Chebyshev’s Theorem and the Empirical Rule

A bell curve is perfectly symmetrical with respect to a vertical line through its peak and is sometimes called a Gauss curve or a normal curve. The second shape a scatter diagram may have is anything but a normal curve as in the next drawing: We can do a lot of good statistics with the normal curve, but virtually none with any other curve. Let us assume that we have recorded the 1000 ages and computed the mean and standard deviation of these ages. Assuming the mean age came out as 40 years and the standard deviation as 6 years we can do the following predictions. Chebyshev’s Theorem In the case of a scatter diagram that seems to be anything but a normal curve, all we can go by is Chebyshev’s theorem. This very important but rarely used theorem states that in those cases where we have a non-normal distribution, the following can be said abut the individual data, which in this case are the ages: †¢ At least 75% of all the ages will lie in the range of [pic]. In our case this means that at least 75% of the people will have an age in the range of [pic] years which simplifies to a range of 28 to 52 years. †¢ At least 88. 9% of all the ages will lie in the range of [pic]. In our case this means that at least 88. 9% of the people will have an age in the range of [pic] years which simplifies to a range of 22 to 58 years. †¢ At least 93. 75% of all the ages will lie in the range of [pic]. In our case this means that at least 93. 75% of the people will have an age in the range of [pic] years which simplifies to a range of 16 to 64 years. †¢ At least 96% of all the ages will lie in the range of [pic]. In our case this means that at least 96% of the people will have an age in the range of [pic] years which simplifies to a range of 10 to 70 years. At least 97. 2% of all the ages will lie in the range of [pic]. In our case this means that at least 97. 2% of the people will have an age in the range of [pic] years which simplifies to a range of 4 to 76 years. How can we calculate these percentages? To calculate the 75%, the 88. 9%, the 93. 75%, etc, we look at the number of standard deviations in the respective intervals. The 75% goes together with me an  ± 1 standard deviation, the 88. 9% with mean  ± 2 standard deviations, the 93. 75% with mean  ± 3 standard deviations, and the 96% with mean  ± 4 standard deviations. In general you can say that the percentage of people with an age in the range of mean  ± k standard deviations can be found by calculating the value of the quantity [pic] and then converting that into a percentage. Summarizing the above we get the following table: |Interval |k |[pic] |% | |[pic] |2 |[pic] |75 | |[pic] |3 |[pic] |88. 9 | |[pic] |4 |[pic] |93. 75 | |[pic] |5 |[pic] |96 | [pic] |6 |[pic] |97. 2 | Do we have to restrict ourselves to whole numbers as values for k? No, we may take any value for k as long as it larger than 1. For instance, for k = 2. 5 we get the result that [pic] in the interval [pic] years Example 1: Students Who Care is a student volunteer program in which college students donate work time in community centers for homeless people. Professor Gill is the faculty sponsor for this student volunteer program. For several years Dr. Gill has kept a record of the total number of work hours volunteered by s student in the program each semester. For students in the program, for each semester the mean number of hours was 29. 1 hours with a standard deviation of 1. 7 hours. Find an interval for the number of hours volunteered in which at least 88. 9% of the students in this program would fit. Solution: From the table above we see that a percentage of 88. 9 will coincide with an interval of [pic] hours. This can be rewritten as an interval from 24 to 34. 2 hours volunteered each semester. Example 2: The East Coast Independent News periodically runs ads in its own classified section offering a month’s free subscription to those who respond. This way management can get a sense about the number of subscribers who read the classified section each day. Careful records have been kept over a period of 2 years. The mean number of responses was 525 with a standard deviation of 30. What is the smallest percentage of responses in the interval between 375 and 675? Solution: The difference between the mean of 525 and the upper limit of this interval is 150. This is 5 standard deviations since[pic]. The same is true for the difference between the mean and the lower limit of this interval. According to the table above this coincides with 96%. The Empirical Rule When the data values seem to have a normal distribution, or approximately so, we can use a much easier theorem than Chebyshev’s. The empirical rule states that in cases where the distribution is normal, the following statements are true: †¢ Approximately 68% of the data values will fall within 1 standard deviation of the mean. †¢ Approximately 95% of the data values will fall within 2 standard deviations of the mean. Approximately 99. 7% of the data values will fall within 3 standard deviations of the mean. Example 3: The average salary for graduates entering the actuarial field is $60,000. If the salaries are normally distributed with a standard deviation of $5000, then what percentage of the graduates will have a salary between $50,000 and $70,000? Solution: Both $50,000 and $70,000 are $10,000 away from the mean of $60,000. This is two standard deviations away from the mean, so 95% of the graduates will have a salary in this interval. [pic] Age No of people [pic]

Highlight problems and issues trade union face in the UK and recommend Essay

Highlight problems and issues trade union face in the UK and recommend practical solutionsto solve them - Essay Example Failure to come on an agreement on the negotiations made, the trade unions are entitled to strike. Capitalism is a form of governance where, the land, labor and other factors of production are privately owned states Clarke (1978). Trade unions can be defined from this perspective as the representatives of the employees’ terms of employment to employers and government from a given policy which is based on the person who privately owns the union. The trade unions here are not government controlled but privately controlled. Trade union from a radical perspective can be defines as the representation of employees from the industrial and commercial sectors only. These are the unions that only focus on the two business sectors but they are not necessarily privately owned. Marxism emphasises on the workers’ rights to enjoy the fruits of their labor. It thus defines trade unions as organs which represent work towards ceasing employees’ harassment at work place and ensuring they enjoy the benefits and results that come from their work. Lionel (1968) states that in the UK, most unions belong to the Trades Union Congress, which represents many unions, and hence has more power over issues effecting workers nationally. Therefore trade unions act as a voice for the employees and often take on the role of challenging and updating contracts and conditions for members and also protect the members from harassment and employment related legal issues. Rosen (1969) states the main sources of power of the trade unions as the government, employers, and employees. These are the key sources of trade unions as they are the determinants of the representatives in the trade unions and the legal issues that it should abide to. Stewart (1986) argues that if a union does not have a good collective bargaining style, then it will lose its members who are the employees. On the other hand, the employers and the

Business Activities, Employment and Inflation Term Paper

Business Activities, Employment and Inflation - Term Paper Example If the injections of an economy (J) are not equal to the withdrawals (W), then there is a clear disequilibrium in the economy. The only factors that bring this equilibrium back into line is the change in the national income (GDP) and the levels of employments (Sloman and Sutcliffe). To understand the above figure better, lets consider that the economy is faced with a state of equilibrium, i.e. the levels of withdrawals and the level of injections is the same. If there is an increase in the injections, and the firms aim at investing more into the company, then the aggregate demand, i.e. Cd will also be higher. Hence to meet up with this demand, the firms will also need to increase the labor and other resources which would in turn lead to higher levels of incomes for the households (Y) (Sloman and Sutcliffe). With an increase in the income of the households, there will be an increase in the expenses as well, which in turn will lead the firms to also sell higher. Higher sales will mean the firms need to produce higher which again would mean more labor, and other resources (Mankiw). This is a multiplied affect that will continue to go on within the economy. This effect is referred to as, ‘Multiplier Effect’ and is defined as, ‘an initial increase in aggregate demand of $Xm leads to an eventual rise in national income that is greater than $Xm’ (Sloman and Sutcliffe). This follows the principle of ‘Cumulative Causation’, which can be defined as, ‘An initial event can cause an ultimate effect which is much larger’ (Sloman and Sutcliffe).

Public Policy Problems In The Environment :: essays research papers

Public policy is defined by Webster’s as the â€Å"The basic policy or set of policies forming the foundation of public laws, especially such policy not yet formally enunciated.† The United States Government has many policies in the area of the environment. The Environmental Protection Agency (EPA) was created in 1970 to help identify environmental problems in our nation, and to set policy on how to deal with those problems. Yet, with so much money spent by the government to deal with problems with the environment, it must be noted that problems still exist, even within the bureaucracy that was meant to help in the first place. During the presidential campaign of the last election, an issue arose concerning the â€Å"energy crisis† that was driving gasoline and oil prices up throughout our country. Vice President Al Gore supported President Clinton’s ideology of waiting for the proper legislative initiatives to pass through Congress, and when the situation merited, provide some limited releases of oil from the national oil reserve. Candidate George W. Bush, on the other had, favored drilling in the government protected lands of Alaska to find future oil reserves so that America would no longer be so dependent on foreign oil. The problem with Bush’s plan, according to Gore, was that this could be devastating to the environment of the scarcely populated Alaskan wilderness. Regardless of the political, legal or moral implications of such drilling, there are problems dealing with multiple types of rationality in this issue. In his book â€Å"Reason in Society,† Paul Diesing describes six major types of rationality. These include technical, economic, social, legal, political and ecological rational. It is easy to comprehend that this environmental issue involves each one of these types of rationality. First of all, the technical rationality is demonstrated through the question of whether or not oil can be found in Alaska, and if it could, would there be enough present to really make a dramatic difference for the consumer? It must also be considered as to how this drilling may effect the environment of this area. Technical rationality also questions whether or not there are ways to drill that can possibly leave the natural resources of this area with as little human interference as possible. Engineers and scientists can try to come up with ways to create a process with which the area will not be devastated by the involvement with man in those areas.

Describe and Compare the Two Forms of Cubism

According to the Tate Gallery’s exposition (1979) Cubism has remained the most important and influential movement of the 20th century, notwithstanding the movement’s short duration. According to Read (1994) the major period for Cubism was from 1907 to 1914, with Picasso and Braque as the main originators of the movement. The rationale for the Tate’s statement is given as â€Å"the artists associated with [Cubism] took some of the most decisive steps towards abstraction†, and this extreme development â€Å"has become the archetype of later revolutionary movements† (p. 84). The movement, according to Read, was the first abstract style of the 20th century, and named by the art critic Louis Vauxcelles, who took up a remark by Matisse about â€Å"Braque’s little cubes† (p. 100). One source (artlex. com) cites Vauxcelles as saying: â€Å"M. Braque scorns form and reduces everything, sites, figures and houses, to geometric schemas and cubes. † One of the most innovative developments is that the creators of Cubism sought to replace a single viewpoint and light source, normal within the western art world since the Renaissance, with a much more complete representation of any object, combining many ‘aspects’. Initially colours were temporarily abandoned and shapes were simplified and flattened. Space was furthermore rendered by means of oblique lines and overlapping forms (The Tate Gallery, 1979). According to Belton (2002, p. 109) Picasso and Braque both struggled with the problem of representing three dimensional objects and figures in the two dimensional medium of painting; â€Å"their solution was to create an abstract form that could display two or more sides of an object simultaneously†. Whilst Picasso’s Demoiselles d’Avignon is generally viewed as the first Cubist painting, Read (1994) argues that the painting might be more usefully viewed as ‘pre-Cubist’, or ‘proto-Cubist’, as it was so heavily influenced by Iberian or African art. Cezanne’s later work is often viewed as the catalyst for the development of Cubism, and Read cites Cezanne’s advice to Bernard â€Å"to deal with nature by means of the cylinder, the sphere and the cone† (p. 100). Cezanne, by trusting his eyes and attempting to express natural, binocular vision, allowed for the ‘truth’ of the shifted viewpoint (Moszynska, 1990). Cubism gives the artist a way of depicting the world in a way that goes beyond what can be seen, and attempts to deal with the energies of objects. According to Read (1994) Cubism could be categorized into various divisions, including ‘analytic’, ‘hermetic’ and ‘synthetic’. This essay will mainly concentrate in the analytic and synthetic forms of Cubism. The term ‘hermetic’ refers to the largely or wholly indecipherable way of representing an object in the flatter type of abstraction, as typical of both Braque’s and Picasso’s later way of working. In this phase the allover pattern became more important. Other sources (including artlex. com) refer to ‘analytic’ cubism as ‘facet’ cubism. Analytical and Synthetic Cubism acquired their names through the comments by art historian Einstein, and in effect are retrospective labels. Einstein wrote that the â€Å"simplistic distortions† employed by Picasso, as typified by his portrait of Gertrude Stein, led to â€Å"a period of analysis and fragmentation and finally to a period of synthesis† (as cited in Foster, Krauss, Bois and Buchloh, 2004, p. 106). The analytical phase of Cubism, as developed by Braque and Picasso, was characterised by a number of different features, starting with the contraction of the painters’ palettes, away from the full colour spectrum to rather monochrome selections, which Foster et al. term ‘abstemious’. The second characteristic is the extreme flattening of the visual space, â€Å"as though a roller had pressed all the volume out of the bodies† (ibid. , p. 106). The third characteristic identified by Foster et al. is the visual vocabulary used to describe â€Å"the physical remains of this explosive process† (p. 06). Foster et al. illustrate these features with Picasso’s portrait of Daniel-Henry Kahnweiler (1910) and Bracque’s painting The Portuguese (The Emigrant, 1911-12). The grey or tonal scale, the traditional tool of representing volume, is used very differently by the Cubists. Kahnweiler, the art dealer who exhibited both Braque and Picasso’s work, identified the â€Å"bringing about the unity of the pictorial object† as the exclusive concern of Cubism (Foster et al. , 2004, p. 107). Kahnweiler’s view as upheld by Greenberg, who saw Analytical Cubism as the fusion between two types of flatness: the â€Å"depicted flatness†, shoving the fragmented objects closer to the surface, and the â€Å"literal flatness† of that surface (ibid. , p. 109). Foster and his colleagues however question this: they note a number of differences between the evident intentions of Braque and Picasso in relation to the flat plane, with Picasso, being more ‘tactile’, more focused on exploring the possibilities of using Cubism for sculpture, and Braque more concerned with transparency. Steinberg too, urged against the blurring of Picasso and Braque’s pictures. The two exponents of Cubism saw themselves as being ‘roped together’ like mountaineers in their exploration of this new way of working, with the ebullient Spaniard referring to Braque as his ‘wife’. However, Braque was loyal to ‘passage’, the practice of visual slippage between adjacent elements, whereas Picasso, according to Foster et al. , had an â€Å"overwhelming concern with a vestigial kind of depth† (ibid, p. 109). Picasso seemed more focused on making depth tactile, as Foster et al. demonstrate with showing Picasso’s central plunging depth in Houses on the hill: Horta de Ebro (1909). They go on to argue that Braque is more concerned with the ‘diaphanous’ quality of Cubism, with the loss of traditional notions of figure and ground The Tate Gallery (1979, p. 85) presents Braque’s Clarinet and a Bottle of Rum on a Mantelpiece (1911) as a crucial point in Cubism, â€Å"when the breaking down of objects had been carried to a point very close to complete abstraction†. After this point Braque and Picasso started to introduce areas of wood-graining, the use of collage, and a re-introduction of colour, thereby representing objects in a more recognisable, but also more symbolic way. According to Gersh-Nesic (n. d. ) Synthetic Cubism integrates â€Å"high† and â€Å"low† art (art made by an artist combined with art made for commercial purposes, such as packaging), and according to some can be considered the first Pop Art. Even before 1912 Braque and Picasso had introduced stenciled lettering into their paintings. These stencils were not fine art, they were used for packaging and pub signs. The stencils draw attention to the surface of the canvas, since the uniform letters appear independent of what's painted underneath them. Two technical innovations exemplify new development in Cubism: papier colle and collage. Papier colle involves sticking coloured paper onto the canvas and was invented by Braque. Collage was developed by Picasso, and involved sticking all sorts of materials, such as leather, newspaper, material and rope, onto the surface. Sticking different materials, such as woodgrain, onto the surface of the painting playfully confused what was real and what was an illusion (Tate, n. d. ). New, provocative questions are raised with the use of collage, namely: what is more realistic, to perfectly simulate the look of a newspaper in oil paint, or to stick actual newspaper onto the canvas? (Tate, n. d. ). Wadley (1970, p. 13) holds that â€Å"technically and conceptually† Synthetic Cubism was â€Å"a denial of the European tradition†, in that the surface was now the furthest point from the spectator, not the nearest. Artists working in a synthetic way started with the terms of painting, and from them composed an image which they could justly claim was more real, â€Å"since it in no sense distorted or imitated something else† (ibid. p. 14). Gris, who was the clearest formulator of cubist theory, stated: I work with the elements of the intellect, with the imagination. I try to make concrete that which is abstract. I proceed from the general to the particular, by which I mean that I start with an abstraction in order to arrive at a true fact. Mine is an art of synthesis, of deduction. (ibid. , p. 129) Gris was called a ‘demon of logic’ by Apollinaire. Indeed, his way of working and thinking was different from his Cubist colleagues. According to Wadley the integration in Gris’ work is tighter than in comparable Picassos or Braques. Gris used collage only in his paintings, and its effect is always to strengthen the rigid division of the surface. [ ]There is no hint of Picasso’s ragged edges and random encounters, nor of Braque’s simple elements floating in a spacious arena. The total effect is of tight concentration. (Wadley, 1970, p. 82) Gris’ meticulous style is evident in how he has painstakingly arranged the letters in ‘Le Matin’ in Man in the Cafe, to correspond with the diagonal columns and echo the horizontal line elow (ibid). Foster et al. (2004) make a ‘semiotic’ reading of Picasso’s use of material in his Violin (1912): the ‘twin’ pieces of newsprint paper represent on the one hand the frontal, opaque (wood of the violin), as well as the transparent, amorphous ‘ground’ (background colour). They go on to claim that a similar visual play of meaning is evident in Picasso’s Bottle of Vieux Marc, Glass and Newspaper (1913), where a piece of wallpaper is used to represent the liquid in the glass, the rim of the glass, and the ground of the table-cloth. The cut out piece used for the liquid (looking like a chef’s top hat) represents transparency, whilst the ‘negative shape’ left by the incision represents the solid stem of the glass. The viewer might be left with the question whether the ever playful Picasso was just enjoying a visual pun, or whether he intended any of this to be read as ‘signs’. The artist himself stated: â€Å"Mathematics, trigonometry, chemistry, psycho-analysis, music and whatnot, have been related to cubism to give it an easier interpretation. All this has been pure literature, not to say nonsense, which brought bad results, blinding people with theories. (Wadley, 1970, p. 128) During its lifetime both Analytical and Synthetic Cubism encompassed and influenced many artists – the most notable of these being Leger, the three Duchamp/Villon brothers, Robert and Sonia Delaunay, Gleizes and Metzinger (who published a book on Cubism). For some of these artists Cubism functioned as a transition, although Picasso would hold that â€Å"Cubism is neither a seed, nor a foetus, but an art dealing primarily with forms, and when a form is realized it is there to live its own life† (Wadley, p. 28). It led artists like Piet Mondrian to what he saw as its logical end, complete abstraction. Cubism may have been short-lived as a movement, but it continues to influence contemporary art to this day. Collage, for instance, has become a widely practiced form of art. And in terms of form, the practice of reducing everything to the ‘cylinder, the sphere and the cone’ was brought to mind on viewing some of Manolo Valdes’ work, in an exhibition in The Hague this summer.

Plato s View On Advantage And Justice - 1212 Words

Plato in book 1 is not himself, he is potrayed by another character since plato is the author of book 1. The word Advantage throughout book 1 to demonstrate how justice can be an advantage for some and a disadvantage for others. According to Plato this is what Justice symbolizes, therefore this statement mentioned above is fully justified. To further support this claim, this paper will focus on the following supporting evidence. Firstly, how Plato’s character uses the word advantage to solidify his opinions on what justice should be and secondly, how Plato’s character view on advantage and justice differs from Socrates opinion. As stated before, Plato’s character emphasizes the word advantage to solidify his opinions on what justice†¦show more content†¦The people with this superior power only focus on what will benefit them, individuals in this society are led to believe that the ones who have gained this sort of power care about the weak when in fact, it’s the complete opposite .The weak are seen as cattle ,only meant to feed the strong. . The ones that rule will always find a way to make themselves look better by using anything and everything to their advantage. Plato’s character elaborates that concept in this quote , â€Å"And each ruling group sets down laws for its own advantage(†¦) in every city the same thing is just , the advantage of the established ruling body. It surely is master; so the man who reason rightly concludes that everywhere justice is the same thing, the advantage of the stronger†(339a). As stated before, this quote explains ho w the few that rule make laws that only help them gain over people who are not able to obtain the same power. Limiting their influences, needs and wants in that society . According to Plato’s character , this is fully justified since the rule makers have worked their way to be where they are in society making their decision just. Previously mentioned in the paragraph above, Plato’s character and Socartes views on justice and advantage differ .To demonstrate this statement throughout the pages (18-34) there is numerous evidence for such a claim. Socrates stated that, â€Å"I said a moment ago that no one willingly chooses to rule and get mixedShow MoreRelatedJustice Is An Important Part Of Our Social And Justice System967 Words   |  4 Pagesthe most important concept studied in class was justice. This is an important concept for me because it is something that we see everyday within our political and justice system. 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We looked at how justice was perceived in Plato’s time and how he himself saw justice. I look towards the platform of Jill Stein the Green Party and their pursuit for justice. Justice overtime has not been applied quality and an idea thatRead MoreEssay about Justice in Platos Republic and Hobbes Leviathan2782 Words   |  12 Pagesand Hobbes Leviathan is justice. For Plato, the goal of his Republic is to discover what justice is and to demonstrate that it is better than injustice. Plato does this by explaining justice in two different ways: through a city or polis and through an individual human beings soul. He uses justice in a city to reveal justice in an individual. For Hobbes, the term justice is used to explain the relationship between morality and self-interest. Hobbes explains justice in relation to obligations

Roberta Phillip Robbins, A Youth And A Gang Violence...

Roberta Phillip-Robbins, a youth and a gang violence prevention specialist for Multnomah County declared she was running for Rep. Lew Frederick’s seat in the Oregon House representing North and Northeast Portland. Her top issues include making sure that the voices of Oregon’s underrepresented communities are better heard(Slovic, 2015). She says she wants to work hard to elevate those voices. The candidate should be responsible for representing the people. She says she has the heart for public service which she refers to the rent that one pays for the room on this earth. She was committed to running, winning and serving the residents of House District 43. By signing this document, I hereby state that I will qualify for said office if elected (Slovic, 2016). Tawna Sanchez is the director of family services at the Native American youth and family center and has served as a commissioner in a body that advises the department of human services. â€Å" I do support increasing taxes on corporations. I do not support IP 28 and have deep concerns that IP 28 will increase costs for those who can least afford it,† (Tawna Sanchez for Oregon HD-43 Facebook, n.d.). Her platform includes addressing human inequality, creating long-term affordable housing and limiting campaign contributions. Her main task will be to deal with the issue of corporate taxes and income inequality, and create long-term affordable housing by creating a responsible department. Both of the candidates have personal