Sum of Squares Formula Shortcut

The calculation of a sample variance or standard deviation is typically stated as a fraction. The numerator of this fraction involves a sum of squared deviations from the mean. In statistics, the formula for this total sum of squares is ÃŽ £ (xi - xÌ„)2 Here the symbol xÌ„ refers to the sample mean, and the symbol ÃŽ £ tells us to add up the squared differences (xi - xÌ„) for all i. While this formula works for calculations, there is an equivalent, shortcut formula that does not require us to first calculate the sample mean. This shortcut formula for the sum of squares is ÃŽ £(xi2)-(ÃŽ £ xi)2/n Here the variable n refers to the number of data points in our sample. Standard Formula Example To see how this shortcut formula works, we will consider an example that is calculated using both formulas. Suppose our sample is 2, 4, 6, 8. The sample mean is (2 4 6 8)/4 20/4 5. Now we calculate the difference of each data point with the mean 5. 2 – 5 -34 – 5 -16 – 5 18 – 5 3 We now square each of these numbers and add them together. (-3)2 (-1)2 12 32 9 1 1 9 20. Shortcut Formula Example Now we will use the same set of data: 2, 4, 6, 8, with the shortcut formula to determine the sum of squares. We first square each data point and add them together: 22 42 62 82 4 16 36 64 120. The next step is to add together all of the data and square this sum: (2 4 6 8)2 400. We divide this by the number of data points to obtain 400/4 100. We now subtract this number from 120. This gives us that the sum of the squared deviations is 20. This was exactly the number that we have already found from the other formula. How Does This Work? Many people will just accept the formula at face value and do not have any idea why this formula works. By using a little bit of algebra, we can see why this shortcut formula is equivalent to the standard, traditional way of calculating the sum of squared deviations. Although there may be hundreds, if not thousands of values in a real-world data set, we will assume that there are only three data values: x1 , x2, x3. What we see here could be expanded to a data set that has thousands of points. We begin by noting that( x1 x2 x3) 3 xÌ„. The expression ÃŽ £(xi - xÌ„)2 (x1 - xÌ„)2 (x2 - xÌ„)2 (x3 - xÌ„)2. We now use the fact from basic algebra that (a b)2 a2 2ab b2. This means that (x1 - xÌ„)2 x12 -2x1 xÌ„ xÌ„2. We do this for the other two terms of our summation, and we have: x12 -2x1 xÌ„ xÌ„2 x22 -2x2 xÌ„ xÌ„2 x32 -2x3 xÌ„ xÌ„2. We rearrange this and have: x12 x22 x32 3xÌ„2 - 2xÌ„(x1 x2 x3) . By rewriting (x1 x2 x3) 3xÌ„ the above becomes: x12 x22 x32 - 3xÌ„2. Now since 3xÌ„2 (x1 x2 x3)2/3, our formula becomes: x12 x22 x32 - (x1 x2 x3)2/3 And this is a special case of the general formula that was mentioned above: ÃŽ £(xi2)-(ÃŽ £ xi)2/n Is It Really a Shortcut? It may not seem like this formula is truly a shortcut. After all, in the example above it seems that there are just as many calculations. Part of this has to do with the fact that we only looked at a sample size that was small. As we increase the size of our sample, we see that the shortcut formula reduces the number of calculations by about half. We do not need to subtract the mean from each data point and then square the result. This cuts down considerably on the total number of operations.

Men Who Were Sent to Alcatraz Essay - 722 Words

Alcatraz began in 1850 as a military prison. Later it became a federal prison. Then on the verge of 30 years it was closed. Then for a while Indians used the island. Alcatraz was used to house the criminals who provoked problems at other jails. The island was located off the coast of California. The island has had many uses over a span of 122 years. Alcatraz was first used as a Military prison. They used it as a Military prison from 1850 to 1934. In 1934 it was changed to a federal prison. The Military cons were the first inmates of the prison. Then after roughly 30 years of federal prison use, Alcatraz was shut down due to deterioration and cost. The next to inhabit the island were Indians. In the Monte Cristo, a charter†¦show more content†¦The men’s uniforms were white and black or black and white shirts with blue pants. Also no women were in the jail, but some lived in the islands apartments, with their father or husband who worked in the jail. Some inmates would go years without seeing a woman or hearing one’s voice. The food and drink provided was on holidays the inmates were served fine meals such as stuffed celery, tom turkey roasted, pumpkin pie, fruit cake, and coffee. The inmates were fed 3 times a day and always had decent food. Most inmates said it was better than other federal prisons. But for drinks, the only fresh water on the island was rain and fog. Alcatraz experienced various deaths on the island. Eight inmates were murdered, 15 died of disease. Also five suicides took place; most men did not have time to attempt suicide because guards and officers were always walking the halls, also no knives, guns,Show MoreRelatedInformative Speech on Alcatraz1097 Words   |  5 PagesPurpose: To inform the class Specific Purpose: To describe to the audience a brief history of Alcatraz. Thesis: Alcatraz has been a popular social topic because of the mystery that surrounds it and the stories exaggerated in movies and television shows. INTRODUCTION I. If you disobey the rules of society, they send you to prison; if you disobey the rules of the prison, they send you to Alcatraz. II. Alcatraz has been a popular social topic because of the mystery that surrounds it and the storiesRead MoreInformative Speech Outline on Alcatraz1081 Words   |  5 Pages* Alcatraz Informative Speech Outline General Purpose: To inform the class Specific Purpose: To describe to the audience a brief history of Alcatraz. Thesis: Alcatraz has been a popular social topic because of the mystery that surrounds it and the stories exaggerated in movies and television shows. INTRODUCTION I. If you disobey the rules of society, they send you to prison; if you disobey the rules of the prison, they send you to Alcatraz. II. Alcatraz has been a popular social topicRead MoreInformative Speech Outline on Alcatraz Essay1090 Words   |  5 Pages* Alcatraz Informative Speech Outline General Purpose: To inform the class Specific Purpose: To describe to the audience a brief history of Alcatraz. Thesis: Alcatraz has been a popular social topic because of the mystery that surrounds it and the stories exaggerated in movies and television shows. INTRODUCTION I. If you disobey the rules of society, they send you to prison; if you disobey the rules of the prison, they send you to Alcatraz. II. Alcatraz has been a popular social topicRead More Alcatraz Essay1246 Words   |  5 Pagesisland standing in the strait. This rock is hidden by the fog and isolated by the chilling waters of the Pacific that flow in and out every day. It has a gloom that hangs about its rocky face most know it as Alcatraz but the men who experienced this island, referred to her as â€Å"The Rock†. 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Who cares if such work is illegal? You don’t think these nitwit coppers have their fingers in other pies? You can even see the weed bulging out of their pockets. And of course they keep it right next to theirRead MoreEssay on Al Capone Biography1495 Words   |  6 Pagesnamed Mary Coughlin who gave birth to their son Albert quot;Sonnyquot; Francis. Coughlin and Capone married later that year. He was first arrested on a disorderly conduct charge while working for fellow gangster Frankie Yale. At this time he also murdered two men to prove his willingness to kill, but he was not tried because of the gangland etiquette of â€Å"silence.† Capone was let off of all charges due to lack of proof. After Capone hospitalized a rival gang member, Yale sent him to Chicago until

Smoking Sure free essay sample

For many, what starts as a teenage attempt to fit in with their peers leads to a life long battle with smoking. One can remember the â€Å"Marlboro Man†, candy cigarettes, and let us not forget the Virginia Slim slogan, â€Å"You’ve Come a Long Way Baby†. Yes, smoking sure has come a long way. The smoking hay day of the 60’s, 70’s and 80’s is gone. One is now ostracized in society if they make the choice to continue to smoke. Anti-smoking campaigns advertising the health associated risks with smoking and government mandated laws surrounding its use are making it harder to continue to smoke. All one needs to do is watch the television or read the newspaper and they will more than likely see a commercial touting the effects that smoking has caused an individual; or read an article about another ban that has been placed on public spaces. Can these anti-smoking campaigns influence one to quit smoking by giving them a more realistic view of the health related risks involved and will smoking bans place such an impediment on smoking that one just decides to quit? Tobacco and its use has been around for centuries, however the health associated risks with using tobacco have not always been recognized. We will write a custom essay sample on Smoking Sure or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page Some early Native Americans used tobacco as part of their religious rituals and by the late 1800’s mass production made it more widespread. The World Health Organization (WHO) has estimated that globally there are 1. 3 billion smokers and the number of tobacco users is anticipated to increase to 1. 7 billion by the year 2020 (â€Å"Smoking†). Tobacco is one of the most addictive legal substances on the market today, with nicotine being the most addictive ingredient found in cigarettes (â€Å"Smoking Bans†). In an article, â€Å"Crackdown on Smoking†, Clark reports that cigarettes â€Å"contain 4000 chemicals, including benzene, ammonia and hydrogen cyanide, many of which are poisonous or cancer-causing to otherwise healthy nonsmokers†. These chemicals can increase a smoker’s risk of heart, stroke, and lung diseases. Research regarding the risks that smoking causes continues to progress. The Center for Disease Control (CDC) continues to bring awareness to the tragedies that occur due to smoking: An estimated 443,000 Americans die each year as a result of smoking-related illnesses, and the CDC estimates that more than 8 million people in the United States currently have diseases caused by smoking, including many resulting from exposure to so-called secondhand or â€Å"passive† smoke. Smoking, the CDC asserts, is the leading cause of preventable death in the United States (â€Å"Smoking Bans†). Even if one chooses not to smoke they are not immune to the consequences smoking has; we are all affected by the choices others make when choosing to smoke. An overwhelming $193 billion dollars was spent on healthcare in 2004 due to smoking (â€Å"Smoking Bans†). One can remember when smoking was an acceptable part of society. Smoking was considered sexy and an indication of high stature if one smoked. During the 1960’s it was not uncommon to see people smoking in restaurants and bars, smoking was even acceptable in the workplace. It’s hard to imagine it now but doctors even smoked while tending to their patients. But those times have changed due in large part to public awareness surrounding the use of tobacco. Public awareness with the risks associated with secondhand smoke has been the catalyst for laws to protect those from it. Clark emphasizes this stating: The impetus for the anti-smoking movement goes beyond the familiar science that links smoking to 434,000 annual deaths from heart disease, cancer, stroke and other maladies. The broader issue is passive smoke – also called secondhand smoke, side stream smoke, involuntary smoking and environmental tobacco smoke. The movement for change started in the 1960’s. Since the 1964 Surgeon General’s report â€Å"Smoking and Health†, the Federal Government has enacted laws throughout the years which have started the course for change for those who smoke. Some of the more significant laws were the 1965 Federal Cigarette Labeling and Advertising Act which required health warning labels on all cigarette packages, and in 1970 all television and radio cigarette advertising were banned. President Bill Clinton in 1996 authorized numerous restrictions which included banning the advertisement of cigarettes within a 1,000 foot radius of schools or playgrounds, and the  industry could no longer give away free cigarettes or promotional items that had a tobacco logo on them. Finally in 2009 Congress passed a bill which gave the FDA extensive control over the industry (â€Å"Smoking Bans†). With teenage smoking on the rise the 1970 advertising ban was in part an attempt to discourage underage smoking. Tobacco companies were targeting unde rage smokers with their savvy advertising ploys which presented an image of being cool or popular. Camel brand cigarettes even used a cartoon character â€Å"Joe Camel†, who became as widely recognized as Mickey Mouse, to help target these underage smokers. To counteract these marketing schemes, anti-smoking groups have established their own marketing tactics to present the long term effects that smoking causes in a more dramatic way. These campaigns show graphic images of people who used to smoke and the devastating consequences it has caused them. The message is clear; smoking is dangerous. Federal laws have had an impact on the anti-smoking movement, however; it is the state and local laws that have caused the biggest impact. These state laws have been the hardest on smokers by banning smoking in public places. New York City Mayor Michael Bloomberg, a huge advocate of the anti-smoking movement has seen how effective these laws can be. In 2002 he signed into law the elimination of smoking in the workplace, raised cigarette taxes, and ran anti-smoking advertising campaigns to educate the public. He did not just impose laws though, he also offered help to those who wanted it by giving away free nicotine patches so those who wanted to quit had a resource to help them. Bloomberg reports that â€Å"there are 300,000 fewer smokers in New York City than there were six years ago†. In partnership with WHO, Bloomberg has developed a strategy called MPOWER, this strategy has six solutions that have proved to save lives: monitor tobacco use and prevention policies, protect people from second-hand smoke, offer to help people quit, warn about the dangers of tobacco, enforce bans on tobacco advertising, promotion and sponsorship, and raise taxes on tobacco. In conclusion, one can see that it is becoming increasingly difficult to continue the choice to smoke. New laws and bans are taking away a person’s right to freely smoke where they wish and the government and anti-smoking crusaders will continue this fight until the battle on smoking is won. This is not an issue that will easily be solved and we must all work together to find the solution to this controversial issue. Bloomberg sums it up by stating, â€Å"Tobacco has become the world’s leading cause of death. If we do nothing, tobacco may kill 1 billion people by the end of this century. But only if we do nothing. Skeptics say that the problem of tobacco use is too culturally entrenched to solve. But part of taking on an entrenched problem-whether it’s in health or education or public safety-involves challenging people’s expectations of what is possible†. Anything is possible if we all work together. One must decide, is this a challenge worth taking?

The Merchant Of Venice - Anti-Semitism Essays - Shylock,

The Merchant of Venice - Anti-Semitism William Shakespeare's The Merchant of Venice contains many examples that insult Jews because they were the minority in London in Shakespeare's time. Although many parts of the play could be interpreted as offensive in modern times, Elizabethan audiences found them comical. The majority of London's population at the time was anti-Semitic because there were very few Jews living there. Shakespeare's The Merchant of Venice supports anti-Semitism actions and thoughts and therefore proves that Shakespeare was an anti-Semite. In the second act, Launcelot is debating with himself whether or not he should seek a new employer. Launcelot's problem is that he works for Shylock, who is Jewish. Launcelot persuades himself that, ?Certainly the Jew [Shylock] is the very devil incarnation (2.2.24) Eventually, Launcelot convinces himself that he would much rather run away than be ruled by a Jew. Launcelot presents this argument to his father: ?I am a Jew if I serve the Jew any longer.? (2.2.104) Before Launcelot accepts a new job with Bassanio as his master, he is reminded that Bassanio is much poorer than Shylock. His reply to Bassanio was, ?You have the grace of / God, sir, and he [Shylock] hath enough.? (2.2.139-40) Lorenzo insults Shylock behind his back when he tells Jessica (Shylock's daughter) that if Shylock ever makes it to heaven, it is only because Jessica converted to Christianity. Lorenzo said, ?If e'er the Jew her father come to heaven, / It will be for his gentle daughter's sake (2.4.36-7) When Lorenzo says this, he is implying that Shylock's faith and his Jewish heritage is not strong enough to get him into heaven. Lorenzo says that if Shylock is saved, it is by his Jessica's sake, because she has chosen Christianity over Judaism. This statement implies that Lorenzo believes that Christianity is the religion that is powerful enough to admit one into heaven; therefore Lorenzo is biased against anyone that is not a Christian, such as Shylock the Jew. Later in the play Jessica is insulted by Launcelot. Launcelot believes that parents' sins are passed down to their children. He also believes that being Jewish is a sin. Launcelot frankly tells Jessica his opinion of her: ?For truly I think you are damned.? (3.5.5) He explains that she is damned by both her father and mother because she was born Jewish. Launcelot tells Jessica that her only hope is that, ?Your father got / you not--that you are not the Jew's daughter.? (3.5.9-10) When Jessica hears this, she defends herself by telling Launcelot, ?I shall be saved by my husband. He hath made / me a Christian.? (3.5.17-8) This conversation between Launcelot and Jessica supports that Shakespeare was anti-Semitic because it states that Jews can only be saved by becoming Christians. Throughout The Merchant of Venice, Shakespeare uses examples of anti-Semitism. An Elizabethan audience would have perceived these situations as humorous because it was their reality. But to a Semite or a modern audience, this play may seem offensive and attacking. Launcelot decided to find a new master because he thought that being around Jews was dangerous to his health and mind. Lorenzo stated that the only way Shylock would get to heaven is by his gentle daughter (because she converted to Christianity). And Launcelot told Jessica that she was damned because she was born Jewish. The three discussed are only a few of the insulting situations that Shakespeare presents for Jews and modern audiences in The Merchant of Venice that prove that he was anti-Semitic.