Copyright and Open Licensing
Learning Objectives
By the end of this section, you will be able to:
- Define copyright and open licenses.
- Explain the purpose of copyright law in the United States.
An open license is a vital component of an open educational resource. Because of this, it is important that you understand how open licenses work within copyright law. This chapter will provide an overview of U.S. copyright law, fair use, and licensing to help you navigate this topic.
Attribution: “What is an open license and how does it work?” by The Council of Chief State School Officers is licensed CC BY 4.0.
Copyright Law
U.S. copyright law protects an author’s rights over their original creative works (e.g., research articles, books and manuscripts, artwork, video and audio recordings, musical compositions, architectural designs, video games, and unpublished creative works).[1] As soon as something is “fixed in a tangible medium of expression,” it is automatically protected by copyright. A resource is considered fixed when:
“its embodiment …by or under the authority of the author, is sufficiently permanent or stable to permit it to be perceived, reproduced, or otherwise communicated for a period of more than transitory duration.”[2]
In other words, an idea for a book you want to write is not protected by copyright, but the first draft of your manuscript is. Copyright protection ensures that the creator of a work has complete control over how their work is reproduced, distributed, performed, displayed, and adapted.[3] You do not need to register your resource with the U.S. Copyright Office for this to come into effect; it is automatic.
Public Domain
Works that are no longer protected by copyright are considered part of the public domain. Items in the public domain can be reused freely for any purpose by anyone, without giving attribution to the author or creator.[4]
Public domain works in the U.S. include works whose creator died 70 years prior, works published before 1924, or works dedicated to the public domain by their rightsholder. The Creative Commons organization created a legal tool called CC 0 to help creators dedicate their work to the public domain by releasing all rights to it.[5]
Licensing
The copyright status of a work determines what you can and cannot do with it.[6] As you begin to explore OER for use in your classroom, it is important that you understand your rights over the works you create and what it means to give those rights away.
Most copyrighted works are under full, “all rights reserved” copyright. This means that they cannot be reused in any way without permission from the work’s rightsholder (usually the creator). One way you can get permission to use someone else’s work is through a license, a statement or contract that allows you to perform, display, reproduce, or adapt a copyrighted work in the circumstances specified within the license. For example, the copyright holder for a popular book might sign a license to provide a movie studio with one-time rights to use their characters in a film.
What About Fair Use?
If an OER is available under a copyright license that restricts certain (re)uses, you can make a fair use assessment for reproducing or adapting that work. However, having explicit permission is preferable. We do not recommend using fully copyrighted works in OER projects without written permission from the work’s rightsholder.
Open Licenses
All OER are made available under some type of open license, a set of authorized permissions from the rightsholder of a work for any and all users. The most popular of these licenses are Creative Commons (CC) licenses, customizable copyright licenses that allow others to reuse, adapt, and re-publish content with few or no restrictions. CC licenses allow creators to explain in plain language how their works can be used by others.[7]
Creative Commons licenses will be explored in more detail in the next chapter. However, there are other open licenses that can be applied to educational materials. A few of these licenses are described below:
- GNU Free Documentation License: a copyleft license that grants the right to copy, redistribute, and modify a resource. It requires all copies and derivatives to be available under the same license. Copies may be sold commercially, but the original document or source code must be made available to the user as well.[8]
- Free Art License: The FAL “grants the right to freely copy, distribute, and transform creative works without infringing the author’s rights.” It is meant to be applied to artistic works, not documents.[9]
If you’re interested in learning more about open licenses, feel free to explore the Free Software Foundation’s information on copyleft licenses, some of the first licenses used for open content.[10]
Why Open Licenses?
 The adaptability and reusability of OER make it so that they are not just free to access, but also free for instructors who want to alter the materials for use in their course. For example, in the figure below an openly licensed image has been traced to make it more readable for users.
One of the tenets of OER laid out early on in the open education movement was the idea of the 5 Rs (originally the 4 Rs) introduced by David Wiley.[11] These five attributes lay out what it means for something to be truly “open,” as the term is used in open education. The 5 Rs include:
- Retain = the right to make, own, and control copies of the content.
- Reuse = the right to use the content in a wide range of ways
- Revise = the right to adapt, adjust, modify, or alter the content itself
- Remix = the right to combine the original or revised content with other open content to create something new
- Redistribute = the right to share copies of the original content, your revisions, or your remixes with others
While the “redistribute” and “revise” rights are the most commonly exercised rights in open education, each of the five plays an important role in the utility of an open educational resource. For example, without the right to “remix” materials, an instructor who teaches an interdisciplinary course would not be able to combine two disparate OER into a new resource that more closely fits their needs.
In the next chapter, we’ll look at Creative Commons licenses and how they facilitate the expression of the 5 Rs in unique ways.
- Copyright Law of the United States, 17 USC §102. https://www.copyright.gov/title17/92chap1.html#102 ↵
- Copyright Law of the United States, 17 USC §101. https://www.copyright.gov/title17/92chap1.html#101 ↵
- Copyright Law of the United States, 17 USC §106. https://www.copyright.gov/title17/92chap1.html#106 ↵
- Of course, standard citation procedures still apply for creative works in the public domain. You cannot claim another's work as your own. ↵
- Peters, Diane. "Improving Access to the Public Domain: The Public Domain Mark." Creative Commons Blog, October 11, 2010. https://creativecommons.org/2010/10/11/improving-access-to-the-public-domain-the-public-domain-mark/ ↵
- Attribution: "Licensing" and "Public Domain" were adapted in part from UH OER Training by Billy Meinke, licensed CC BY 4.0. ↵
- By assigning an open license to your work, you allow any user to exercise the rights allowed under the license, and cannot restrict reuse by certain individuals or parties without changing the license itself. ↵
- Free Sotware Foundation. "GNU Free Documentation License." 2008. https://www.gnu.org/licenses/fdl.html ↵
- Copyleft Attitude. "Free Art License 1.3." 2007. http://artlibre.org/licence/lal/en/ ↵
- Free Software Foundation. "What is Copyleft?." Accessed June 29, 2019. https://www.gnu.org/copyleft/copyleft.html ↵
- Wiley, David. "Defining the 'Open' in Open Content and Open Educational Resources." Open Content blog, 2014. http://opencontent.org/definition/ ↵
A copyright license which grants permission for all users to access, reuse, and redistribute a work with few or no restrictions.
A license permits users to certain rights over a copyrighted work. These can be exclusive (allowed for individual groups) or nonexclusive (allowed for all users). Licenses can be restricted by certain factors such as purpose, territory, duration, and media (Source: Findlaw.com).
A set of open licenses that allow creators to clearly mark how others can reuse their work through a set of four badge-like components: Attribution, Share-Alike, Non-Commercial, and No Derivatives.


Facts About the Chi-Square Distribution
Learning Outcomes
- Interpret the chi-square probability distribution as the sample size changes
The notation for the chi-square distribution is [latex]\displaystyle\chi\sim\chi^2_{df}[/latex], where df = degrees of freedom which depends on how chi-square is being used. (If you want to practice calculating chi-square probabilities then use [latex]\displaystyle{df}=n-1[/latex]. The degrees of freedom for the three major uses are each calculated differently.)
For the χ2 distribution, the population mean is μ = df and the population standard deviation is [latex]\displaystyle\sigma_{\chi^2}=\sqrt{2(df)}[/latex].
The random variable is shown as χ2, but may be any upper case letter.
The random variable for a chi-square distribution with k degrees of freedom is the sum of k independent, squared standard normal variables.
[latex]\displaystyle\chi^2=(Z_1)^2+(Z_2)^2+\dots+(Z_k)^2[/latex]
- The curve is nonsymmetrical and skewed to the right.
- There is a different chi-square curve for each df.
- The test statistic for any test is always greater than or equal to zero.
- When df > 90, the chi-square curve approximates the normal distribution. For [latex]\displaystyle{X}\sim\chi^2_{1,000}[/latex] the mean, [latex]\displaystyle\mu=df=1,000[/latex] and the standard deviation, [latex]\displaystyle\sigma=\sqrt{2(1,000)}[/latex]. Therefore, [latex]\displaystyle{X}\sim{N}(1,000, 44.7)[/latex], approximately.
- The mean, μ, is located just to the right of the peak.
References
Data from Parade Magazine.
“HIV/AIDS Epidemiology Santa Clara County.”Santa Clara County Public Health Department, May 2011.
Concept Review
The chi-square distribution is a useful tool for assessment in a series of problem categories. These problem categories include primarily (i) whether a data set fits a particular distribution, (ii) whether the distributions of two populations are the same, (iii) whether two events might be independent, and (iv) whether there is a different variability than expected within a population.
An important parameter in a chi-square distribution is the degrees of freedom df in a given problem. The random variable in the chi-square distribution is the sum of squares of df standard normal variables, which must be independent. The key characteristics of the chi-square distribution also depend directly on the degrees of freedom.
The chi-square distribution curve is skewed to the right, and its shape depends on the degrees of freedom df. For df > 90, the curve approximates the normal distribution. Test statistics based on the chi-square distribution are always greater than or equal to zero. Such application tests are almost always right-tailed tests.
Formula Review
[latex]\displaystyle\chi^2=(Z_1)^2+(Z_2)^2+\dots(Z_{df})^2[/latex] chi-square distribution random variable
[latex]\displaystyle\mu_{\chi^2}=df[/latex] chi-square distribution population mean
[latex]\displaystyle\sigma_{\chi^2}=\sqrt{2(df)}[/latex] Chi-Square distribution population standard deviation


DNA Replication in Eukaryotes
The essential steps of replication are the same as in prokaryotes. Starting replication is more complex in eukaryotes. At the origin of replication, a pre-replication complex is made with other initiator proteins. Other proteins are then recruited to start the replication process. The overall process is the same, although differently named enzymes fulfill the same function. For example, DNA pol III is used for the majority of replication in prokaryotes, while in eukaryotes the leading strand is continuously synthesized by the enzyme pol δ, the lagging strand is synthesized by pol ε. We are focusing on the enzymes used in prokaryotic replication, so don’t worry about these name differences.
Here are the important differences between prokaryotic and eukaryotic replication:
Table 1: Differences between prokaryotic and eukaryotic replication
| Property | Prokaryotes | Eukaryotes |
|---|---|---|
| Origin of replication | Single | Multiple |
| Rate of replication | 1000 nucleotides/s | 50 to 100 nucleotides/s |
| DNA polymerase types | 5 | 14 |
| DNA packaging | supercoiling | wound around histones |
| Telomerase | Not present | Present |
Origins and rate of replication
Eukaryotic genomes are much more complex and larger in size than prokaryotic genomes. The human genome has three billion base pairs per haploid set of chromosomes, and 6 billion base pairs are replicated during the S phase of the cell cycle. This means that there must be multiple origins of replication on the eukaryotic chromosome in order for all the DNA to be replicated in a timely manner; humans can have up to 100,000 origins of replication. The rate of replication is approximately 100 nucleotides per second, much slower than prokaryotic replication.
DNA polymerase types
The number of DNA polymerases in eukaryotes is much more than prokaryotes: 14 are known, of which five are known to have major roles during replication and have been well studied. They are known as pol α, pol β, pol γ, pol δ, and pol ε. I won’t ever ask you the names of these polymerases – learn the names of the prokaryotic polymerases.
DNA packaging
Eukaryotic DNA is wound around proteins known as histones to form structures called nucleosomes. The DNA must be made accessible in order for DNA replication to proceed. The chromatin (the complex between DNA and proteins) may undergo some chemical modifications, so that the DNA may be able to slide off the histones or otherwise be accessible to the enzymes of the DNA replication machinery. Prokaryotes do not packaged their DNA by wrapping it around histones.
Telomere replication
Unlike prokaryotic chromosomes, eukaryotic chromosomes are linear. As you’ve learned, the enzyme DNA pol can add nucleotides only in the 5′ to 3′ direction. In the leading strand, synthesis continues until the end of the chromosome is reached. On the lagging strand, DNA is synthesized in short stretches, each of which is initiated by a separate primer. When the replication fork reaches the end of the linear chromosome, there is no place for a primer to be made for the DNA fragment to be copied at the end of the chromosome. These ends thus remain unpaired, and over time these ends may get progressively shorter as cells continue to divide.
The ends of the linear chromosomes are known as telomeres, which have repetitive sequences that do not code for a particular gene. These telomeres protect the genes that are located on the chromosome from getting deleted as cells continue to divide. In humans, a six base pair sequence, TTAGGG, is repeated 100 to 1000 times. The discovery of the enzyme telomerase (Figure 1) helped in the understanding of how chromosome ends are maintained. The telomerase enzyme contains a catalytic part and a built-in RNA template. It attaches to the end of the chromosome, and complementary bases to the RNA template are added on the 3′ end of the DNA strand. Once the 3′ end of the lagging strand template is sufficiently elongated, DNA polymerase can add the nucleotides complementary to the ends of the chromosomes. Thus, the ends of the chromosomes are replicated.
Telomerase is typically active in germ cells and adult stem cells. It is not active in adult somatic cells. For her discovery of telomerase and its action, Elizabeth Blackburn (Figure 2) received the Nobel Prize for Medicine and Physiology in 2009.
Telomerase and Aging
Cells that undergo cell division continue to have their telomeres shortened because most somatic cells do not make telomerase. This essentially means that telomere shortening is associated with aging. With the advent of modern medicine, preventative health care, and healthier lifestyles, the human life span has increased, and there is an increasing demand for people to look younger and have a better quality of life as they grow older.
In 2010, scientists found that telomerase can reverse some age-related conditions in mice. This may have potential in regenerative medicine (Jaskelioff, 2011). Telomerase-deficient mice were used in these studies; these mice have tissue atrophy, stem cell depletion, organ system failure, and impaired tissue injury responses. Telomerase reactivation in these mice caused extension of telomeres, reduced DNA damage, reversed neurodegeneration, and improved the function of the testes, spleen, and intestines. Thus, telomere reactivation may have potential for treating age-related diseases in humans.
Cancer is characterized by uncontrolled cell division of abnormal cells. The cells accumulate mutations, proliferate uncontrollably, and can migrate to different parts of the body through a process called metastasis. Scientists have observed that cancerous cells have considerably shortened telomeres and that telomerase is active in these cells. Interestingly, only after the telomeres were shortened in the cancer cells did the telomerase become active. If the action of telomerase in these cells can be inhibited by drugs during cancer therapy, then the cancerous cells could potentially be stopped from further division.
References
Unless otherwise noted, images on this page are licensed under CC-BY 4.0 by OpenStax.
OpenStax, Concepts of Biology. OpenStax CNX. May 18, 2016 http://cnx.org/contents/s8Hh0oOc@9.10:2ousESf0@5/DNA-Replication
Jaskelioff et al., 2011 Telomerase reactivation reverses tissue degeneration in aged telomerase-deficient mice. Nature 469: 102-7.


8.6 Halfway Houses
David Carter; Michelle Holcomb; and Kate McLean
So-called “halfway houses” have long been used to control/house offenders. Dating back to the early 1800s in England and Ireland, halfway houses first began to appear in the U.S. around 1820, in Massachusetts. Initially, they were designed to help an offender “get back on their feet” and were funded by non-profit organizations like the Salvation Army. At present, halfway houses are typically used as “way stations” for offenders coming out of prisons but have also been used as an intermediate sanction for probationers. At their core, halfway houses are meant to be places where individuals can get back on their feet, “half-way” out of prison, while enjoying the support – and supervision – of trained personnel. In this way, halfway houses embody many of the contradictions of parole.
The International Halfway House Association breaks down halfway houses into four groups (for-profit, non-for-profit, state-funded, and federally-funded) along two dimensions (supportive and interventive). Halfway houses that serve only a minimal correctional function (functioning mainly as a residence for those reintegrating back into society) are generally labeled supportive. Interventive halfway houses, by contrast, typically offer multiple treatment modalities and may have up to 500 beds. Most halfway houses fall somewhere in the middle of these two poles.
Halfway houses can be privately owned or publicly funded. Each halfway house has a different mission, something that the residential facility focuses on, like vocational training, drug/alcohol treatment, individual or group counseling, education, job placement assistance, etc. Some halfway houses only accept males or females. Some facilities accept the dependent children of the offender in order to provide room and board along with group counseling.
Halfway House Success
Because of the great variability in halfway houses, researchers have found them difficult to assess. This is because it is difficult to make general statements about such a diverse group of facilities, and gathering a representative sample is difficult. Perhaps due to these research limitations, studies have found that halfway houses may increase recidivism, reduce recidivism, or have no effect on recidivism. When disaggregated by halfway house type, programs known to deploy evidence-based interventions have a stronger impact on recidivism. Systematic research on halfway houses may also be complicated by the different types of organizations that operate and fund them. Depending on whether they are public or private, halfway houses may operate according to very different models of care and have different staffing requirements and levels of resources. Much like more formal correctional institutions, halfway houses may provide much-needed treatment services – or function as chaotic “no-man’s lands” that are hardly safer than many prisons.


Spencer Fox Eccles School of Medicine
46 Research Reflection by Gabriel LaBrin
Gabriel LaBrin
Faculty Mentor: Valerie Vaughn (Internal Medicine, University of Utah)
Entering into this summer, I was hopeful to participate in an experience that would tie together research and populations vulnerable to being harmed by our health system. I was lucky enough to find Dr, Valerie Vaughn is for these research and assisting these exact populations, especially in the realm of antibiotic stewardship. This research experience has perfectly encapsulated the components I had hoped for, allowing me to conduct extensive research on whether or not there are psychological biases present in the development of clinical guidelines, which if evident could be very detrimental to all populations but especially vulnerable ones.
Spending time sorting through hundreds of recommendations, creating data acquisition tools through repeated trial and error, and learning how to interpret said data were not easy tasks. And yet, they have been vital in growing my capability for research in addition to my awareness of the need for well-conducted research with a goal to help improve care for all. After this research experience, I look forward to pursuing additional research opportunities throughout the rest of my academic career and into my professional career.
Through this experience, I was able to work as a full-time researcher in the Healthcare, Educators, Leaders & Innovators Complex (HELIX) building. Dr. Valerie Vaughn has been a phenomenal mentor. She is an incredibly hard worker who seamlessly intertwines passion and enthusiasm with academic and clinical work. She has inspired me this past summer to work hard and to allow my own passion and the passion of those around me to motivate and inspire me to help others.
I have been able to shadow Dr. Vaughn in numerous situations, attending weekly Hospital Internal Medicine Department meetings, weekly Internal Medicine Grand Rounds meetings, and quarterly Utah Quality Assurance Laboratory meetings, allowing me to observe her daily professional life as well as make connections with other medical professionals, whom I would continue to reach out to for insight throughout my research experience.
Working with passionate and caring researchers and professionals who strive to serve those around them in every way possible has given me hope and excitement to do the same now and in the future in my own interactions.


8 Byron, “Of His Mother’s Treatment of Him”
I thought, my dear Augusta, that your opinion of my meek mama would coincide with mine; her temper is so variable, and, when inflamed, so furious, that I dread our meeting; not but I dare say that I am troublesome enough, but I always endeavor to be as dutiful as possible. She is very strenuous, and so tormenting in her entreaties and commands, with regard to my reconciliation with that detestable Lord G. that I suppose she has a penchant for his Lordship; but I am confident that he does not return it, for he rather dislikes her than otherwise, at least as far as I can judge. But she has an excellent opinion of her personal attractions, sinks her age a good six years, avers that when I was born she was only eighteen, when you, my dear sister, know as well as I know that she was of age when she married my father, and that I was not born for three years afterward.  But I am now coming to what must shock you as well as it does me. When she has occasion to lecture me (not very seldom you will think no doubt) she does not do it in a manner that commands respect or in an impressive style.  But she flies into a fit of frenzy, upbraids me as if I was the most undutiful wretch in existence, rakes up the ashes of my father, abuses him, says I shall be a true Byrrone, which is the worst epithet she could invent.
Am I to call this woman mother? Because by nature’s law she has authority over me, am I to be trampled upon in this manner? Am I to be goaded with insult, loaded with obloquy, and suffer my feelings to be outraged on the most trivial occasions? I owe her respect as a son, but I renounce her as a friend. What an example does she show me. I hope in God I shall never follow it. I have not told you all, nor can I; I respect you as a female, nor altho I ought to confide in you as a sister, will I shock you with the repetition of the scenes which you may judge of by the sample I have given you, and which to all but you are buried in oblivion. Would they were so in my mind! I am afraid they never will. And can I, my dear sister, look up to this mother, with that respect, that affection I ought? Am I to be eternally subject to her caprice? I hope not—indeed, a few short years will emancipate me from the shackles I now wear, and then perhaps she will govern her passion better than at present.
You mistake me if you think I dislike Lord Carlisle. I respect him and might like him did I know him better. For him too my mother has an antipathy, why I know not. I am afraid he will be of little use to me in separating me from her, which she would oppose with all her might. But I dare say he will assist me if he would, so I take the will for the deed, and am obliged to him in exactly the same manner as if he succeeded in his efforts. I am in great hopes that at Christmas I will be with Mr. Hanson during the vacation. I shall do all I can to avoid a visit to my mother wherever she is. It is the first duty of a parent to impress precepts of obedience in their children, but her method is so violent, so capricious, that the patience of Job, the versatility of a member of the House of Commons would not support it. I revere Dr. Drury much more than I do her, yet he is never violent, never outrageous: I dread offending him, not however through fear, but the respect I bear him makes me unhappy when I am under his displeasure. My mother’s precepts never convey instruction, never fix upon my mind; to be sure they are calculated to inculcate obedience, so are chains and tortures, but tho they may restrain for a time the mind revolts from such treatment.
Not that Mrs. Byron ever injures my sacred person. I am rather too old for that, but her words are of that rough texture which offend more than personal ill usage. “A talkative woman is like an adder’s tongue,” so says one of the prophets, but which I can’t tell, and very likely you don’t wish to know, but he was a true one whoever he was.
The postage of your letters, my dear Augusta, don’t fall upon me; but if they did it would make no difference, for I am generally in cash and should think the trifle I paid for your epistles the best laid out I ever spent in my life. Write soon. Remember me to Lord Carlisle, and believe me, I am ever
Your affectionate brother and friend,
Byrone.[30]


Main Body
Leadership Guide to Conflict and Conflict Management
Fadi Smiley
“A part of effective leadership is caring for and supporting one another, even when there is conflict or a difference of opinion.” -Ty Howard
Introduction
Conflict may occur between people or within groups in all kinds of situations. Due to the wide range of differences among people, the lack of conflict may signal the absence of effective interaction. Conflict should not be considered good or bad, rather it may be viewed as a necessity to help build meaningful relationships between people and groups. The means and how the conflict is handled will determine whether it is productive or devastating. Conflict has a potential to create positive opportunities and advancement towards a common goal, however, conflict can also devastate relationships and lead to negative outcomes ((Kazimoto, 2013; Fisher, 2000; Evans, 2013).
Today’s healthcare leaders are taught to lead change, development, and transformation in organizations. Leadership may be described as the ability to emphasize the pursuit of goals and motivate others to pursue them as well. Northouse states that leadership is a process whereby an individual influences a group of individuals to achieve a common goal (Northouse, 2016). Others characterize leadership as the ability to inspire trust, build relationships, encourage followers.
An under-reported aspect that is not commonly discussed among leadership qualities is the ability to handle conflict (Guttman, 2004). Guttman explains that there might be two reasons as to why there is little recognition of conflict management in leaders. One is called rationalistic fallacy, and Guttman explains that most of the literature available focuses on arming leaders with all necessary leadership concepts and success will just follow, almost as if it is assumed that leaders will automatically know how to manage conflict. Secondly, Guttman explains that leaders may have a fatalistic attitude towards conflict. Leaders may look at conflict as situation that will never be resolved, so why bother addressing it? We should focus on what can be addressed and changed (Guttman, 2004).
Conflict management is a skill that leaders must be able to employ when needed to help foster a productive working environment (Guttman, 2004). There is a realization that conflict management should be a skill that leaders need to give priority to learning and mastering (Kazimoto, 2013). The inability of a leader to deal with conflict will not only lead to negative outcomes but may also undermine the credibility of the leader (Kazimoto, 2013). Whereas if a leader is able to establish an atmosphere of cooperation and foster teamwork, making it clear that this is his/her value system, there is a likelihood that this value system will be adopted by the entire organization (Guttman, 2004). Therefore, it is very important that we discuss and address conflict management as a leadership skill.
This chapter will discuss the definition of conflict and its sources, describe conflict management and resolution, and discuss a guide for leaders to use to help them effectively manage and resolve conflict. We discuss the different types of conflict that can exist and describe the different conflict management modes that can be used to address them. Lastly, we will analyze the relationship between leadership and conflict management through a literature review. By reading this chapter, I hope that readers will understand conflict, the role it plays within teams and organizations, and the importance of developing conflict management skills for leaders.
Defining Conflict
What is conflict? The answer to this question varies, depending on the source. The Webster Dictionary defines conflict as “the competitive or opposing action of incompatibles: an antagonistic state or action.” For some, the definition of conflict may involve war, military fight, or political dispute. For others, conflict involves a disagreement that arises when two or more people or parties pursue a common goal. Conflict means different things to different people, making it very difficult to come up with a universal or true definition. To complicate this even further, when one party may feel that they are in a conflict situation, the other party may think that they are just in a simple discussion about differing opinions (Fisher, 2000; Evans, 2013; Conflict, 2011).
To fully understand conflict and how to manage it, we first need to establish a definition that will allow us to effectively discuss conflict management and its use by today’s leaders. Conflict can be described as a disagreement among two entities that may be portrayed by antagonism or hostility. This is usually fueled by the opposition of one party to another to reach an objective that is different from the other, even though both parties are working towards a common goal (Fisher, 2000; Evans, 2013). To help us better understand what conflict is, we need to analyze its possible sources. According to American psychologist Daniel Katz, conflict may arise from 3 different sources: economic, value, and power. (Evans, 2013)
- Economic Conflict involves competing motives to attain scarce resources. This type of conflict typically occurs when behavior and emotions of each party are aimed at increasing their own gain. Each party involved may come into conflict as a result of them trying to attain the most of these resources. An example of this is when union and management conflict on how to divide and distribute company funds (Fisher, 2000; Evans, 2013).
- Value Conflict involves incompatibility in the ways of life. This type of conflict includes the different preferences and ideologies that people may have as their principles. This type of conflict is very difficult to resolve because the differences are belief-based and not fact-based. An example of this is demonstrated in international war in which each side asserts its own set of beliefs (Fisher, 2000; Evans, 2013).
- Power Conflict occurs when each party tries to exert and maintain its maximum influence in the relationship and social setting. For one party to have influence over the other, one party must be stronger (in terms of influence) than the other. This will result in a power struggle that may end in winning, losing, or a deadlock with continuous tension between both parties. This type of conflict may occur between individuals, groups, or nations. This conflict will come into play when one party chooses to take a power approach to the relationship. The key word here is “chooses.” The power conflict is a choice that is made by one party to exert its influence on the other. It is also important to note that power may enter all types of conflict since the parties are trying to control each other (Fisher, 2000; Evans, 2013).
According to Ana Shetach, an organizational consultant in team process and development, conflict can be a result from every aspect such as attitude, race, gender, looks, education, opinions, feelings, religion and cultures. Conflict may also arise from differences in values, affiliations, roles, positions, and status. Even though it seems that there is a vast array of sources for conflict, most conflict is not of a pure type and typically is a mixture of several sources (Shetach, 2012).
Conflict is an inevitable part of life and occurs naturally during our daily activities. There will always be differences of opinions or disagreements between individuals and/or groups. Conflict is a basic part of the human experience and can influence our actions or decisions in one way or another. It should not be viewed as an action that always results in negative outcomes but instead as an opportunity for learning and growth which may lead to positive outcomes. We can reach positive outcomes through effective conflict management and resolution, which will be discussed in more detail later in the chapter (Evans, 2013).
Since conflict can result in emotions that can make a situation uncomfortable, it is often avoided. Feelings such as guilt, anger, anxiety, and fear can be a direct result of conflict, which can cause individuals to avoid it all together. Conflict can be a good thing and avoiding it to preserve a false impression of harmony can cause even more damage (Loehr, 2017a). If we analyze the CPP Global Human Capital Report, we see evidence that conflict can lead to positive outcomes within the workplace environment. This research project asked 5000 individuals about their experiences with conflict in the workplace environment. They reported, that as a result of conflict:
- 41% of respondents had better understanding of others
- 33% of respondents had improved working relationships
- 29% of respondents found a better solution to a problem
- 21% of respondents saw higher performance in the team
- 18% of respondents felt increased motivation (CPP Global Human Capital Report, 2008)
Based on this report, we can conclude that conflict can lead to positive outcomes and increased productivity, depending on the conflict itself (Loehr, 2017a). Approximately 76% of the respondents reported that conflict resulted in some type of positive outcome. This speaks volumes to the ideology that conflict within the workplace is something that should be welcomed and not avoided (CPP Global Human Capital Report, 2008).
Conflict can occur in various ways in the human experience, whether it is within one-self between differing ideas or between people. Even though this chapter will focus on the conflict at the social level, it is important that we review all the different levels of conflict that may exist. The levels of conflict that we will discuss include interpersonal, intrapersonal, intergroup, and intragroup conflict (Loehr, 2017a; Fisher, 2000; Evans, 2013).
Levels of Conflict
- Interpersonal Conflict. This level of conflict occurs when two individuals have differing goals or approaches in their relationship. Each individual has their own type of personality, and because of this, there will always be differences in choices and opinions. Compromise is necessary for managing this type of conflict and can eventually help lead to personal growth and developing relationships with others. If interpersonal conflict is not addressed, it can become destructive to the point where a mediator (leader) may be needed (Loehr, 2017a; Fisher, 2000; Evans, 2013).
- Intrapersonal Conflict. This level of conflict occurs within an individual and takes place within the person’s mind. This is a physiological type of conflict that can involve thoughts and emotions, desires, values, and principles. This type of conflict can be difficult to resolve if the individual has trouble interpreting their own inner battles. It may lead to symptoms that can become physically apparent, such as anxiety, restlessness, or even depression. This level of conflict can create other levels of conflict if the individual is unable to come to a resolution on their own. An individual who is unable to come to terms on their own inner conflicts may allow this to affect their relationships with other individuals and therefore create interpersonal conflict. Typically, it is best for an individual dealing with intrapersonal conflict to communicate with others who may help them resolve their conflict and help relieve them of the situation (Loehr, 2017a; Fisher, 2000; Evans, 2013).
- Intergroup Conflict. This level of conflict occurs when two different groups or teams within the same organization have a disagreement. This may be a result of competition for resources, differences in goals or interests, or even threats to group identity. This type of conflict can be very destructive and escalate very quickly if not resolved effectively. This can ultimately lead to high costs for the organization. On the other hand, intergroup conflict can lead to remarkable progress towards a positive outcome for the organization if it is managed appropriately (Loehr, 2017a; Fisher, 2000; Evans, 2013).
- Intragroup Conflict. This level of conflict can occur between two individuals who are within the same group or team. Similar to interpersonal conflict, disagreements between team members typically are a result of different personalities. Within a team, conflict can be very beneficial as it can lead to progress to accomplishing team objectives and goals. However, if intragroup conflict is not managed correctly, it can disrupt the harmony of the entire team and result in slowed productivity (Loehr, 2017a; Fisher, 2000; Evans, 2013).
Regardless of the level of conflict that takes place, there are several methods that can be employed to help manage conflicts. And with the seemingly infinite triggers for conflict, management of conflict is a constant challenge for leaders. To help address this, we will next discuss what conflict management is and then later examine the role of leadership in conflict management and resolution.
Conflict Management
Conflict Management may be defined as the process of reducing negative outcomes of conflict while increasing the positive. Effectively managed conflicts can lead to a resolution that will result in positive outcomes and productivity for the team and/or organization (Loehr, 2017b; Evans, 2013). Leaders need to be able to manage conflict when it occurs, and their ability to manage them is critical to the success of the individuals and/or teams involved (Guttman, 2004). There are several models available for leaders to use when determining their conflict management behavior. So where do leaders begin when they want to recognize their own conflict management styles? In this section, we will cover a popular method of conflict management styles, the Thomas-Kilmann Conflict Mode Instrument, that will help us answer this question (Loehr, 2017b; Evans, 2013).
The Thomas-Kilmann Conflict Mode Instrument (TKI)
The Thomas-Kilmann Conflict Mode Instrument is an assessment tool that helps measure an individual’s behavior in conflict situations. The assessment takes less than 15 minutes to complete and provides feedback to an individual about how effectively they can use five different conflict-handling modes. TKI helps leaders understand how individual or team dynamics are affected by each of the modes, as well as helping leaders decide on which mode to employ in different conflict situations (Kilmann & Thomas, n.d.).
The TKI is based on two dimensions of behavior that help characterize the five different conflict-handling modes. The first dimension is assertiveness, and this describes the extent to which a person will try to fulfill their own concerns. The second is cooperativeness, and this describes the extent to which a person will try to fulfil others’ concerns. The five conflict-handling modes fall within a scale of assertiveness and cooperativeness as shown in the figure below. They include: avoiding, accommodating, competing, collaborating, and compromising (Loehr, 2017b; Kilmann & Thomas, n.d.).
(source:killmanndiagnostics.com)
The TKI Five Conflict-Handling Modes (Avoiding, Accomodating, Competing, Collaborating, and Compromising)
Avoiding
This mode is low assertiveness and low cooperative. The leader withdraws from the conflict, and therefore no one wins. They do not pursue their own concerns nor the concerns of others. The leader may deal with the conflict in a passive attitude in hopes that the situation just “resolves itself.” In many cases, avoiding conflict may be effective and beneficial, but on the other hand, it prevents the matter from being resolved and can lead to larger issues. Situations when this mode is useful include: when emotions are elevated and everyone involved needs time to calm down so that productive discussions can take place, the issue is of low importance, the team is able to resolve the conflict without participation from leadership, there are more important matters that need to be addressed, and the benefit of avoiding the conflict outweighs the benefit of addressing it. This mode should not be used when the conflict needs to be resolved in a timely manner and when the reason for ignoring the conflict is just that (Loehr, 2017a; Mediate.com; Kilmann & Thomas, n.d.).
Accommodating
This mode is low assertiveness and high cooperation. The leader ignores their own concerns in order to fulfill the concerns of others. They are willing to sacrifice their own needs to “keep the peace” within the team. Therefore, the leader loses and the other person or party wins. This mode can be effective, as it can yield an immediate solution to the issue but may also reveal the leader as a “doormat” who will accommodate to anyone who causes conflict. Situations when this mode is useful include: when an individual realizes they are wrong and accepts a better solution, when the issue is more important to the other person or party which can be seen as a good gesture and builds social credits for future use, when damage may result if the leader continues to push their own agenda, when a leader wants to allow the team to develop and learn from their own mistakes, and when harmony needs to be maintained to avoid trouble within the team. This mode should not be used when the outcome is critical to the success of the team and when safety is an absolute necessity to the resolution of the conflict (Loehr, 2017b; Mediate.com; Kilmann & Thomas, n.d.).
Competing
This mode is high assertiveness and low cooperation. The leader fulfills their own concerns at the expense of others. The leader uses any appropriate power they have to win the conflict. This is a powerful and effective conflict-handling mode and can be appropriate and necessary in certain situations. The misuse of this mode can lead to new conflict; therefore, leaders who use this conflict-handling mode need to be mindful of this possibility so that they are able to reach a productive resolution. Situations when this mode is useful include: an immediate decision is needed, an outcome is critical and cannot be compromised, strong leadership needs to be demonstrated, unpopular actions are needed, when company or organizational welfare is at stake, and when self-interests need to be protected. This mode should be avoided when: relationships are strained and may lead to retaliation, the outcome is not very important to the leader, it may result in weakened support and commitment from followers, and when the leader is not very knowledgeable of the situation (Loehr, 2017b; Mediate.com; Kilmann & Thomas, n.d.).
Collaborating
This mode is high assertiveness and high cooperation. In this mode both individuals or teams win the conflict. The leader works with the team to ensure that a resolution is met that fulfills both of their concerns. This mode will require a lot of time, energy and resources to identify the underlying needs of each party. This mode is often described as “putting an idea on top of an idea on top of an idea” to help develop the best resolution to a conflict that will satisfy all parties involved. The best resolution in this mode is typically a solution to the conflict that would not have been produced by a single individual. Many leaders encourage collaboration because not only can it lead to positive outcomes, but more importantly it can result in stronger team structure and creativity. Situations when this mode is useful include: the concerns of parties involved are too important to be compromised, to identify and resolve feelings that have been interfering with team dynamics, improve team structure and commitment, to merge ideas from individuals with different viewpoints on a situation, and when the objective is to learn. This mode should be avoided in situations where time, energy and resources are limited, a quick and vital decision needs to be made, and the conflict itself is not worth the time and effort (Loehr, 2017b; Mediate.com; Kilmann & Thomas, n.d.).
Compromising
This mode is moderate assertiveness and moderate cooperative. It is often described as “giving up more than one would want” to allow for each individual to have their concerns partially fulfilled. This can be viewed as a situation where neither person wins or losses, but rather as an acceptable solution that is reached by either splitting the difference between the two positions, trading concerns, or seeking a middle ground. Leaders who use this conflict-handling mode may be able to produce acceptable outcomes but may put themselves in a situation where team members will take advantage of the them. This can be a result of the team knowing that their leader will compromise during negotiations. Compromising can also lead to a less optimal outcome because less effort is needed to use this mode. Situations when this mode is effective include: a temporary and/or quick decision to a complex issue is needed, the welfare of the organization will benefit from the compromise of both parties, both parties are of equal power and rank, when other modes of conflict-handling are not working, and when the goals are moderately important and not worth the time and effort. This mode should be avoided when partial satisfaction of each party’s concerns may lead to propagation of the issue or when a leader recognizes that their team is taking advantage of their compromising style (Loehr, 2017b; Mediate.com; Kilmann & Thomas, n.d.).
Personal Reflections on the Thomas-Kilmann Conflict Mode Instrument
I chose the TKC Instrument because I felt it encompassed all aspects of conflict behavior and does a thorough job of explaining those behaviors. When compared to other models, the TKI model is more specific in the description of conflict behaviors as well. The TKI model has been around for well over 30 years, and I feel it does a very good job of breaking down a complex theory of conflict styles into a format that is easily understood and can be used by anyone.
Leaders should be capable of using all five conflict-handling modes and should not limit themselves to using only one mode during times of conflict (Loehr, 2017b). Leaders must be able to adapt to different conflict situations and recognize which type of conflict-handling mode is best to employ given the conflict at hand (Mediate.com). The use of these modes can result in positive or negative resolutions and it is imperative that today’s leaders understand how to effectively employ them (Loehr, 2017b; Mediate.com; Kilmann & Thomas, n.d.).
Leadership and Conflict Management
The leader’s role in managing conflict can have a significant impact on how they are resolved within the workplace or organization. Leaders spend about 24% of their time resolving conflicts, however the process to approaching conflict management relates to a great extent to their leadership style (Guttman, 2004). Leaders who use conflict management skills can provide guidance and direction towards conflict resolution. A common trait of leaders is they are able to build teams that work well together and help to set the tone for the organization. They must be able to facilitate the resolution of conflicts through effective conflict management (Guttman, 2004; Doucet, Poitras & Chenevert, 2009).
Leaders exhibit a variety of characteristics and traits that allow them to be great leaders, but does it help them when it comes to conflict management? I believe that it does. These same traits can help leaders dealt with conflict. The ability to recognize one’s own leadership style will ultimately help describe how a leader handles conflict. Peter Northouse states that “it is up to the leader to assess what action, if any, is needed and then intervene with the specific leadership function to meet the demand of the situation.” To be an effective leader, one needs to respond with the action that is required of the situation” (Northouse, 2016). I feel this demonstrates that the job of a leader is to analyze a conflict and facilitate the situation to produce a resolution that can be positive and productive. Northouse reassures us that any leader can draw on his/her leadership skills to employ appropriate conflict management strategies (Northouse, 2016).
A study conducted by Zhang et al. analyzed the relationship between transformational leadership and conflict management. Zhang et al. looked at how transformational leadership affects team coordination and performance through conflict management. What they found was that transformational leaders who used conflict management methods were able influence their teams to establish stronger identities, discuss their disagreements and frustrations outwardly, and work out solutions that benefited the team (Zhang, Cao, Tjosvold, 2011). I feel that this study helps to confirm that leaders must be able to possess conflict management skills to effectively run a productive team and organization. This study also shows us that there may be a possibility that certain leadership styles are more effective at conflict management. This is not definite but hopefully there will be more studies done to determine this.
Earlier in the chapter, we discussed the different types of conflict-handling modes that leaders may possess, but it is also necessary to briefly discuss the leadership skills and behaviors needed to effectively manage conflicts. Leadership skills needed to be effective at conflict management can be categorized to show which skills match up with five of the TKI conflict-handling modes. The avoiding mode requires leadership skills such as: to be able to withdraw from a conflict or sidestep issues, have the ability to leave issues unresolved, and to have a sense of timing. The accommodating mode requires skills such as: being able to obey orders, set your own concerns aside, selflessness, and the ability to yield for the greater good. The competing mode requires skills such as: standing your ground, debating, using influence, stating your position clearly, and stressing your feelings. The collaborating mode requires skills such as: active listening, identifying concerns, analyzing input, and confrontation. The compromising mode requires skills such as: negotiating and finding the middle ground, making concessions, and assessing value (Kilmann & Thomas, n.d.; Understanding Conflict and Conflict Management, n.d.). Behaviors that allow leaders to be effective at conflict management include (Guttman, 2004):
- Be Candid. Leaders cannot hesitate to put issues on the table to be discussed
- Be Receptive. Leaders need to make sure that team members understand that it is ok for conflict to exist and that everyone’s opinion will be discussed
- Depersonalize. Leaders must be able to remove personal feeling from the mix and view conflict as a team issue
- Learn to Listen. Leaders must listen carefully and make sure that they provide feedback as well
- Be clear. Leaders need to make sure that all team members understand how decisions will be made to resolve the conflict
- Out-law Triangulation. Leaders must prevent team members from “ganging-up” on others that they may disagree with
- Be Accountable. Leaders must make sure that they follow through on their actions but also hold others to their actions as well
- Recognize and Reward. Leaders must be able to recognize successful conflict management and then reward it
Effective leaders know how to bring conflict situations out into the open so that all parties involved can begin to work towards a resolution that will benefit everyone. They manage conflicts in way that it is seen as an opportunity to build productive relationships (Guttman, 2004; Kazimoto, 2013).
Conclusion
The means in which conflict is managed will determine whether the outcome will be positive and productive, or negative and destructive. Leaders are taught to lead change, development, and transformation in organizations. One way leaders can accomplish their goals is through effective conflict management. Conflict can be described as a disagreement among two parties that is usually portrayed as antagonism or hostility. Conflict can arise from three different sources: economic, value, and power conflicts. Conflict can also occur at different levels of the human experience, which include: interpersonal, intrapersonal, intergroup, and intragroup conflict levels.
Conflict management is the process of reducing negative outcomes while increasing the positive. Leaders must be able to utilize conflict management skills to provide direction and guidance towards a resolution. Leaders can use the Thomas-Kilmann Conflict Mode Instrument to help them recognize and understand their own conflict handling modes. The TKI model also helps leaders understand which conflict handling modes are most appropriate for each particular conflict situation.
Zhang et al. shows us that transformational leaders are able to effectively influence their teams to workout solutions to their conflicts. This shows us that transformational leaders are able to utilize conflict management effectively. Peter Northouse also tells us that an effective leader needs to be able to respond to a situation with an action that is required of that particular situation. Therefore, leaders must be able draw on their leadership skills to effectively employ conflict management strategies.
Conflict is certain to occur regardless of the setting and individuals involved. For conflict to result in a positive outcome, leaders and teams must recognize that conflict not only exists but is a necessity. Understanding conflict allows leaders to manage it more effectively and can provide a path to accomplishing positive outcomes. Conflict management can be an active force that will allow leaders grow healthy relationships within their organizations which can ultimately result in effective productivity.
Conflict management must be a part of a leader’s toolbox and be deployed when conflict arises within a team or organization. If conflict is not addressed in a timely manner, it can not only affect the moral of the team/organization but can create larger issues later. Once this happens it may be more difficult to resolve then it would have been if the conflict was addressed immediately and effectively. Leaders must be able to recognize that conflict can cause negative issues within their team or organization. If they are able to pull on their leadership skills and recognize which conflict-handling mode is required for each situation, they can create an opportunity to improve team structure and dynamics, and ultimately achieve their goal of changing, developing, and transforming organizations.


Examining Sample Assignment 4: Reflection
We’ve looked at writing assignments that emphasize different aspects of Bloom’s Taxonomy, with one exception: metacognitive knowledge. Yet, I have said that metacognition shows up frequently in certain types of assignments.
This assignment is a final reflection for a portfolio in my first-year writing class. Students submit two revised final papers, plus a reflection. I’m only giving you the instructions for the reflection part of the assignment.
| Assignment | Write a reflection that introduces your portfolio and discusses your work in this course for the term. |
| Audience | Your professor or any other writing teacher interested in your best work and your own understanding of your writing |
| Length | 750-1000 words |
Your reflection introduces and explains both of the papers in your portfolio and your writing in general. That is, this document is both a self-assessment and a reflection on your writing this semester. It should include in some form the following information:
- The larger part should focus on the specific essays you have selected and how those pieces demonstrate that you have met the goals for the course. Consider questions like the following:
- How do these essays demonstrate that you have met the course goals?
- What are the strengths and weaknesses of the essays?
- To what extent are you satisfied with these essays as representative of your writing ability?
- What have you learned from working with these essays?
- In addition, you should discuss your writing process generally and the progress you have made this semester:
- How has your writing changed, and where do you see those changes?
- What new ways of writing have you tried, and what did you learn from those attempts?
- Have you learned anything in particular about your writing process?
I strongly recommend that you refer directly and substantively to the course goals in your reflection, as these are the concepts that have been driving your work (and my planning) for the term.
You may feel free to cover issues and ideas beyond these questions, and you should not feel the need to answer all of them (although you must address both major parts above). You should choose the elements that are most relevant to your writing for the semester and deal specifically with those. In addition, you should feel free to talk about problems and successes common to both of the essays you have chosen to include.
Examining the Verbs in Key Sentences
Here are the sentences that let us know what we are supposed to be doing in this assignment:
Write a reflection that introduces your portfolio and discusses your work in this course for the term.
The larger part should focus on the specific essays you have selected and how those pieces demonstrate that you have met the goals for the course.
In addition, you should discuss your writing process generally and the progress you have made this semester….
This assignment has two other elements that make it different from the ones we’ve examined in previous chapters. First, it includes a lot of questions guiding us through thinking about the course and our work, but as the assignment says, we do not have to answer all of the questions. Second, and related to the first point, there are a number of statements that indicate we have a range of choices:
- Choose which questions you’d like to deal with, including none at all.
- Add aspects of the course and your writing that aren’t suggested by the questions if you’d like.
- Feel free to talk about both successes and issues in your writing.
I’m not marking the verbs here, but these elements help us understand the task.
Applying Bloom
What does this assignment ask us to do? We actually have a sentence that names the task in two parts:
…[T]his document is both a self-assessment and a reflection on your writing this semester.
The verb here (“is”) doesn’t help us, but this sentence identifies the two parts of the work: assess and reflect if we convert those nouns into verbs.
Before jumping into the next section, take what you know about the task in the sample assignment and see which types of knowledge and which cognitive processes you believe the assignment is looking for.
After you read the rest of this chapter, decide whether or not you agree with my analysis.
Kinds of Cognitive Processes
Let’s look at those verbs to understand the cognitive processes involved. We are being asked to reflect as we introduce our work and discuss it. We should focus on the essays we have selected for our portfolio and how those demonstrate that we have met the course goals. We should discuss our process.
Remembering is more important here than in the other assignments we’ve examined. Not everything that we reflect on will exist in some form that we can look up. And if we look at the grid of Bloom’s Taxonomy, we see that at the intersection of remember and metacognitive knowledge, we are doing tasks like identifying strategies—in this case, strategies that have worked for us, or not. Still, remembering isn’t central to this assignment.
How much we demonstrate our understanding and how much we analyze depends on the questions that we choose to focus on in our reflection. These aren’t explicitly part of the assignment, though. Similarly, creating is a part of the assignment primarily through the freedom we have to choose what we would like to present and how we do that.
Applying, however, is part of this assignment because we are asked to demonstrate how we have met the course goals using our portfolio essays. Whenever we are asked to use one thing to help explain another, we’re in the realm of application.
Evaluating is a primary task in this assignment—and in many reflective assignments. Reflections ask you to judge, for example, your progress or your work or your learning. It is not enough to describe your work; you must also assess it.
Kinds of Knowledge
And the key sentences, as always, also point to the kinds of knowledge we are supposed to demonstrate in the assignment.
Any time a professor is asking you to reflect, they are asking for metacognitive knowledge. You might reference specific factual elements or describe key concepts or processes, but in a reflection, these types of knowledge are all in the service of metacognition. What have you learned? How do you know that you have learned that? How does this learning connect with other ideas and information that you have? Why are these ideas and connections important? These are metacognitive questions.
In this assignment, we will also use factual knowledge (for example, about the course goals or our own writing), and when we talk about our writing process, we would use our procedural knowledge of what writing processes can look like. We might even use conceptual knowledge as we explain how we learned about characteristics of a good thesis statement or a strong body paragraph.
But these are all secondary. Metacognition is central here.
Putting It Together
This assignment asks us to evaluate our work according to the course goals and according to our own sense of what has been most important to our learning. These rely heavily on metacognitive knowledge and on the tasks of applying and evaluating.
Thus, the intellectual work of this assignment is to demonstrate our own awareness of our performance in the course, specifically examined in relation to an external system (the course goals) and an internal one (our own sense of our accomplishments).
A kind of knowledge that is self-aware and that focuses on how a thinker thinks. It is sometimes called "thinking about thinking."


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Book Contents Navigation
Acknowledgment of Country
Accessibility Information
Acknowledgments
About the Authors
Introduction
Section 1.1: Data and Types of Statistical Variables
Section 1.2: Descriptive Statistics
Section 1.3: Missing Data
Section 1.4: Checking Values
Section 1.5: Normality
Section 1.6: Outliers
Section 1.7: Chapter One Self-Test
Section 2.1: p Values
Section 2.2: Significance
Section 2.3: Confidence Intervals
Section 2.4: Effect Sizes
Section 2.5: Statistical Power
Section 2.6: Chapter Two Self-Test
Section 3.1: Looking at Group Differences
Section 3.2: Between Versus Within Groups Analysis
Section 3.3: Independent T-test Assumptions, Interpretation, and Write Up
Section 3.4: Paired T-test Assumptions, Interpretation, and Write Up
Section 3.5: Chapter Three Self-Test
Section 4.1: Examining Relationships
Section 4.2: Correlation Assumptions, Interpretation, and Write Up
Section 4.3: Chapter Four Self-Test
Section 5.1: The Linear Model
Section 5.2: Simple Regression Assumptions, Interpretation, and Write Up
Section 5.3: Multiple Regression Explanation, Assumptions, Interpretation, and Write Up
Section 5.4: Hierarchical Regression Explanation, Assumptions, Interpretation, and Write Up
Section 5.5: Chapter Five Self-Test
Section 6.1: Between Versus Within Group Analyses
Section 6.2: One-Way ANOVA Assumptions, Interpretation, and Write Up
Section 6.3 Repeated Measures ANOVA Assumptions, Interpretation, and Write Up
Section 6.4: Chapter Six Self-Test
Section 7.1: Mediation and Moderation Models
Section 7.2: Mediation Assumptions, The PROCESS Macro, Interpretation, and Write Up
Section 7.3: Moderation Models, Assumptions, Interpretation, and Write Up
Section 7.4: Chapter Seven Self-Test
Section 8.1: Factor Analysis Definitions
Section 8.2: EFA versus CFA
Section 8.3: EFA Steps with Factor Extraction
Section 8.4: EFA Determining the Number of Factors
Section 8.5: EFA Interpretation
Section 8.6: EFA Write Up
Section 8.7: Scale Reliability
Section 8.8: Chapter Eight Self-Test
Section 9.1: Nonparametric Definitions
Section 9.2: Choosing Appropriate Tests
Section 9.3: Comparing Two Independent Conditions: The Mann– Whitney U Test
Section 9.4: Comparing Two Dependent Conditions or Paired Samples – Wilcoxon Sign-Rank Test
Section 9.5: Differences Between Several Independent Groups: The Kruskal–Wallis Test
Section 9.6: Chapter Nine Self-Test
References
Please review your learning by completing the below self-test:
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Statistics for Research Students Copyright © 2022 by University of Southern Queensland is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.


Chapter 10 Rotational Motion and Angular Momentum
10.6 Collisions of Extended Bodies in Two Dimensions
Summary
- Observe collisions of extended bodies in two dimensions.
- Examine collision at the point of percussion.
Bowling pins are sent flying and spinning when hit by a bowling ball—angular momentum as well as linear momentum and energy have been imparted to the pins. (See Figure 1). Many collisions involve angular momentum. Cars, for example, may spin and collide on ice or a wet surface. Baseball pitchers throw curves by putting spin on the baseball. A tennis player can put a lot of top spin on the tennis ball which causes it to dive down onto the court once it crosses the net. We now take a brief look at what happens when objects that can rotate collide.
Consider the relatively simple collision shown in Figure 2, in which a disk strikes and adheres to an initially motionless stick nailed at one end to a frictionless surface. After the collision, the two rotate about the nail. There is an unbalanced external force on the system at the nail. This force exerts no torque because its lever armis zero. Angular momentum is therefore conserved in the collision. Kinetic energy is not conserved, because the collision is inelastic. It is possible that momentum is not conserved either because the force at the nail may have a component in the direction of the disk’s initial velocity. Let us examine a case of rotation in a collision in Example 1.
Example 1: Rotation in a Collision
Suppose the disk in Figure 2 has a mass of 50.0 g and an initial velocity of 30.0 m/s when it strikes the stick that is 1.20 m long and 2.00 kg.
(a) What is the angular velocity of the two after the collision?
(b) What is the kinetic energy before and after the collision?
(c) What is the total linear momentum before and after the collision?
Strategy for (a)
We can answer the first question using conservation of angular momentum as noted. Because angular momentum is Iω, we can solve for angular velocity.
Solution for (a)
Conservation of angular momentum states
where primed quantities stand for conditions after the collision and both momenta are calculated relative to the pivot point. The initial angular momentum of the system of stick-disk is that of the disk just before it strikes the stick. That is,
where I is the moment of inertia of the disk and ω is its angular velocity around the pivot point. Now, I=mr2 (taking the disk to be approximately a point mass) and ω=v/r, so that
After the collision,
It is ω′ that we wish to find. Conservation of angular momentum gives
Rearranging the equation yields
where I′ is the moment of inertia of the stick and disk stuck together, which is the sum of their individual moments of inertia about the nail. Chapter 10.3 Making Connections gives the formula for a rod rotating around one end to be I=Mr2/3. Thus,
Entering known values in this equation yields,
The value of I′ is now entered into the expression for ω′, which yields
Strategy for (b)
The kinetic energy before the collision is the incoming disk’s translational kinetic energy, and after the collision, it is the rotational kinetic energy of the two stuck together.
Solution for (b)
First, we calculate the translational kinetic energy by entering given values for the mass and speed of the incoming disk.
After the collision, the rotational kinetic energy can be found because we now know the final angular velocity and the final moment of inertia. Thus, entering the values into the rotational kinetic energy equation gives
Strategy for (c)
The linear momentum before the collision is that of the disk. After the collision, it is the sum of the disk’s momentum and that of the center of mass of the stick.
Solution of (c)
Before the collision, then, linear momentum is
After the collision, the disk and the stick’s center of mass move in the same direction. The total linear momentum is that of the disk moving at a new velocity v′=rω′ plus that of the stick’s center of mass,
which moves at half this speed because Thus,
Gathering similar terms in the equation yields,
so that
Substituting known values into the equation,
Discussion
First note that the kinetic energy is less after the collision, as predicted, because the collision is inelastic. More surprising is that the momentum after the collision is actually greater than before the collision. This result can be understood if you consider how the nail affects the stick and vice versa. Apparently, the stick pushes backward on the nail when first struck by the disk. The nail’s reaction (consistent with Newton’s third law) is to push forward on the stick, imparting momentum to it in the same direction in which the disk was initially moving, thereby increasing the momentum of the system.
The above example has other implications. For example, what would happen if the disk hit very close to the nail? Obviously, a force would be exerted on the nail in the forward direction. So, when the stick is struck at the end farthest from the nail, a backward force is exerted on the nail, and when it is hit at the end nearest the nail, a forward force is exerted on the nail. Thus, striking it at a certain point in between produces no force on the nail. This intermediate point is known as the percussion point.
An analogous situation occurs in tennis as seen in Figure 3.  If you hit a ball much farther down, for example, on the shaft of the racquet, the handle is pushed into your palm. And if you hit the ball at the racquet’s percussion point (what some people call the “sweet spot”), then little or no force is exerted on your hand, and there is less vibration, reducing chances of a tennis elbow. The same effect occurs for a baseball bat.
Check Your Understanding
1: Is rotational kinetic energy a vector? Justify your answer.
Section Summary
- Angular momentum L is analogous to linear momentum and is given by L=Iω.
- Angular momentum is changed by torque, following the relationship
- Angular momentum is conserved if the net torque is zero L = constant (net τ=0) or L=L′(net τ=0). This equation is known as the law of conservation of angular momentum, which may be conserved in collisions.
Conceptual Questions
1: Describe two different collisions—one in which angular momentum is conserved, and the other in which it is not. Which condition determines whether or not angular momentum is conserved in a collision?
2: Suppose an ice hockey puck strikes a hockey stick that lies flat on the ice and is free to move in any direction. Which quantities are likely to be conserved: angular momentum, linear momentum, or kinetic energy (assuming the puck and stick are very resilient)?
3: While driving his motorcycle at highway speed, a physics student notices that pulling back lightly on the right handlebar tips the cycle to the left and produces a left turn. Explain why this happens.
Problems & Exercises
Problems & Exercises
1: Repeat Example 1 in which the disk strikes and adheres to the stick 0.100 m from the nail.
2: Repeat Example 1 in which the disk originally spins clockwise at 1000 rpm and has a radius of 1.50 cm.
3: Twin skaters approach one another as shown in Figure 4 and lock hands. (a) Calculate their final angular velocity, given each had an initial speed of 2.50 m/s relative to the ice. Each has a mass of 70.0 kg, and each has a center of mass located 0.800 m from their locked hands. You may approximate their moments of inertia to be that of point masses at this radius. (b) Compare the initial kinetic energy and final kinetic energy.
4: Suppose a 0.250-kg ball is thrown at 15.0 m/s to a motionless person standing on ice who catches it with an outstretched arm as shown in Figure 5.
(a) Calculate the final linear velocity of the person, given his mass is 70.0 kg.
(b) What is his angular velocity if each arm is 5.00 kg? You may treat the ball as a point mass and treat the person’s arms as uniform rods (each has a length of 0.900 m) and the rest of his body as a uniform cylinder of radius 0.180 m. Neglect the effect of the ball on his center of mass so that his center of mass remains in his geometrical center.
(c) Compare the initial and final total kinetic energies.
5: Repeat Example 1 in which the stick is free to have translational motion as well as rotational motion.
Solutions
Check Your Understanding
1: No, energy is always scalar whether motion is involved or not. No form of energy has a direction in space and you can see that rotational kinetic energy does not depend on the direction of motion just as linear kinetic energy is independent of the direction of motion.
Problems & Exercises
1: (a) (b) (c)
3: (a) $$\boldsymbol{3.13\textbf{ rad/s}}$$ (b) $$\boldsymbol{\textbf{Initial KE }=438\textbf{ J, final KE }= 438\textbf{ J}}$$
5: (a) $$\boldsymbol{1.70\textbf{ rad/s}}$$ (b) $$\boldsymbol{\textbf{Initial KE }= 22.5\textbf{ J, final KE }= 2.04\textbf{ J}}$$ (c)


Misuses of Apostrophe
Jared Aragona
Misuses of apostrophe: Don’t use an apostrophe for -s ending that are not meant to indicate possession, such as with plurals, or for any other reason than those listed above.
Incorrect: The horse’s ran around the field.
Correct: The horses ran around the field.
Correct: The horse’s saddle is brown.

Correct: I found three babies in the baby’s room.
The MLCKRB (Master List Code Key and Rule Book): An English Grammar & Style Handbook by Jared Aragona, CC BY 4.0


Introduction to Production, Costs, and Industry Structure
BRING IT HOME
Amazon
In less than two decades, Amazon.com has transformed the way consumers sell, buy, and even read. Prior to Amazon, independent bookstores with limited inventories in small retail locations primarily sold books. There were exceptions, of course. Borders and Barnes & Noble offered larger stores in urban areas. In the last decade, however, independent bookstores have mostly disappeared, Borders has gone out of business, and Barnes & Noble is struggling. Online delivery and purchase of books has overtaken the more traditional business models. How has Amazon changed the book selling industry? How has it managed to crush its competition?
A major reason for the giant retailer’s success is its production model and cost structure, which has enabled Amazon to undercut the competitors’ prices even when factoring in the cost of shipping. Read on to see how firms great (like Amazon) and small (like your corner deli) determine what to sell, at what output, and price.
Chapter Overview
In this chapter, you will learn about:
- Explicit and Implicit Costs, and Accounting and Economic Profit
- Production in the Short Run
- Costs in the Short Run
- Production in the Long Run
- Costs in the Long Run
This chapter is the first of four chapters that explores the theory of the firm. This theory explains how firms behave. What does that mean? Let’s define what we mean by the firm. A firm (or producer or business) combines inputs of labor, capital, land, and raw or finished component materials to produce outputs. If the firm is successful, the outputs are more valuable than the inputs. This activity of production goes beyond manufacturing (i.e., making things). It includes any process or service that creates value, including transportation, distribution, wholesale and retail sales.
Production involves a number of important decisions that define a firm’s behavior. These decisions include, but are not limited to:
- What product or products should the firm produce?
- How should the firm produce the products (i.e., what production process should the firm use)?
- How much output should the firm produce?
- What price should the firm charge for its products?
- How much labor should the firm employ?
The answers to these questions depend on the production and cost conditions facing each firm. That is the subject of this chapter. The answers also depend on the market structure for the product(s) in question. Market structure is a multidimensional concept that involves how competitive the industry is. We define it by questions such as these:
- How much market power does each firm in the industry possess?
- How similar is each firm’s product to the products of other firms in the industry?
- How difficult is it for new firms to enter the industry?
- Do firms compete on the basis of price, advertising, or other product differences?
Figure 6.2 illustrates the range of different market structures, which we will explore in Perfect Competition, Monopoly, and Monopolistic Competition and Oligopoly.
Let’s examine how firms determine their costs and desired profit levels. Then we will discuss the origins of cost, both in the short and long run. Private enterprise, which can be private individual or group business ownership, characterizes the U.S. economy. In the U.S. system, we have the option to organize private businesses as sole proprietorships (one owner), partners (more than one owner), and corporations (legal entities separate from the owners.
When people think of businesses, often corporate giants like Walmart, Microsoft, or General Motors come to mind. However, firms come in all sizes, as Table 6.1 shows. The vast majority of American firms have fewer than 20 employees. As of 2015, the U.S. Census Bureau counted 6.8 million firms with employees in the U.S. economy. Just over half of all the workers in private firms are at large firms that employ more than 500 workers. Another 33% of workers in the U.S. economy are at firms with fewer than 100 workers. These small-scale businesses include everything from dentists and lawyers to businesses that mow lawns or clean houses. Table 6.1 does not include a separate category for the millions of small “non-employer” businesses, where a single owner or a few partners are not officially paid wages or a salary, but simply receive whatever they can earn.
|
Number of Employees |
Firms (% of total firms) |
Number of Paid Employees (% of total employment) |
|
Total |
6,872,350 |
124,067,053 |
|
1-4 |
2,896,318 (42.1%) |
5,873,209 (4.7%) |
|
5–9 |
1,015,610 (14.8%) |
6,610,246 (5.3%) |
|
10–19 |
648,289 (9.4%) |
8,293,511 (6,7%) |
|
20–99 |
696,747 (10.1%) |
20,640,504 (16.6%) |
|
100–499 |
363,684 (5.3%) |
17,502,382 (14.1%) |
|
< 500 |
5,620,648 (81.8%) |
58,919,852 (47.5%) |
|
500 or more |
1,251,702 (18.2%) |
65,147,201 (52.5%) |


Facts about the F Distribution
Learning Outcomes
- Discuss two uses for the F distribution: one-way ANOVA and the test of two variances
Here are some facts about the F distribution.
- The curve is not symmetrical but skewed to the right.
- There is a different curve for each set of dfs.
- The F statistic is greater than or equal to zero.
- As the degrees of freedom for the numerator and for the denominator get larger, the curve approximates the normal.
- Other uses for the F distribution include comparing two variances and two-way Analysis of Variance. Two-Way Analysis is beyond the scope of this chapter.
try it
MRSA, or Staphylococcus aureus, can cause a serious bacterial infections in hospital patients. This table shows various colony counts from different patients who may or may not have MRSA.
| Conc = 0.6 | Conc = 0.8 | Conc = 1.0 | Conc = 1.2 | Conc = 1.4 |
|---|---|---|---|---|
| 9 | 16 | 22 | 30 | 27 |
| 66 | 93 | 147 | 199 | 168 |
| 98 | 82 | 120 | 148 | 132 |
Plot of the data for the different concentrations:
Test whether the mean number of colonies are the same or are different. Construct the ANOVA table (by hand or by using a TI-83, 83+, or 84+ calculator), find the p-value, and state your conclusion. Use a 5% significance level.
While there are differences in the spreads between the groups, the differences do not appear to be big enough to cause concern.
We test for the equality of mean number of colonies:
H0 : μ1 = μ2 = μ3 = μ4 = μ5Ha: μi ≠ μj some i ≠ j
The one-way ANOVA table results are shown in below.
| Source of Variation | Sum of Squares (SS) | Degrees of Freedom (df) | Mean Square (MS) | F |
|---|---|---|---|---|
| Factor (Between) | 10,233 | 5 – 1 = 4 | [latex]\displaystyle\frac{{{10},{233}}}{{4}}={2},{558.25}[/latex] | [latex]\displaystyle\frac{{{2},{558.25}}}{{{4},{194.9}}}={0.6099}[/latex] |
| Error (Within) | 41,949 | 15 – 5 = 10 | ||
| Total | 52,182 | 15 – 1 = 14 | [latex]\displaystyle\frac{{{41},{949}}}{{10}}={4},{194.9}[/latex] |
Distribution for the test: F4,10Probability Statement: p-value = P(F > 0.6099) = 0.6649.
Compare α and the p-value: α = 0.05, p-value = 0.669, α < p-value
Make a decision: Since α < p-value, we do not reject H0.
Conclusion: At the 5% significance level, there is insufficient evidence from these data that different levels of tryptone will cause a significant difference in the mean number of bacterial colonies formed.
Example
Four sororities took a random sample of sisters regarding their grade means for the past term. The results are shown in the table.
Mean Grades for Four Sororities
| Sorority 1 | Sorority 2 | Sorority 3 | Sorority 4 |
|---|---|---|---|
| 2.17 | 2.63 | 2.63 | 3.79 |
| 1.85 | 1.77 | 3.78 | 3.45 |
| 2.83 | 3.25 | 4.00 | 3.08 |
| 1.69 | 1.86 | 2.55 | 2.26 |
| 3.33 | 2.21 | 2.45 | 3.18 |
Using a significance level of 1%, is there a difference in mean grades among the sororities?
Solution:
Let μ1, μ2, μ3, μ4 be the population means of the sororities. Remember that the null hypothesis claims that the sorority groups are from the same normal distribution. The alternate hypothesis says that at least two of the sorority groups come from populations with different normal distributions. Notice that the four sample sizes are each five.
Note
This is an example of a balanced design, because each factor (i.e., sorority) has the same number of observations.
H0: μ1 = μ2 = μ3 = μ4
Ha: Not all of the means μ1, μ2, μ3, μ4 are equal.
Distribution for the test: F3,16
where k = 4 groups and n = 20 samples in total
df(num)= k – 1 = 4 – 1 = 3
df(denom) = n – k = 20 – 4 = 16
Calculate the test statistic: F = 2.23
Graph:
Probability statement: p-value = P(F > 2.23) = 0.1241
Compare α and the p-value: α = 0.01
p-value = 0.1241
α < p-value
Make a decision: Since α < p-value, you cannot reject H0.
Conclusion: There is not sufficient evidence to conclude that there is a difference among the mean grades for the sororities.
Using a Calculator
Put the data into lists L1, L2, L3, and L4. Press STAT
and arrow over to TESTS
. Arrow down to F:ANOVA
. Press ENTER
and Enter (L1,L2,L3,L4
).
The calculator displays the F statistic, the p-value and the values for the one-way ANOVA table:
F = 2.2303
p = 0.1241 (p-value)
Factor df = 3
SS = 2.88732
MS = 0.96244
Error df = 16
SS = 6.9044
MS = 0.431525
try it
Four sports teams took a random sample of players regarding their GPAs for the last year. The results are shown below:
GPAs for Four Sports Teams
| Basketball | Baseball | Hockey | Lacrosse |
|---|---|---|---|
| 3.6 | 2.1 | 4.0 | 2.0 |
| 2.9 | 2.6 | 2.0 | 3.6 |
| 2.5 | 3.9 | 2.6 | 3.9 |
| 3.3 | 3.1 | 3.2 | 2.7 |
| 3.8 | 3.4 | 3.2 | 2.5 |
Use a significance level of 5%, and determine if there is a difference in GPA among the teams.
With a p-value of 0.9271, we decline to reject the null hypothesis. There is not sufficient evidence to conclude that there is a difference among the GPAs for the sports teams.
Example
A fourth grade class is studying the environment. One of the assignments is to grow bean plants in different soils. Tommy chose to grow his bean plants in soil found outside his classroom mixed with dryer lint. Tara chose to grow her bean plants in potting soil bought at the local nursery. Nick chose to grow his bean plants in soil from his mother’s garden. No chemicals were used on the plants, only water. They were grown inside the classroom next to a large window. Each child grew five plants. At the end of the growing period, each plant was measured, producing the data (in inches) in this table.
| Tommy’s Plants | Tara’s Plants | Nick’s Plants |
|---|---|---|
| 24 | 25 | 23 |
| 21 | 31 | 27 |
| 23 | 23 | 22 |
| 30 | 20 | 30 |
| 23 | 28 | 20 |
Does it appear that the three media in which the bean plants were grown produce the same mean height? Test at a 3% level of significance.
Solution:
This time, we will perform the calculations that lead to the F’statistic. Notice that each group has the same number of plants, so we will use the formula [latex]\displaystyle{F}'=\frac{{{n}\cdot{{s}_{\overline{{x}}}^{{ {2}}}}}}{{{{s}_{{\text{pooled}}}^{{2}}}}}[/latex].
First, calculate the sample mean and sample variance of each group.
| Tommy’s Plants | Tara’s Plants | Nick’s Plants | |
|---|---|---|---|
| Sample Mean | 24.2 | 25.4 | 24.4 |
| Sample Variance | 11.7 | 18.3 | 16.3 |
Next, calculate the variance of the three group means (Calculate the variance of 24.2, 25.4, and 24.4). Variance of the group means = 0.413 = [latex]\displaystyle{{s}_{\overline{{x}}}^{{ {2}}}}[/latex]
Then [latex]\displaystyle{M}{S}_{{\text{between}}}={n}{{s}_{\overline{{x}}}^{{ {2}}}}={({5})}{({0.413})} text{ where } {n}={5}[/latex] is the sample size (number of plants each child grew).
Calculate the mean of the three sample variances (Calculate the mean of 11.7, 18.3, and 16.3). Mean of the sample variances = 15.433 = [latex]\displaystyle{{s}_{{\text{pooled}}}^{{2}}}[/latex]
Then [latex]\displaystyle{M}{S}_{{\text{within}}}={{s}_{{\text{pooled}}}^{{2}}}={15.433}[/latex].
The F statistic (or F ratio) is
[latex]\displaystyle{F}=\frac{{{M}{S}_{{\text{between}}}}}{{{M}{S}_{{\text{within}}}}}=\frac{{{n}{{s}_{\overline{{x}}}^{{ {2}}}}}}{{{{s}_{{\text{pooled}}}^{{2}}}}}=\frac{{{({5})}{({0.413})}}}{{15.433}}={0.134}[/latex]
The dfs for the numerator = the number of groups – 1 = 3 – 1 = 2.
The dfs for the denominator = the total number of samples – the number of groups = 15 – 3 = 12
The distribution for the test is F2,12 and the F statistic is F = 0.134
The p-value is P(F > 0.134) = 0.8759.
Decision: Since α = 0.03 and the p-value = 0.8759, do not reject H0. (Why?)
Conclusion: With a 3% level of significance, from the sample data, the evidence is not sufficient to conclude that the mean heights of the bean plants are different.
Using a Calculator
To calculate the p-value:
- Press
2nd DISTR
- Arrow down to
Fcdf
(and pressENTER
. - Enter 0.134,
E99
, 2, 12) - Press
ENTER
The p-value is 0.8759.
try it
Another fourth grader also grew bean plants, but this time in a jelly-like mass. The heights were (in inches) 24, 28, 25, 30, and 32. Do a one-way ANOVA test on the four groups. Are the heights of the bean plants different? Use the same method as shown in Example 2.
- F = 0.9496
- p-value = 0.4402
From the sample data, the evidence is not sufficient to conclude that the mean heights of the bean plants are different.
References
Data from a fourth grade classroom in 1994 in a private K – 12 school in San Jose, CA.
Hand, D.J., F. Daly, A.D. Lunn, K.J. McConway, and E. Ostrowski. A Handbook of Small Datasets: Data for Fruitfly Fecundity. London: Chapman & Hall, 1994.
Hand, D.J., F. Daly, A.D. Lunn, K.J. McConway, and E. Ostrowski. A Handbook of Small Datasets. London: Chapman & Hall, 1994, pg. 50.
Hand, D.J., F. Daly, A.D. Lunn, K.J. McConway, and E. Ostrowski. A Handbook of Small Datasets. London: Chapman & Hall, 1994, pg. 118.
“MLB Standings – 2012.” Available online at http://espn.go.com/mlb/standings/_/year/2012.
Mackowiak, P. A., Wasserman, S. S., and Levine, M. M. (1992), “A Critical Appraisal of 98.6 Degrees F, the Upper Limit of the Normal Body Temperature, and Other Legacies of Carl Reinhold August Wunderlich,” Journal of the American Medical Association, 268, 1578-1580.
Concept Review
The graph of the F distribution is always positive and skewed right, though the shape can be mounded or exponential depending on the combination of numerator and denominator degrees of freedom. The F statistic is the ratio of a measure of the variation in the group means to a similar measure of the variation within the groups. If the null hypothesis is correct, then the numerator should be small compared to the denominator. A small F statistic will result, and the area under the F curve to the right will be large, representing a large p-value. When the null hypothesis of equal group means is incorrect, then the numerator should be large compared to the denominator, giving a large F statistic and a small area (small p-value) to the right of the statistic under the F curve.
When the data have unequal group sizes (unbalanced data), then techniques need to be used for hand calculations. In the case of balanced data (the groups are the same size) however, simplified calculations based on group means and variances may be used. In practice, of course, software is usually employed in the analysis. As in any analysis, graphs of various sorts should be used in conjunction with numerical techniques. Always look of your data!
OpenStax, Statistics, “Facts About the F Distribution,” licensed under a CC BY 3.0 license.