When the Standard Deviation Is Equal to Zero

When the Standard Deviation Is Equal to Zero The sample standard deviation is a descriptive statistic that measures the spread of a quantitative data set. This number can be any non-negative real number. Since zero is a nonnegative real number, it seems worthwhile to ask, â€Å"When will the sample standard deviation be equal to zero?† This occurs in the very special and highly unusual case when all of our data values are exactly the same. We will explore the reasons why. Description of the Standard Deviation Two important questions that we typically want to answer about a data set include: What is the center of the dataset?How spread out is the set of data? There are different measurements, called descriptive statistics that answer these questions. For example, the center of the data, also known as the average, can be described in terms of the mean, median or mode. Other statistics, which are less well-known, can be used such as the midhinge or the trimean. For the spread of our data, we could use the range, the interquartile range or the standard deviation. The standard deviation is paired with the mean to quantify the spread of our data. We can then use this number to compare multiple data sets. The greater our standard deviation is, then the greater the spread is. Intuition So let’s consider from this description what it would mean to have a standard deviation of zero. This would indicate that there is no spread at all in our data set. All of the individual data values would be clumped together at a single value. Since there would only be one value that our data could have, this value would constitute the mean of our sample. In this situation, when all of our data values are the same, there would be no variation whatsoever. Intuitively it makes sense that the standard deviation of such a data set would be zero. Mathematical Proof The sample standard deviation is defined by a formula. So any statement such as the one above should be proved by using this formula. We begin with a data set that fits the description above: all values are identical, and there are n values equal to x. We calculate the mean of this data set and see that it is   x (x x . . . x)/n nx/n x. Now when we calculate the individual deviations from the mean, we see that all of these deviations are zero. Consequently, the variance and also the standard deviation are both equal to zero too. Necessary and Sufficient We see that if the data set displays no variation, then its standard deviation is zero. We may ask if the converse of this statement is also true. To see if it is, we will use the formula for standard deviation again. This time, however, we will set the standard deviation equal to zero. We will make no assumptions about our data set, but will see what setting s 0 implies Suppose that the standard deviation of a data set is equal to zero. This would imply that the sample variance s2 is also equal to zero. The result is the equation: 0 (1/(n - 1)) ∑ (xi - x )2 We multiply both sides of the equation by n - 1 and see that the sum of the squared deviations is equal to zero. Since we are working with real numbers, the only way for this to occur is for every one of the squared deviations to be equal to zero. This means that for every i, the term (xi - x )2 0. We now take the square root of the above equation and see that every deviation from the mean must be equal to zero. Since for all i, xi - x 0 This means that every data value is equal to the mean. This result along with the one above allows us to say that the sample standard deviation of a data set is zero if and only if all of its values are identical.

Biography of Pompey the Great, Roman Statesman

Biography of Pompey the Great, Roman Statesman Pompey the Great (September 29, 106 BCE–September 28, 48  BCE) was one of the main Roman military leaders and statesmen during the final decades of the Roman Republic. He made a political alliance with Julius Caesar, married his daughter, and then fought against him for control of the empire. A skilled warrior, Pompey became known as Pompey the Great. Fast Facts: Pompey the Great Known For: Pompey was a Roman military commander and statesman who was part of the First Triumvirate with Marcus Licinius Crassus and Julius Caesar.Also Known As: Pompey, Gnaeus Pompeius MagnusBorn: September 29, 106 BCE in Picenum, Roman RepublicDied: September 28, 48 BCE in Pelusium, EgyptSpouse(s): Antistia  (m. 86-82 BCE), Aemilia Scaura  (m. 82-79 BCE), Mucia Tertia  (m. 79-61 BCE), Julia  (m. 59-54 BCE), Cornelia Metella  (m. 52-48 BCE)Children: Gnaeus Pompeius, Pompeia Magna, Sextus Pompeius Early Life Unlike Caesar, whose Roman heritage was long and illustrious, Pompey came from a non-Latin family in Picenum (in northern Italy), with money. His father,  Gnaeus Pompeius Strabo, was a member of the Roman Senate. At 23, following in his fathers footsteps, Pompey entered the political scene by raising troops to help Roman general Sulla liberate Rome from the Marians. Marius and Sulla had been at odds ever since Marius took credit for a victory in Africa that his subordinate Sulla had engineered. Their struggles led to many Roman deaths and unthinkable violations of Roman law, such as bringing an army into the city itself. Pompey was a Sullan and a supporter of the conservative Optimates. A novus homo, or new man, Marius was Julius Caesars uncle and a supporter of the populist group known as the Populares. Pompey fought Marius men in Sicily and Africa. For his bravery in battle, he was given the title Pompey the Great (Pompeius Magnus). Sertorian War and Third Mithridatic War Civil war continued in Rome when Quintus Sertorius, one of the Populares, launched an attack against the Sullans in the Western Roman Empire. Pompey was sent to assist the Sullans  in the fighting, which lasted from 80 BCE to 72 BCE. Pompey was a skilled strategist; he used his forces to draw out the enemy and attack them when they least suspected it. In 71 BCE, he helped Roman leaders suppress the slave uprising led by Spartacus, and he later played a role in the defeat of the pirate menace. When he invaded the country of Pontus, in Asia Minor, in 66 BCE, Mithridates, who had long been a thorn in Romes side, fled to the Crimea where he arranged for his own death. This meant the Mithridatic wars were finally over; Pompey could take credit for another victory. On behalf of Rome, Pompey also took control of Syria in 64 BCE​ and captured Jerusalem. When he returned to Rome in 61 BCE, he held a triumphal celebration. The First Triumvirate Along with Marcus Licinius Crassus and Julius Caesar, Pompey formed what is known as the First Triumvirate, which became the dominating force in Roman politics. Together, these three rulers were able to seize power from some of the Optimates and resist the power of the Roman nobles in the Senate. Like Pompey, Caesar was a skilled and highly respected military leader; Crassus was the wealthiest man in the Roman Empire. The alliances between the three men, however, were personal, tenuous, and short-lived. Crassus was not happy that Pompey had taken credit for overcoming the Spartans, but with Caesar mediating, he agreed to the arrangement for political ends. When Pompeys wife Julia (Caesars daughter) died, one of the main links broke. Crassus, a less capable military leader than the other two, was killed in military action in Parthia. Civil War After the dissolution of the First Triumvirate, tensions began to escalate between Pompey and Caesar. Some Roman leaders, including those who had previously resisted the authority of Pompey and Caesar, decided to back Pompey in an election for consul, fearing that the failure to do so would create a power vacuum in Rome. Pompey then married Cornelia, the daughter of the Roman consul Metellus Scipio. For a time, Pompey controlled much of the Roman Empire while Caesar continued his campaigns abroad. In 51 BCE, Pompey made moves to relieve Caesar of his command. He promised to give up his own armies as well; however, some scholars claim that this was merely a ploy to hurt public opinion of Caesar, who no one expected would surrender his forces. Negotiations continued unsuccessfully for some time, with neither commander willing to make military concessions, and eventually the conflict turned into outright war. The Great Roman Civil War- also known as Caesars Civil War- lasted four years, from 49 to 45 BCE. It came to an end with Caesars decisive victory at the Battle of Munda. Death Pompey and Caesar first faced each other as enemy commanders after Caesar, defying orders from Rome, crossed the Rubicon. Caesar was the victor of the battle at Pharsalus in Greece, where he was outnumbered by Pompeys forces. After the defeat, Pompey fled to Egypt, where he was killed and his head cut off so that it could be sent to Caesar. Legacy Even though he turned against Caesar, Pompey was widely admired by his countrymen for his role in the conquest of various territories. He was especially admired by the nobles, and statues of him were placed in Rome as a tribute to his military and political accomplishments. His image was printed on silver coins in 40 BCE. Pompey has been depicted in a number of films and television series, including Julius Caesar, Rome, Ancient Rome: The Rise and Fall of an Empire, and Spartacus: War of the Damned. Sources Fields, Nic.  Warlords of Republican Rome: Caesar versus Pompey. Casemate, 2010.Gillespie, William Ernest.  Caesar, Cicero and Pompey: the Roman Civil War. 1963.Morrell, Kit.  Pompey, Cato, and the Governance of the Roman Empire. Oxford University Press, 2017.Seager, Robin.  Pompey, a Political Biography. University of California Press, 1979.

Financial planning Assignment Example | Topics and Well Written Essays - 1250 words

Financial planning - Assignment Example In the absence of the business knowledge, the company would not capable of competing effectively. 2. Investors are instrumental to economic growth. Dissatisfaction of the investors elicits disengagement from doing business in a given location (Taylor, 2012). The consequences of such moves include deterioration of economic growth. Thus, the interest of the clients comes first when designing policies that would affect business environment. Insensitive policies client’s demand often restrain investments. Similarly, clients often prefer to dissociate with businesses that fail to address their concerns. Good business policies affect investors as well as their clients. Conflict of interest is a critical concern to business growth because it affects the way investors would do business. Many investors are willing to engage in businesses that have limiting conflict of interests. 3. First, the business will be uncompetitive in the target market. Inappropriate advice limits the ability to address the needs of the consumers in target market resulting to lack of competitiveness. Second, inappropriate advice precipitates business failure because the company would be incapable of meeting the expectations of the clients (Taylor, 2012). The viability of the business largely depends on appropriate measures that it takes when handling the needs of the consumers. Lastly, inappropriate advice elicit inappropriate responses, for example, if the business does not receive appropriate advice on how to manage address the concerns of the employees it might end up loosing talented workers. Largely, the advise should be tailored towards addressing the business needs. 4. This strategy would change image of the business and consumer brand promise. The design of products tends to influence the image of the company in the eyes of the consumers. Improving the company’s products with respect to clients needs is one the mechanism of

INDIVIDUAL COURSEWORK Essay Example | Topics and Well Written Essays - 1500 words

INDIVIDUAL COURSEWORK - Essay Example Various British designers are accredited with the reputation and image par excellence and would create opportunities for augmenting the possibilities of being inducted in reputed professional companies overseas or abroad (Andriopoulos, C., & Dawson, P. 2009). There is a leading company in United Kingdom that is privately owned and involved in clothing as its primary business. The company is also designing and an expert in retailing its brands of label range wears of various segments regardless of age. Mr. David Reiss the owner of the company Reiss took over from his father and created an opportunity of narrowing the gap in mass market clothes of high street as well as designer wears. It is pertinent to note that narrowing down the gap translates into provision of good quality as well as stylish clothes for prices that are amazingly affordable. The strategy following the aftermath of success creates vistas for opening up expansion in Dublin the capital of Ireland. The strategy can be created by launching wide ranges of women’s wear and globalize the business along with Dublin. Even though the efforts are on for making penetration into various countries the headway has been successful in claiming establishment of 27 outlets of retai l as well as concessions numbering to three in the United Kingdom until now. Mr. Reiss continues to be endowed with the capacity of a guiding influence with strength that is all pervasive in the company. The company Reiss is capable of creating a good brand value and an entry by creating an opportunity to internationalize the company in a manner that would be conducive for strategizing the brands and the segments. There has been successful expansion into the fields of new segments of wears including opportunities for children’s wears. This strategy should be further improved to create wide ranges of brands for encompassing a new

Government Regulatory Agencies Essay Example for Free

Government Regulatory Agencies Essay Government Regulatory Agencies and Impact on Consumer Choices By Trisha Robinson, Chastity Hafer, George Ward, Quagina Jackson Serphy HCS 490 Health care Consumer- Trends and Marketing For: Cindy Perkins, DC, MBA 05/01/2011 The Food Drug Administration (FDA) The Food Drug Administration is in charge of promoting and the protection of public health safety by the supervision and regulation of food safety, dietary supplements, tobacco products, prescription and over the counter drugs you get from the pharmacy. The FDA also regulates vaccines, blood transfusions, biopharmaceuticals, medical devices, veterinary products, and cosmetics (fda. gov). The FDA is in charge of advancing the public health by the speed of new ideas or methods that make medicines and food affordable and by the producing of strong or desired effect, more safer, and by given the public a more accountable scientific information that the public can use when using medication and foods and to lessen the use of tobacco to improve the quality of life. The FDA makes sure that laws are enforced by monitoring companies of claims that the company makes about their products. If the FDA thinks the company has violated the law a warning letter will be sent to the company. The letter states that the company is in violation and the company has 15 working days to respond and say how this will be fixed. Companies that do not respond will be fined or their product will be seized. The health care product or service category selected Service of choice:Â  Regulatory Information Regulatory Information is a service, which The Food and Drug Administration (FDA) offers an array of information on rules and regulations, which allows consumers, industries, and health care professionals to know the safety, availability, and effectiveness of drugs, biological, medical, and electronic products, and cosmetics. In addition, the service provides information on non-regulated categories such as advertising, consumer products (household goods), alcohol, drug abuse, meat and poultry, health insurance, drugs of abuse, pesticides, restaurants and grocery stores, and water. It offers information on areas such as safety, labeling, product licensing, product approval, manufacturing and performance standards on medical devices and radioactive products, and animal drugs, livestock feeds, and food. Rationale for choosing FDA A. Familiar with the regulations of the FDA. 1. Protecting the public’s health B. Interested in the operations of the FDA and how they develop their regulations. C. The services FDA protects. 1. Responsible for advancing the public health by helping to speed innovations that make medicines more effective.

Travel as Experience in Jane Eyre :: Jane Eyre Essays

Travel as Experience in Jane Eyre In his essay "The Progress of Error" William Cowper writes: Returning he proclaims by many a grace, By shrugs and strange contortions of his face, How much a dunce, that has been sent to roam, Excels a dunce, that has been kept at home. (Buzard 99) In the novel, we are presented with the tale of Jane Eyre and her travels around the English countryside. What she has seen and done are not considered extraordinary but rather common to a woman of her social standing. On the other hand, Rochester as a man of wealth and land has traveled the world and seen the sights of many nations. He has been to the new world and has also completed the Grand Tour of Europe that so many aristocrats before him have done. Yet when he returns home jaded, he finds in the plainest of women something that he had not found in his countless expeditions. When Jane is betrayed by Rochester, she leaves on her own tour with only a hope of survival without him. She eventually returns from her trek and has learned what she truly desires is to be with Rochester. Rochester’s advantageous trip abroad does not deliver the hope and satisfaction that the Grand Tour promises. On the other hand, Jane’s inconvenient journey around her homeland proves reveal ing to her independent nature. These details closely mirror the questions that arise when the value of travel as a learning experience is considered. Ultimately, Jane learns that where one goes is less important than how one spends the time. We see both sides of this argument in their first real conversation. While trying to explain why he finds her so interesting and at the same time must condescend to her, Rochester tells Jane that "†¦I have battled through a varied experience wit+h many men of many nations, and roamed over half the globe, while you have lived quietly with one set of people in one house" (140; ch. 14). Rochester believes this view of conquering more of the world makes him a stronger, better person. Jane retaliates that Rochester shouldn’t feel superior just because "†¦you have seen more of the world than I have – your claim to superiority depends on the use you have made of your time and experience" (140; ch. 14). Rochester responds by admitting he has "made an indifferent, not to say a bad use of both advantages" (140; ch.

Oedipus: a Tragic Hero

Oedipus: A Tragic Hero Aristotle’s tragic hero is one of the most recognizable types of heroes among literature. A tragic hero combines five major points all of which have to do with the hero’s stature in society, his faults, how these faults effect him, the punishment his faults gets him, and how he reacts to this punishment. Aristotle explained that the story of Oedipus the King, written by Sophocles, is a perfect example of a tragic hero. In the play, Oedipus is given a prophecy in which he is told that he will kill his father then marry his mother. As in many Greek plays, Oedipus tries to run from his prophecy and ends up fulfilling exactly what it is foretold. Through the play we see that Oedipus posses many of the characteristics of a tragic hero such as: he is of noble stature, his downfall was his own fault, and that his punishment was not wholly deserved. The first trait that Oedipus possesses that makes him a tragic hero is that he is a man of noble stature. Within the first lines of the play, Oedipus immediately supports this in saying, â€Å"Here I am myself-/you know me, the world knows my fame:/I am Oedipus† (7-8). This quote shows that Oedipus is so confident in people knowing who he is that he is willing to put this out there, though he is not being overly confident. This quote is later given reason to when we hear from other characters such as the Priest when he explains things that Oedipus has done such as, â€Å"You freed us from the Sphinx, you came to Thebes/and cut us loose from the bloody tribute we had paid/that harsh, brutal singer† (44-46). The Priest depicts when Oedipus defeated the Sphinx, which is the task that gains him the position of king of Thebes. The next characteristic that Oedipus possesses that makes him a tragic hero is that his downfall is of his own doing. The act that ultimately contributes to Oedipus’ downfall is that of when he kills his father and he explains to Jocasta that he killed everyone, â€Å"I killed them all-every mother’s son! † (898). Since he is talking about his father’s caravan it shows that he, indeed killed his father even though he was unaware that it was his father at the time. That is only the beginning of his downfall though as he earlier told the Chorus that anyone harboring the murderer would be cursed, even if it was himself, â€Å"†¦if by any chance/he proves to be an intimate of our house†¦may the curse I just called down strike me! † (284-287). This shows just how serious that Oedipus was about punishing this person and even though he doesn’t understand that it is himself, and takes awhile for him to realize, that he was responsible for what happened to him. The final discussed characteristic that Oedipus possesses is that his punishment was not wholly deserved. There are three main punishments that Oedipus has to go through because of his crime. Those punishments are his exile from Thebes, the suicide of Jocasta, and the loss of his sight. However, the punishment that pushes everything over the edge is when Oedipus strikes out his own eyes. When he appears to the Chorus he comes out and speaks of how horrid he feels for everything in saying, â€Å"The blackest things/a man can do, I have done them all! † (1541-1542). During this very scene the Chorus speaks, not to Oedipus directly, but to each other and the literally shrink away from him. It is obvious that they don’t believe he deserves this through their words, â€Å"O the terror–/the suffering, for all the world to see/the worst terror that ever met my eyes,/what madness swept over you? † (1432-1435). The Chorus continues to speak of how Oedipus cursed himself by the gods and that this is his punishment, but cannot understand what god would force such a punishment on any man. Works Cited Mifflin, Houghton. Language of Literature World Literature. Boston: Houghton Mifflin Company, 2005. Print.