample must be representative of the population. A sample that is not representative of the population is biased. Biased samples that are not representative of the population give results that are inaccurate and not valid.
Self-selected samples: Responses only by people who choose to respond, such as call-in surveys, are often unreliable.
Sample size issues: Samples that are too small may be unreliable. Larger samples are better, if possible. In some situations, having small samples is unavoidable and can still be used to draw conclusions. Examples: crash testing cars or medical testing for rare conditions
Undue influence: collecting data or asking questions in a way that influences the response
Non-response or refusal of subject to participate: The collected responses may no longer be representative of the population. Often, people with strong positive or negative opinions may answer surveys, which can affect the results.
Causality: A relationship between two variables does not mean that one causes the other to occur. They may be related (correlated) because of their relationship through a different variable.
Self-funded or self-interest studies: A study performed by a person or organization in order to support their claim. Is the study impartial? Read the study carefully to evaluate the work. Do not automatically assume that the study is good, but do not automatically assume the study is bad either. Evaluate it on its merits and the work done.
Misleading use of data: improperly displayed graphs, incomplete data, or lack of context
Confounding: When the effects of multiple factors on a response cannot be separated. Confounding makes it difficult or impossible to draw valid conclusions about the effect of each factor.
If we were to examine two samples representing the same population, even if we used random sampling methods for the samples, they would not be exactly the same. Just as there is variation in data, there is variation in samples. As you become accustomed to sampling, the variability will begin to seem natural.
A local radio station has a fan base of 20,000 listeners. The station wants to know if its audience would prefer more music or more talk shows. Asking all 20,000 listeners is an almost impossible task.
The station uses convenience sampling and surveys the first 200 people they meet at one of the station’s music concert events. 24 people said they’d prefer more talk shows, and 176 people said they’d prefer more music.
Do you think that this sample is representative of (or is characteristic of) the entire 20,000 listener population?
Variation is present in any set of data. For example, 16-ounce cans of beverage may contain more or less than 16 ounces of liquid. In one study, eight 16 ounce cans were measured and produced the following amount (in ounces) of beverage:
15.8; 16.1; 15.2; 14.8; 15.8; 15.9; 16.0; 15.5
Measurements of the amount of beverage in a 16-ounce can may vary because different people make the measurements or because the exact amount, 16 ounces of liquid, was not put into the cans. Manufacturers regularly run tests to determine if the amount of beverage in a 16-ounce can falls within the desired range.
Be aware that as you take data, your data may vary somewhat from the data someone else is taking for the same purpose. This is completely natural. However, if two or more of you are taking the same data and get very different results, it is time for you and the others to reevaluate your data-taking methods and your accuracy.
It was mentioned previously that two or more samples from the same population, taken randomly, and having close to the same characteristics of the population will likely be different from each other. Suppose Doreen and Jung both decide to study the average amount of time students at their college sleep each night. Doreen and Jung each take samples of 500 students. Doreen uses systematic sampling and Jung uses cluster sampling. Doreen's sample will be different from Jung's sample. Even if Doreen and Jung used the same sampling method, in all likelihood their samples would be different. Neither would be wrong, however.
Think about what contributes to making Doreen’s and Jung’s samples different.
If Doreen and Jung took larger samples (i.e. the number of data values is increased), their sample results (the average amount of time a student sleeps) might be closer to the actual population average. But still, their samples would be, in all likelihood, different from each other. This variability in samples cannot be stressed enough.
The size of a sample (often called the number of observations, usually given the symbol n) is important. The examples you have seen in this book so far have been small. Samples of only a few hundred observations, or even smaller, are sufficient for many purposes. In polling, samples that are from 1,200 to 1,500 observations are considered large enough and good enough if the survey is random and is well done. Later we will find that even much smaller sample sizes will give very good results. You will learn why when you study confidence intervals.
Be aware that many large samples are biased. For example, call-in surveys are invariably biased, because people choose to respond or not.
Once you have a set of data, you will need to organize it so that you can analyze how frequently each datum occurs in the set. However, when calculating the frequency, you may need to round your answers so that they are as precise as possible.
The way a set of data is measured is called its level of measurement. Correct statistical procedures depend on a researcher being familiar with levels of measurement. Not every statistical operation can be used with every set of data. Data can be classified into four levels of measurement. They are (from lowest to highest level):
Nominal scale level
Ordinal scale level
Interval scale level
Ratio scale level
Data that is measured using a nominal scale is qualitative (categorical). Categories, colors, names, labels and favorite foods along with yes or no responses are examples of nominal level data. Nominal scale data are not ordered. For example, trying to classify people according to their favorite food does not make any sense. Putting pizza first and sushi second is not meaningful.
Smartphone companies are another example of nominal scale data. The data are the names of the companies that make smartphones, but there is no agreed upon order of these brands, even though people may have personal preferences. Nominal scale data cannot be used in calculations.
Data that is measured using an ordinal scale is similar to nominal scale data but there is a big difference. The ordinal scale data can be ordered. An example of ordinal scale data is a list of the top five national parks in the United States. The top five national parks in the United States can be ranked from one to five but we cannot measure differences between the data.
Another example of using the ordinal scale is a cruise survey where the responses to questions about the cruise are “excellent,” “good,” “satisfactory,” and “unsatisfactory.” These responses are ordered from the most desired response to the least desired. But the differences between two pieces of data cannot be measured. Like the nominal scale data, ordinal scale data cannot be used in calculations.
Data that is measured using the interval scale is similar to ordinal level data because it has a definite ordering but there is a difference between data. The differences between interval scale data can be measured though the data does not have a starting point.
Temperature scales like Celsius (C) and Fahrenheit (F) are measured by using the interval scale. In both temperature measurements, 40° is equal to 100° minus 60°. Differences make sense. But 0 degrees does not because, in both scales, 0 is not the absolute lowest temperature. Temperatures like −10° F and −15° C exist and are colder than 0.
Interval level data can be used in calculations, but one type of comparison cannot be done. 80° C is not four times as hot as 20° C (nor is 80° F four times as hot as 20° F). There is no meaning to the ratio of 80 to 20 (or four to one).
Data that is measured using the ratio scale takes care of the ratio problem and gives you the most information. Ratio scale data is like interval scale data, but it has a 0 point and ratios can be calculated. For example, four multiple choice statistics final exam scores are 80, 68, 20 and 92 (out of a possible 100 points). The exams are machine-graded.
The data can be put in order from lowest to highest: 20, 68, 80, 92.
The differences between the data have meaning. The score 92 is more than the score 68 by 24 points. Ratios can be calculated. The smallest score is 0. So 80 is four times 20. The score of 80 is four times better than the score of 20.
Twenty students were asked how many hours they worked per day. Their responses, in hours, are as follows: 5; 6; 3; 3; 2; 4; 7; 5; 2; 3; 5; 6; 5; 4; 4; 3; 5; 2; 5; 3.
Table 1.5 lists the different data values in ascending order and their frequencies.
|
Data value |
Frequency |
|
2 |
3 |
|
3 |
5 |
|
4 |
3 |
|
5 |
6 |
|
6 |
2 |
|
7 |
1 |
Table 1.5 Frequency Table of Student Work Hours
A frequency is the number of times a value of the data occurs. According to Table 1.5, there are three students who work two hours, five students who work three hours, and so on. The sum of the values in the frequency column, 20, represents the total number of students included in the sample.
A relative frequency is the ratio (fraction or proportion) of the number of times a value of the data occurs in the set of all outcomes to the total number of outcomes. To find the relative frequencies, divide each frequency by the total number of students in the sample–in this case, 20. Relative frequencies can be written as fractions, percents, or decimals.
|
Data value |
Frequency |
Relative frequency |
|
2 |
3 |
or 0.15 |
|
3 |
5 |
or 0.25 |
|
4 |
3 |
or 0.15 |
|
5 |
6 |
or 0.30 |
|
6 |
2 |
or 0.10 |
|
7 |
1 |
or 0.05 |
Table 1.6 Frequency Table of Student Work Hours with Relative Frequencies
The sum of the values in the relative frequency column of Table 1.6 is , or 1.
Cumulative relative frequency is the accumulation of the previous relative frequencies. To find the cumulative relative frequencies, add all the previous relative frequencies to the relative frequency for the current row, as shown in Table 1.7.
|
Heights (inches) |
Frequency |
Relative frequency |
Cumulative relative frequency |
|
2 |
3 |
or 0.15 |
0.15 |
|
3 |
5 |
or 0.25 |
0.15 + 0.25 = 0.40 |
|
4 |
3 |
or 0.15 |
0.40 + 0.15 = 0.55 |
|
5 |
6 |
or 0.30 |
0.55 + 0.30 = 0.85 |
|
6 |
2 |
or 0.10 |
0.85 + 0.10 = 0.95 |
|
7 |
1 |
or 0.05 |
0.95 + 0.05 = 1.00 |
Table 1.7 Frequency Table of Student Work Hours with Relative and Cumulative Relative Frequencies
The last entry of the cumulative relative frequency column is one, indicating that one hundred percent of the data has been accumulated.
Table 1.8 represents the heights, in inches, of a sample of 100 male semiprofessional soccer players.
|
Heights (inches) |
Frequency |
Relative frequency |
Cumulative relative frequency |
|
59.95–61.95 |
5 |
= 0.05 |
0.05 |
|
61.95–63.95 |
3 |
= 0.03 |
0.05 + 0.03 = 0.08 |
|
63.95–65.95 |
15 |
= 0.15 |
0.08 + 0.15 = 0.23 |
|
65.95–67.95 |
40 |
= 0.40 |
0.23 + 0.40 = 0.63 |
|
67.95–69.95 |
17 |
= 0.17 |
0.63 + 0.17 = 0.80 |
|
69.95–71.95 |
12 |
= 0.12 |
0.80 + 0.12 = 0.92 |
|
71.95–73.95 |
7 |
= 0.07 |
0.92 + 0.07 = 0.99 |
|
73.95–75.95 |
1 |
= 0.01 |
0.99 + 0.01 = 1.00 |
|
Total = 100 |
Total = 1.00 |
Table 1.8 Frequency Table of Soccer Player Height
The data in this table have been grouped into the following intervals:
59.95 to 61.95 inches
61.95 to 63.95 inches
63.95 to 65.95 inches
65.95 to 67.95 inches
67.95 to 69.95 inches
69.95 to 71.95 inches
71.95 to 73.95 inches
73.95 to 75.95 inches
In this sample, there are five players whose heights fall within the interval 59.95–61.95 inches, three players whose heights fall within the interval 61.95–63.95 inches, 15 players whose heights fall within the interval 63.95–65.95 inches, 40 players whose heights fall within the interval 65.95–67.95 inches, 17 players whose heights fall within the interval 67.95–69.95 inches, 12 players whose heights fall within the interval 69.95–71.95, seven players whose heights fall within the interval 71.95–73.95, and one player whose heights fall within the interval 73.95–75.95. All heights fall between the endpoints of an interval and not at the endpoints.
|
Rainfall (inches) |
Frequency |
Relative frequency |
Cumulative relative frequency |
|
2.95–4.97 |
6 |
= 0.12 |
0.12 |
|
4.97–6.99 |
7 |
= 0.14 |
0.12 + 0.14 = 0.26 |
|
6.99–9.01 |
15 |
= 0.30 |
0.26 + 0.30 = 0.56 |
|
9.01–11.03 |
8 |
= 0.16 |
0.56 + 0.16 = 0.72 |
|
11.03–13.05 |
9 |
= 0.18 |
0.72 + 0.18 = 0.90 |
|
13.05–15.07 |
5 |
= 0.10 |
0.90 + 0.10 = 1.00 |
|
Total = 50 |
Total = 1.00 |
Table 1.9
From Table 1.9, find the percentage of rainfall that is less than 9.01 inches.
From Table 1.9, find the percentage of rainfall that is between 6.99 and 13.05 inches.
Table 1.9 represents the amount, in inches, of annual rainfall in a sample of towns. What fraction of towns surveyed get between 11.03 and 13.05 inches of rainfall each year?
Table 1.12 contains the total number of fatal motor vehicle traffic crashes in the United States for the period from 1994 to 2011.
|
Year |
Total number of crashes |
Year |
Total number of crashes |
|
1994 |
36,254 |
2004 |
38,444 |
|
1995 |
37,241 |
2005 |
39,252 |
|
1996 |
37,494 |
2006 |
38,648 |
|
1997 |
37,324 |
2007 |
37,435 |
|
1998 |
37,107 |
2008 |
34,172 |
|
1999 |
37,140 |
2009 |
30,862 |
|
2000 |
37,526 |
2010 |
30,296 |
|
2001 |
37,862 |
2011 |
29,757 |
|
2002 |
38,491 |
Total |
653,782 |
|
2003 |
38,477 |
Table 1.12
Answer the following questions.
What is the frequency of deaths measured from 2000 through 2004?
What percentage of deaths occurred after 2006?
What is the relative frequency of deaths that occurred in 2000 or before?
What is the percentage of deaths that occurred in 2011?
What is the cumulative relative frequency for 2006? Explain what this number tells you about the data.
Does aspirin reduce the risk of heart attacks? Is one brand of fertilizer more effective at growing roses than another? Is fatigue as dangerous to a driver as the influence of alcohol? Questions like these are answered using randomized experiments. In this module, you will learn important aspects of experimental design. Proper study design ensures the production of reliable, accurate data.
The purpose of an experiment is to investigate the relationship between two variables. When one variable causes change in another, we call the first variable the independent variable or explanatory variable. The affected variable is called the dependent variable or response variable: stimulus, response. In a randomized experiment, the researcher manipulates values of the explanatory variable and measures the resulting changes in the response variable. The different values of the explanatory variable are called treatments. An experimental unit is a single object or individual to be measured.
You want to investigate the effectiveness of vitamin E in preventing disease. You recruit a group of subjects and ask them if they regularly take vitamin E. You notice that the subjects who take vitamin E exhibit better health on average than those who do not. Does this prove that vitamin E is effective in disease prevention? It does not. There are many differences between the two groups compared in addition to vitamin E consumption. People who take vitamin E regularly often take other steps to improve their health: exercise, diet, other vitamin supplements, choosing not to smoke. Any one of these factors could be influencing health. As described, this study does not prove that vitamin E is the key to disease prevention.
Statistics is, or should be, about scientific investigation and how to do it better, but many statisticians believe it is a branch of mathematics. Now I agree that the physicist, the chemist, the engineer, and the statistician can never know too much mathematics, but their objectives should be better physics, better chemistry, better engineering, and in the case of statistics, better scientific investigation. Whether in any given study this implies more or less mathematics is incidental.
George E. P. Box
Additional variables that can cloud a study are called lurking variables. In order to prove that the explanatory variable is causing a change in the response variable, it is necessary to isolate the explanatory variable. The researcher must design her experiment in such a way that there is only one difference between groups being compared: the planned treatments. This is accomplished by the random assignment of experimental units to treatment groups. When subjects are assigned treatments randomly, all of the potential lurking variables are spread equally among the groups. At this point the only difference between groups is the one imposed by the researcher. Different outcomes measured in the response variable, therefore, must be a direct result of the different treatments. In this way, an experiment can prove a cause-and-effect connection between the explanatory and response variables.
The power of suggestion can have an important influence on the outcome of an experiment. Studies have shown that the expectation of the study participant can be as important as the actual medication. In one study of performance-enhancing drugs, researchers noted:
Results showed that believing one had taken the substance resulted in [performance] times almost as fast as those associated with consuming the drug itself. In contrast, taking the drug without knowledge yielded no significant performance increment.
McClung, M. Collins, D. “Because I know it will!”: placebo effects of an ergogenic aid on athletic performance. Journal of Sport & Exercise Psychology. 2007 Jun. 29(3):382-94. Web. April 30, 2013.
When participation in a study prompts a physical response from a participant, it is difficult to isolate the effects of the explanatory variable. To counter the power of suggestion, researchers set aside one treatment group as a control group. This group is given a placebo treatment–a treatment that cannot influence the response variable. The control group helps researchers balance the effects of being in an experiment with the effects of the active treatments. Of course, if you are participating in a study and you know that you are receiving a pill which contains no actual medication, then the power of suggestion is no longer a factor. Blinding in a randomized experiment preserves the power of suggestion. When a person involved in a research study is blinded, he does not know who is receiving the active treatment(s) and who is receiving the placebo treatment. A double-blind experiment is one in which both the subjects and the researchers involved with the subjects are blinded.
Average also called mean or arithmetic mean; a number that describes the central tendency of the data
Blinding not telling participants which treatment a subject is receiving
Categorical Variable variables that take on values that are names or labels
Cluster Sampling a method for selecting a random sample and dividing the population into groups (clusters); use simple random sampling to select a set of clusters. Every individual in the chosen clusters is included in the sample.
Continuous Random Variable a random variable (RV) whose outcomes are measured; the height of trees in the forest is a continuous RV.
Control Group a group in a randomized experiment that receives an inactive treatment but is otherwise managed exactly as the other groups
Convenience Sampling a nonrandom method of selecting a sample; this method selects individuals that are easily accessible and may result in biased data.
Cumulative Relative Frequency The term applies to an ordered set of observations from smallest to largest. The cumulative relative frequency is the sum of the relative frequencies for all values that are less than or equal to the given value.
Data a set of observations (a set of possible outcomes); most data can be put into two groups: qualitative (an attribute whose value is indicated by a label) or quantitative (an attribute whose value is indicated by a number). Quantitative data can be separated into two subgroups: discrete and continuous. Data is discrete if it is the result of counting (such as the number of students of a given ethnic group in a class or the number of books on a shelf). Data is continuous if it is the result of measuring (such as distance traveled or weight of luggage)
Discrete Random Variable a random variable (RV) whose outcomes are counted
Double-blinding the act of blinding both the subjects of an experiment and the researchers who work with the subjects
Experimental Unit any individual or object to be measured
Explanatory Variable the independent variable in an experiment; the value controlled by researchers
Frequency the number of times a value of the data occurs
Informed Consent Any human subject in a research study must be cognizant of any risks or costs associated with the study. The subject has the right to know the nature of the treatments included in the study, their potential risks, and their potential benefits. Consent must be given freely by an informed, fit participant.
Institutional Review Board a committee tasked with oversight of research programs that involve human subjects
Lurking Variable a variable that has an effect on a study even though it is neither an explanatory variable nor a response variable
Mathematical Models a description of a phenomenon using mathematical concepts, such as equations, inequalities, distributions, etc.
Nonsampling Error an issue that affects the reliability of sampling data other than natural variation; it includes a variety of human errors including poor study design, biased sampling methods, inaccurate information provided by study participants, data entry errors, and poor analysis.
Numerical Variable variables that take on values that are indicated by numbers
Observational Study a study in which the independent variable is not manipulated by the researcher
Parameter a number that is used to represent a population characteristic and that generally cannot be determined easily
Placebo an inactive treatment that has no real effect on the explanatory variable
Population all individuals, objects, or measurements whose properties are being studied
Probability a number between zero and one, inclusive, that gives the likelihood that a specific event will occur
Proportion the number of successes divided by the total number in the sample
Qualitative Data See Data.
Quantitative Data See Data.
Random Assignment the act of organizing experimental units into treatment groups using random methods
Random Sampling a method of selecting a sample that gives every member of the population an equal chance of being selected.
Relative Frequency the ratio of the number of times a value of the data occurs in the set of all outcomes to the number of all outcomes to the total number of outcomes
Representative Sample a subset of the population that has the same characteristics as the population
Response Variable the dependent variable in an experiment; the value that is measured for change at the end of an experiment
Sample a subset of the population studied
Sampling Bias not all members of the population are equally likely to be selected
Sampling Error the natural variation that results from selecting a sample to represent a larger population; this variation decreases as the sample size increases, so selecting larger samples reduces sampling error.
Sampling with Replacement Once a member of the population is selected for inclusion in a sample, that member is returned to the population for the selection of the next individual.
Sampling without Replacement A member of the population may be chosen for inclusion in a sample only once. If chosen, the member is not returned to the population before the next selection.
Simple Random Sampling a straightforward method for selecting a random sample; give each member of the population a number. Use a random number generator to select a set of labels. These randomly selected labels identify the members of your sample.
Statistic a numerical characteristic of the sample; a statistic estimates the corresponding population parameter.
Statistical Models a description of a phenomenon using probability distributions that describe the expected behavior of the phenomenon and the variability in the expected observations.
Stratified Sampling a method for selecting a random sample used to ensure that subgroups of the population are represented adequately; divide the population into groups (strata). Use simple random sampling to identify a proportionate number of individuals from each stratum.
Survey a study in which data is collected as reported by individuals.
Systematic Sampling a method for selecting a random sample; list the members of the population. Use simple random sampling to select a starting point in the population. Let k = (number of individuals in the population)/(number of individuals needed in the sample). Choose every kth individual in the list starting with the one that was randomly selected. If necessary, return to the beginning of the population list to complete your sample.
Treatments different values or components of the explanatory variable applied in an experiment
Variable a characteristic of interest for each person or object in a population
The mathematical theory of statistics is easier to learn when you know the language. This module presents important terms that will be used throughout the text.
Data are individual items of information that come from a population or sample. Data may be classified as qualitative (categorical), quantitative continuous, or quantitative discrete.
Because it is not practical to measure the entire population in a study, researchers use samples to represent the population. A random sample is a representative group from the population chosen by using a method that gives each individual in the population an equal chance of being included in the sample. Random sampling methods include simple random sampling, stratified sampling, cluster sampling, and systematic sampling. Convenience sampling is a nonrandom method of choosing a sample that often produces biased data.
Samples that contain different individuals result in different data. This is true even when the samples are well-chosen and representative of the population. When properly selected, larger samples model the population more closely than smaller samples. There are many different potential problems that can affect the reliability of a sample. Statistical data needs to be critically analyzed, not simply accepted.
Some calculations generate numbers that are artificially precise. It is not necessary to report a value to eight decimal places when the measures that generated that value were only accurate to the nearest tenth. Round off your final answer to one more decimal place than was present in the original data. This means that if you have data measured to the nearest tenth of a unit, report the final statistic to the nearest hundredth.
In addition to rounding your answers, you can measure your data using the following four levels of measurement.
Nominal scale level: data that cannot be ordered nor can it be used in calculations
Ordinal scale level: data that can be ordered; the differences cannot be measured
Interval scale level: data with a definite ordering but no starting point; the differences can be measured, but there is no such thing as a ratio.
Ratio scale level: data with a starting point that can be ordered; the differences have meaning and ratios can be calculated.
When organizing data, it is important to know how many times a value appears. How many statistics students study five hours or more for an exam? What percent of families on our block own two pets? Frequency, relative frequency, and cumulative relative frequency are measures that answer questions like these.
A poorly designed study will not produce reliable data. There are certain key components that must be included in every experiment. To eliminate lurking variables, subjects must be assigned randomly to different treatment groups. One of the groups must act as a control group, demonstrating what happens when the active treatment is not applied. Participants in the control group receive a placebo treatment that looks exactly like the active treatments but cannot influence the response variable. To preserve the integrity of the placebo, both researchers and subjects may be blinded. When a study is designed properly, the only difference between treatment groups is the one imposed by the researcher. Therefore, when groups respond differently to different treatments, the difference must be due to the influence of the explanatory variable.
“An ethics problem arises when you are considering an action that benefits you or some cause you support, hurts or reduces benefits to others, and violates some rule.” (Andrew Gelman, “Open Data and Open Methods,” Ethics and Statistics, http://www.stat.columbia.edu/~gelman/research/published/ChanceEthics1.pdf (accessed May 1, 2013).) Ethical violations in statistics are not always easy to spot. Professional associations and federal agencies post guidelines for proper conduct. It is important that you learn basic statistical procedures so that you can recognize proper data analysis.
Find a real life example from your own research or the Further reading section, and write a case study to outline each of the key terms used in this chapter, including the relevant statistics for each key term.
The Data and Story Library, http://lib.stat.cmu.edu/DASL/Stories/CrashTestDummies.html (accessed May 1, 2013).
Gallup-Healthways Well-Being Index. http://www.well-beingindex.com/default.asp (accessed May 1, 2013).
Gallup-Healthways Well-Being Index. http://www.well-beingindex.com/methodology.asp (accessed May 1, 2013).
Gallup-Healthways Well-Being Index. http://www.gallup.com/poll/146822/gallup-healthways-index-questions.aspx (accessed May 1, 2013).
Data from http://www.bookofodds.com/Relationships-Society/Articles/A0374-How-George-Gallup-Picked-the-President
Dominic Lusinchi, “’President’ Landon and the 1936 Literary Digest Poll: Were Automobile and Telephone Owners to Blame?” Social Science History 36, no. 1: 23-54 (2012), http://ssh.dukejournals.org/content/36/1/23.abstract (accessed May 1, 2013).
“The Literary Digest Poll,” Virtual Laboratories in Probability and Statistics http://www.math.uah.edu/stat/data/LiteraryDigest.html (accessed May 1, 2013).
“Gallup Presidential Election Trial-Heat Trends, 1936–2008,” Gallup Politics http://www.gallup.com/poll/110548/gallup-presidential-election-trialheat-trends-19362004.aspx#4 (accessed May 1, 2013).
The Data and Story Library, http://lib.stat.cmu.edu/DASL/Datafiles/USCrime.html (accessed May 1, 2013).
LBCC Distance Learning (DL) program data in 2010-2011, http://de.lbcc.edu/reports/2010-11/future/highlights.html#focus (accessed May 1, 2013).
Data from San Jose Mercury News
Lookup London. History of the Scottish Widows fund. accessed 2 December 2022 https://lookup.london/history-scottish-widows/
St Andrews University. Jacob Bernoulli Biography. https://mathshistory.st-andrews.ac.uk/Biographies/Bernoulli_Jacob/
“State & County QuickFacts,” U.S. Census Bureau. http://quickfacts.census.gov/qfd/download_data.html (accessed May 1, 2013).
“State & County QuickFacts: Quick, easy access to facts about people, business, and geography,” U.S. Census Bureau. http://quickfacts.census.gov/qfd/index.html (accessed May 1, 2013).
“Table 5: Direct hits by mainland United States Hurricanes (1851-2004),” National Hurricane Center, http://www.nhc.noaa.gov/gifs/table5.gif (accessed May 1, 2013).
“Levels of Measurement,” http://infinity.cos.edu/faculty/woodbury/stats/tutorial/Data_Levels.htm (accessed May 1, 2013).
Courtney Taylor, “Levels of Measurement,” about.com, http://statistics.about.com/od/HelpandTutorials/a/Levels-Of-Measurement.htm (accessed May 1, 2013).
David Lane. “Levels of Measurement,” Connexions, http://cnx.org/content/m10809/latest/ (accessed May 1, 2013).
“Vitamin E and Health,” Nutrition Source, Harvard School of Public Health, http://www.hsph.harvard.edu/nutritionsource/vitamin-e/ (accessed May 1, 2013).
Stan Reents. “Don’t Underestimate the Power of Suggestion,” athleteinme.com, http://www.athleteinme.com/ArticleView.aspx?id=1053 (accessed May 1, 2013).
Ankita Mehta. “Daily Dose of Aspiring Helps Reduce Heart Attacks: Study,” International Business Times, July 21, 2011. Also available online at http://www.ibtimes.com/daily-dose-aspirin-helps-reduce-heart-attacks-study-300443 (accessed May 1, 2013).
The Data and Story Library, http://lib.stat.cmu.edu/DASL/Stories/ScentsandLearning.html (accessed May 1, 2013).
M.L. Jackson et al., “Cognitive Components of Simulated Driving Performance: Sleep Loss effect and Predictors,” Accident Analysis and Prevention Journal, Jan no. 50 (2013), http://www.ncbi.nlm.nih.gov/pubmed/22721550 (accessed May 1, 2013).
“Earthquake Information by Year,” U.S. Geological Survey. http://earthquake.usgs.gov/earthquakes/eqarchives/year/ (accessed May 1, 2013).
“Fatality Analysis Report Systems (FARS) Encyclopedia,” National Highway Traffic and Safety Administration. http://www-fars.nhtsa.dot.gov/Main/index.aspx (accessed May 1, 2013).
Data from www.businessweek.com (accessed May 1, 2013).
Data from www.forbes.com (accessed May 1, 2013).
“America’s Best Small Companies,” http://www.forbes.com/best-small-companies/list/ (accessed May 1, 2013).
U.S. Department of Health and Human Services, Code of Federal Regulations Title 45 Public Welfare Department of Health and Human Services Part 46 Protection of Human Subjects revised January 15, 2009. Section 46.111:Criteria for IRB Approval of Research.
“April 2013 Air Travel Consumer Report,” U.S. Department of Transportation, April 11 (2013), http://www.dot.gov/airconsumer/april-2013-air-travel-consumer-report (accessed May 1, 2013).
Lori Alden, “Statistics can be Misleading,” econoclass.com, http://www.econoclass.com/misleadingstats.html (accessed May 1, 2013).
Maria de los A. Medina, “Ethics in Statistics,” Based on “Building an Ethics Module for Business, Science, and Engineering Students” by Jose A. Cruz-Cruz and William Frey, Connexions, http://cnx.org/content/m15555/latest/ (accessed May 1, 2013).
“Access to electricity (% of population),” The World Bank, 2013. Available online at http://data.worldbank.org/indicator/EG.ELC.ACCS.ZS?order=wbapi_data_value_2009%20wbapi_data_value%20wbapi_data_value-first&sort=asc (accessed May 15, 2015).
“Distance Education.” Wikipedia. Available online at http://en.wikipedia.org/wiki/Distance_education (accessed May 15, 2013).
“NBA Statistics – 2013,” ESPN NBA, 2013. Available online at http://espn.go.com/nba/statistics/_/seasontype/2 (accessed May 15, 2013).
Newport, Frank. “Americans Still Enjoy Saving Rather than Spending: Few demographic differences seen in these views other than by income,” GALLUP® Economy, 2013. Available online at http://www.gallup.com/poll/162368/americans-enjoy-saving-rather-spending.aspx (accessed May 15, 2013).
Pryor, John H., Linda DeAngelo, Laura Palucki Blake, Sylvia Hurtado, Serge Tran. The American Freshman: National Norms Fall 2011. Los Angeles: Cooperative Institutional Research Program at the Higher Education Research Institute at UCLA, 2011. Also available online at http://heri.ucla.edu/PDFs/pubs/TFS/Norms/Monographs/TheAmericanFreshman2011.pdf (accessed May 15, 2013).
“The World FactBook,” Central Intelligence Agency. Available online at https://www.cia.gov/library/publications/the-world-factbook/geos/af.html (accessed May 15, 2013).
“What are the key statistics about pancreatic cancer?” American Cancer Society, 2013. Available online at http://www.cancer.org/cancer/pancreaticcancer/detailedguide/pancreatic-cancer-key-statistics (accessed May 15, 2013).
“Millennials: A Portrait of Generation Next,” PewResearchCenter. Available online at http://www.pewsocialtrends.org/files/2010/10/millennials-confident-connected-open-to-change.pdf (accessed May 15, 2013).
“Millennials: Confident. Connected. Open to Change.” Executive Summary by PewResearch Social & Demographic Trends, 2013. Available online at http://www.pewsocialtrends.org/2010/02/24/millennials-confident-connected-open-to-change/ (accessed May 15, 2013).
“Prevalence of HIV, total (% of populations ages 15-49),” The World Bank, 2013. Available online at http://data.worldbank.org/indicator/SH.DYN.AIDS.ZS?order=wbapi_data_value_2011+wbapi_data_value+wbapi_data_value-last&sort=desc (accessed May 15, 2013).
Pryor, John H., Linda DeAngelo, Laura Palucki Blake, Sylvia Hurtado, Serge Tran. The American Freshman: National Norms Fall 2011. Los Angeles: Cooperative Institutional Research Program at the Higher Education Research Institute at UCLA, 2011. Also available online at http://heri.ucla.edu/PDFs/pubs/TFS/Norms/Monographs/TheAmericanFreshman2011.pdf (accessed May 15, 2013).
“Summary of the National Risk and Vulnerability Assessment 2007/8: A profile of Afghanistan,” The European Union and ICON-Institute. Available online at http://ec.europa.eu/europeaid/where/asia/documents/afgh_brochure_summary_en.pdf (accessed May 15, 2013).
“The World FactBook,” Central Intelligence Agency. Available online at https://www.cia.gov/library/publications/the-world-factbook/geos/af.html (accessed May 15, 2013).
“UNICEF reports on Female Literacy Centers in Afghanistan established to teach women and girls basic resading [sic] and writing skills,” UNICEF Television. Video available online at http://www.unicefusa.org/assets/video/afghan-female-literacy-centers.html (accessed May 15, 2013).
“ATL Fact Sheet,” Department of Aviation at the Hartsfield-Jackson Atlanta International Airport, 2013. Available online at http://www.atl.com/about-atl/atl-factsheet/ (accessed February 6, 2019).
Center for Disease Control and Prevention. “Teen Drivers: Fact Sheet,” Injury Prevention & Control: Motor Vehicle Safety, October 2, 2012. Available online at http://www.cdc.gov/Motorvehiclesafety/Teen_Drivers/teendrivers_factsheet.html (accessed May 15, 2013).
“Children and Childrearing,” Ministry of Health, Labour, and Welfare. Available online at http://www.mhlw.go.jp/english/policy/children/children-childrearing/index.html (accessed May 15, 2013).
“Eating Disorder Statistics,” South Carolina Department of Mental Health, 2006. Available online at http://www.state.sc.us/dmh/anorexia/statistics.htm (accessed May 15, 2013).
“Giving Birth in Manila: The maternity ward at the Dr Jose Fabella Memorial Hospital in Manila, the busiest in the Philippines, where there is an average of 60 births a day,” theguardian, 2013. Available online at http://www.theguardian.com/world/gallery/2011/jun/08/philippines-health#/?picture=375471900&index=2 (accessed May 15, 2013).
“How Americans Use Text Messaging,” Pew Internet, 2013. Available online at http://pewinternet.org/Reports/2011/Cell-Phone-Texting-2011/Main-Report.aspx (accessed May 15, 2013).
Lenhart, Amanda. “Teens, Smartphones & Testing: Texting volume is up while the frequency of voice calling is down. About one in four teens say they own smartphones,” Pew Internet, 2012. Available online at http://www.pewinternet.org/~/media/Files/Reports/2012/PIP_Teens_Smartphones_and_Texting.pdf (accessed May 15, 2013).
“One born every minute: the maternity unit where mothers are THREE to a bed,” MailOnline. Available online at http://www.dailymail.co.uk/news/article-2001422/Busiest-maternity-ward-planet-averages-60-babies-day-mothers-bed.html (accessed May 15, 2013).
Vanderkam, Laura. “Stop Checking Your Email, Now.” CNNMoney, 2013. Available online at http://management.fortune.cnn.com/2012/10/08/stop-checking-your-email-now/ (accessed May 15, 2013).
“World Earthquakes: Live Earthquake News and Highlights,” World Earthquakes, 2012. http://www.world-earthquakes.com/index.php?option=ethq_prediction (accessed May 15, 2013).
Average also called mean or arithmetic mean; a number that describes the central tendency of the data
Blinding not telling participants which treatment a subject is receiving
Categorical Variable variables that take on values that are names or labels
Cluster Sampling a method for selecting a random sample and dividing the population into groups (clusters); use simple random sampling to select a set of clusters. Every individual in the chosen clusters is included in the sample.
Continuous Random Variable a random variable (RV) whose outcomes are measured; the height of trees in the forest is a continuous RV.
Control Group a group in a randomized experiment that receives an inactive treatment but is otherwise managed exactly as the other groups
Convenience Sampling a nonrandom method of selecting a sample; this method selects individuals that are easily accessible and may result in biased data.
Cumulative Relative Frequency The term applies to an ordered set of observations from smallest to largest. The cumulative relative frequency is the sum of the relative frequencies for all values that are less than or equal to the given value.
Data a set of observations (a set of possible outcomes); most data can be put into two groups: qualitative (an attribute whose value is indicated by a label) or quantitative (an attribute whose value is indicated by a number). Quantitative data can be separated into two subgroups: discrete and continuous. Data is discrete if it is the result of counting (such as the number of students of a given ethnic group in a class or the number of books on a shelf). Data is continuous if it is the result of measuring (such as distance traveled or weight of luggage)
Discrete Random Variable a random variable (RV) whose outcomes are counted
Double-blinding the act of blinding both the subjects of an experiment and the researchers who work with the subjects
Experimental Unit any individual or object to be measured
Explanatory Variable the independent variable in an experiment; the value controlled by researchers
Frequency the number of times a value of the data occurs
Informed Consent Any human subject in a research study must be cognizant of any risks or costs associated with the study. The subject has the right to know the nature of the treatments included in the study, their potential risks, and their potential benefits. Consent must be given freely by an informed, fit participant.
Institutional Review Board a committee tasked with oversight of research programs that involve human subjects
Lurking Variable a variable that has an effect on a study even though it is neither an explanatory variable nor a response variable
Mathematical Models a description of a phenomenon using mathematical concepts, such as equations, inequalities, distributions, etc.
Nonsampling Error an issue that affects the reliability of sampling data other than natural variation; it includes a variety of human errors including poor study design, biased sampling methods, inaccurate information provided by study participants, data entry errors, and poor analysis.
Numerical Variable variables that take on values that are indicated by numbers
Observational Study a study in which the independent variable is not manipulated by the researcher
Parameter a number that is used to represent a population characteristic and that generally cannot be determined easily
Placebo an inactive treatment that has no real effect on the explanatory variable
Population all individuals, objects, or measurements whose properties are being studied
Probability a number between zero and one, inclusive, that gives the likelihood that a specific event will occur
Proportion the number of successes divided by the total number in the sample
Qualitative Data See Data.
Quantitative Data See Data.
Random Assignment the act of organizing experimental units into treatment groups using random methods
Random Sampling a method of selecting a sample that gives every member of the population an equal chance of being selected.
Relative Frequency the ratio of the number of times a value of the data occurs in the set of all outcomes to the number of all outcomes to the total number of outcomes
Representative Sample a subset of the population that has the same characteristics as the population
Response Variable the dependent variable in an experiment; the value that is measured for change at the end of an experiment
Sample a subset of the population studied
Sampling Bias not all members of the population are equally likely to be selected
Sampling Error the natural variation that results from selecting a sample to represent a larger population; this variation decreases as the sample size increases, so selecting larger samples reduces sampling error.
Sampling with Replacement Once a member of the population is selected for inclusion in a sample, that member is returned to the population for the selection of the next individual.
Sampling without Replacement A member of the population may be chosen for inclusion in a sample only once. If chosen, the member is not returned to the population before the next selection.
Simple Random Sampling a straightforward method for selecting a random sample; give each member of the population a number. Use a random number generator to select a set of labels. These randomly selected labels identify the members of your sample.
Statistic a numerical characteristic of the sample; a statistic estimates the corresponding population parameter.
Statistical Models a description of a phenomenon using probability distributions that describe the expected behavior of the phenomenon and the variability in the expected observations.
Stratified Sampling a method for selecting a random sample used to ensure that subgroups of the population are represented adequately; divide the population into groups (strata). Use simple random sampling to identify a proportionate number of individuals from each stratum.
Survey a study in which data is collected as reported by individuals.
Systematic Sampling a method for selecting a random sample; list the members of the population. Use simple random sampling to select a starting point in the population. Let k = (number of individuals in the population)/(number of individuals needed in the sample). Choose every kth individual in the list starting with the one that was randomly selected. If necessary, return to the beginning of the population list to complete your sample.
Treatments different values or components of the explanatory variable applied in an experiment
Variable a characteristic of interest for each person or object in a population
|
Degrees of freedom in the numerator |
||||||||||
|---|---|---|---|---|---|---|---|---|---|---|
|
Degrees of freedom |
p |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
|
1 |
.100 |
39.86 |
49.50 |
53.59 |
55.83 |
57.24 |
58.20 |
58.91 |
59.44 |
59.86 |
|
.050 |
161.45 |
199.50 |
215.71 |
224.58 |
230.16 |
233.99 |
236.77 |
238.88 |
240.54 |
|
|
.025 |
647.79 |
799.50 |
864.16 |
899.58 |
921.85 |
937.11 |
948.22 |
956.66 |
963.28 |
|
|
.010 |
4052.2 |
4999.5 |
5403.4 |
5624.6 |
5763.6 |
5859.0 |
5928.4 |
5981.1 |
6022.5 |
|
|
.001 |
405284 |
500000 |
540379 |
562500 |
576405 |
585937 |
592873 |
598144 |
602284 |
|
|
2 |
.100 |
8.53 |
9.00 |
9.16 |
9.24 |
9.29 |
9.33 |
9.35 |
9.37 |
9.38 |
|
.050 |
18.51 |
19.00 |
19.16 |
19.25 |
19.30 |
19.33 |
19.35 |
19.37 |
19.38 |
|
|
.025 |
38.51 |
39.00 |
39.17 |
39.25 |
39.30 |
39.33 |
39.36 |
39.37 |
39.39 |
|
|
.010 |
98.50 |
99.00 |
99.17 |
99.25 |
99.30 |
99.33 |
99.36 |
99.37 |
99.39 |
|
|
.001 |
998.50 |
999.00 |
999.17 |
999.25 |
999.30 |
999.33 |
999.36 |
999.37 |
999.39 |
|
|
3 |
.100 |
5.54 |
5.46 |
5.39 |
5.34 |
5.31 |
5.28 |
5.27 |
5.25 |
5.24 |
|
.050 |
10.13 |
9.55 |
9.28 |
9.12 |
9.01 |
8.94 |
8.89 |
8.85 |
8.81 |
|
|
.025 |
17.44 |
16.04 |
15.44 |
15.10 |
14.88 |
14.73 |
14.62 |
14.54 |
14.47 |
|
|
.010 |
34.12 |
30.82 |
29.46 |
28.71 |
28.24 |
27.91 |
27.67 |
27.49 |
27.35 |
|
|
.001 |
167.03 |
148.50 |
141.11 |
137.10 |
134.58 |
132.85 |
131.58 |
130.62 |
129.86 |
|
|
4 |
.100 |
4.54 |
4.32 |
4.19 |
4.11 |
4.05 |
4.01 |
3.98 |
3.95 |
3.94 |
|
.050 |
7.71 |
6.94 |
6.59 |
6.39 |
6.26 |
6.16 |
6.09 |
6.04 |
6.00 |
|
|
.025 |
12.22 |
10.65 |
9.98 |
9.60 |
9.36 |
9.20 |
9.07 |
8.98 |
8.90 |
|
|
.010 |
21.20 |
18.00 |
16.69 |
15.98 |
15.52 |
15.21 |
14.98 |
14.80 |
14.66 |
|
|
.001 |
74.14 |
61.25 |
56.18 |
53.44 |
51.71 |
50.53 |
49.66 |
49.00 |
48.47 |
|
|
5 |
.100 |
4.06 |
3.78 |
3.62 |
3.52 |
3.45 |
3.40 |
3.37 |
3.34 |
3.32 |
|
.050 |
6.61 |
5.79 |
5.41 |
5.19 |
5.05 |
4.95 |
4.88 |
4.82 |
4.77 |
|
|
.025 |
10.01 |
8.43 |
7.76 |
7.39 |
7.15 |
6.98 |
6.85 |
6.76 |
6.68 |
|
|
.010 |
16.26 |
13.27 |
12.06 |
11.39 |
10.97 |
10.67 |
10.46 |
10.29 |
10.16 |
|
|
.001 |
47.18 |
37.12 |
33.20 |
31.09 |
29.75 |
28.83 |
28.16 |
27.65 |
27.24 |
|
|
6 |
.100 |
3.78 |
3.46 |
3.29 |
3.18 |
3.11 |
3.05 |
3.01 |
2.98 |
2.96 |
|
.050 |
5.99 |
5.14 |
4.76 |
4.53 |
4.39 |
4.28 |
4.21 |
4.15 |
4.10 |
|
|
.025 |
8.81 |
7.26 |
6.60 |
6.23 |
5.99 |
5.82 |
5.70 |
5.60 |
5.52 |
|
|
.010 |
13.75 |
10.92 |
9.78 |
9.15 |
8.75 |
8.47 |
8.26 |
8.10 |
7.98 |
|
|
.001 |
35.51 |
27.00 |
23.70 |
21.92 |
20.80 |
20.03 |
19.46 |
19.03 |
18.69 |
|
|
7 |
.100 |
3.59 |
3.26 |
3.07 |
2.96 |
2.88 |
2.83 |
2.78 |
2.75 |
2.72 |
|
.050 |
5.59 |
4.74 |
4.35 |
4.12 |
3.97 |
3.87 |
3.79 |
3.73 |
3.68 |
|
|
.025 |
8.07 |
6.54 |
5.89 |
5.52 |
5.29 |
5.12 |
4.99 |
4.90 |
4.82 |
|
|
.010 |
12.25 |
9.55 |
8.45 |
7.85 |
7.46 |
7.19 |
6.99 |
6.84 |
6.72 |
|
|
.001 |
29.25 |
21.69 |
18.77 |
17.20 |
16.21 |
15.52 |
15.02 |
14.63 |
14.33 |
Table A1 F critical values
|
Degrees of freedom in the numerator |
||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
|
Degrees of freedom |
p |
10 |
12 |
15 |
20 |
25 |
30 |
40 |
50 |
60 |
120 |
1000 |
|
1 |
.100 |
60.19 |
60.71 |
61.22 |
61.74 |
62.05 |
62.26 |
62.53 |
62.69 |
62.79 |
63.06 |
63.30 |
|
.050 |
241.88 |
243.91 |
245.95 |
248.01 |
249.26 |
250.10 |
251.14 |
251.77 |
252.20 |
253.25 |
254.19 |
|
|
.025 |
968.63 |
976.71 |
984.87 |
993.10 |
998.08 |
1001.4 |
1005.6 |
1008.1 |
1009.8 |
1014.0 |
1017.7 |
|
|
.010 |
6055.8 |
6106.3 |
6157.3 |
6208.7 |
6239.8 |
6260.6 |
6286.8 |
6302.5 |
6313.0 |
6339.4 |
6362.7 |
|
|
.001 |
605621 |
610668 |
615764 |
620908 |
624017 |
626099 |
628712 |
630285 |
631337 |
633972 |
636301 |
|
|
2 |
.100 |
9.39 |
9.41 |
9.42 |
9.44 |
9.45 |
9.46 |
9.47 |
9.47 |
9.47 |
9.48 |
9.49 |
|
.050 |
19.40 |
19.41 |
19.43 |
19.45 |
19.46 |
19.46 |
19.47 |
19.48 |
19.48 |
19.49 |
19.49 |
|
|
.025 |
39.40 |
39.41 |
39.43 |
39.45 |
39.46 |
39.46 |
39.47 |
39.48 |
39.48 |
39.49 |
39.50 |
|
|
.010 |
99.40 |
99.42 |
99.43 |
99.45 |
99.46 |
99.47 |
99.47 |
99.48 |
99.48 |
99.49 |
99.50 |
|
|
.001 |
999.40 |
999.42 |
999.43 |
999.45 |
999.46 |
999.47 |
999.47 |
999.48 |
999.48 |
999.49 |
999.50 |
|
|
3 |
.100 |
5.23 |
5.22 |
5.20 |
5.18 |
5.17 |
5.17 |
5.16 |
5.15 |
5.15 |
5.14 |
5.13 |
|
.050 |
8.79 |
8.74 |
8.70 |
8.66 |
8.63 |
8.62 |
8.59 |
8.58 |
8.57 |
8.55 |
8.53 |
|
|
.025 |
14.42 |
14.34 |
14.25 |
14.17 |
14.12 |
14.08 |
14.04 |
14.01 |
13.99 |
13.95 |
13.91 |
|
|
.010 |
27.23 |
27.05 |
26.87 |
26.69 |
26.58 |
26.50 |
26.41 |
26.35 |
26.32 |
26.22 |
26.14 |
|
|
.001 |
129.25 |
128.32 |
127.37 |
126.42 |
125.84 |
125.45 |
124.96 |
124.66 |
124.47 |
123.97 |
123.53 |
|
|
4 |
.100 |
3.92 |
3.90 |
3.87 |
3.84 |
3.83 |
3.82 |
3.80 |
3.80 |
3.79 |
3.78 |
3.76 |
|
.050 |
5.96 |
5.91 |
5.86 |
5.80 |
5.77 |
5.75 |
5.72 |
5.70 |
5.69 |
5.66 |
5.63 |
|
|
.025 |
8.84 |
8.75 |
8.66 |
8.56 |
8.50 |
8.46 |
8.41 |
8.38 |
8.36 |
8.31 |
8.26 |
|
|
.010 |
14.55 |
14.37 |
14.20 |
14.02 |
13.91 |
13.84 |
13.75 |
13.69 |
13.65 |
13.56 |
13.47 |
|
|
.001 |
48.05 |
47.41 |
46.76 |
46.10 |
45.70 |
45.43 |
45.09 |
44.88 |
44.75 |
44.40 |
44.09 |
|
|
5 |
.100 |
3.30 |
3.27 |
3.24 |
3.21 |
3.19 |
3.17 |
3.16 |
3.15 |
3.14 |
3.12 |
3.11 |
|
.050 |
4.74 |
4.68 |
4.62 |
4.56 |
4.52 |
4.50 |
4.46 |
4.44 |
4.43 |
4.40 |
4.37 |
|
|
.025 |
6.62 |
6.52 |
6.43 |
6.33 |
6.27 |
6.23 |
6.18 |
6.14 |
6.12 |
6.07 |
6.02 |
|
|
.010 |
10.05 |
9.89 |
9.72 |
9.55 |
9.45 |
9.38 |
9.29 |
9.24 |
9.20 |
9.11 |
9.03 |
|
|
.001 |
26.92 |
26.42 |
25.91 |
25.39 |
25.08 |
24.87 |
24.60 |
24.44 |
24.33 |
24.06 |
23.82 |
|
|
6 |
.100 |
2.94 |
2.90 |
2.87 |
2.84 |
2.81 |
2.80 |
2.78 |
2.77 |
2.76 |
2.74 |
2.72 |
|
.050 |
4.06 |
4.00 |
3.94 |
3.87 |
3.83 |
3.81 |
3.77 |
3.75 |
3.74 |
3.70 |
3.67 |
|
|
.025 |
5.46 |
5.37 |
5.27 |
5.17 |
5.11 |
5.07 |
5.01 |
4.98 |
4.96 |
4.90 |
4.86 |
|
|
.010 |
7.87 |
7.72 |
7.56 |
7.40 |
7.30 |
7.23 |
7.14 |
7.09 |
7.06 |
6.97 |
6.89 |
|
|
.001 |
18.41 |
17.99 |
17.56 |
17.12 |
16.85 |
16.67 |
16.44 |
16.31 |
16.21 |
15.98 |
15.77 |
|
|
7 |
.100 |
2.70 |
2.67 |
2.63 |
2.59 |
2.57 |
2.56 |
2.54 |
2.52 |
2.51 |
2.49 |
2.47 |
|
.050 |
3.64 |
3.57 |
3.51 |
3.44 |
3.40 |
3.38 |
3.34 |
3.32 |
3.30 |
3.27 |
3.23 |
|
|
.025 |
4.76 |
4.67 |
4.57 |
4.47 |
4.40 |
4.36 |
4.31 |
4.28 |
4.25 |
4.20 |
4.15 |
|
|
.010 |
6.62 |
6.47 |
6.31 |
6.16 |
6.06 |
5.99 |
5.91 |
5.86 |
5.82 |
5.74 |
5.66 |
|
|
.001 |
14.08 |
13.71 |
13.32 |
12.93 |
12.69 |
12.53 |
12.33 |
12.20 |
12.12 |
11.91 |
11.72 |
Table A2 F critical values (continued)
OpenStax is part of Rice University, which is a 501(c)(3) nonprofit charitable corporation. As an educational initiative, it's our mission to improve educational access and learning for everyone. Through our partnerships with philanthropic foundations and our alliance with other educational resource companies, we're breaking down the most common barriers to learning. Because we believe that everyone should and can have access to knowledge.
Looking for more information about OpenStax? Visit our FAQ page.
We publish high-quality, peer-reviewed, openly licensed college textbooks that are absolutely free online and low cost in print. Seriously. We've also developed a low-cost, research-based courseware that gives students the tools they need to complete their course the first time around. Check out our current library of textbooks and explore OpenStax Tutor.
Our textbooks are being used in 60% of college and universities in the U.S. and over 100 countries, but there’s still room to grow. Alongside our efforts to expand our library, we have been taking steps toward improving student learning and advancing research in learning science. Access is just the beginning – we want to make educational tools better for a new generation of learners. Explore our map to see where we're headed.