The Importance of the Central Limit Theorem

The central limit theorem is a result from probability theory. This theorem shows up in a number of places in the field of statistics. Although the central limit theorem can seem abstract and devoid of any application, this theorem is actually quite important to the practice of statistics. So what exactly is the importance of the central limit theorem? It all has to do with the distribution of our population. This theorem allows you to simplify problems in statistics by allowing you to work with a distribution that is approximately normal. Statement of the Theorem The statement of the central limit theorem can seem quite technical but can be understood if we think through the following steps. We begin with a simple random sample with n individuals from a population of interest. From this sample, we can easily form a sample mean that corresponds to the mean of what measurement we are curious about in our population. A sampling distribution for the sample mean is produced by repeatedly selecting simple random samples from the same population and of the same size, and then computing the sample mean for each of these samples. These samples are to be thought of as being independent of one another. The central limit theorem concerns the sampling distribution of the sample means. We may ask about the overall shape of the sampling distribution. The central limit theorem says that this sampling distribution is approximately normal—commonly known as a bell curve. This approximation improves as we increase the size of the simple random samples that are used to produce the sampling distribution. There is a very surprising feature concerning the central limit theorem. The astonishing fact is that this theorem says that a normal distribution arises regardless of the initial distribution. Even if our population has a skewed distribution, which occurs when we examine things such as incomes or people’s weights, a sampling distribution for a sample with a sufficiently large sample size will be normal. Central Limit Theorem in Practice The unexpected appearance of a normal distribution from a population distribution that is skewed (even quite heavily skewed) has some very important applications in statistical practice. Many practices in statistics, such as those involving hypothesis testing or confidence intervals, make some assumptions concerning the population that the data was obtained from. One assumption that is initially made in a statistics course is that the populations that we work with are normally distributed. The assumption that data is from a normal distribution simplifies matters  but seems a little unrealistic. Just a little work with some real-world data shows that outliers, ​skewness, multiple peaks and asymmetry show up quite routinely. We can get around the problem of data from a population that is not normal. The use of an appropriate sample size and the central limit theorem help us to get around the problem of data from populations that are not normal. Thus, even though we might not know the shape of the distribution where our data comes from, the central limit theorem says that we can treat the sampling distribution as if it were normal. Of course, in order for the conclusions of the theorem to hold, we do need a sample size that is large enough. Exploratory data analysis can help us to determine how large of a sample is necessary for a given situation.

On Confucius and Confucianism - 1665 Words

Name Date Confuscius was a man of hard work and calm demeanor. As a child he grew up in poverty and became disgusted with the way the rich lived their everyday lives. As he grew older and learned more of the world, his wisdom grew. His wisdom grew so much so that his students wrote down, through Analects, his words. It is through these words one might suggest Confucius was a sage, or a perfect man. That was not the case. His life was that of a hardwoking individual who lived and died. He himself was not one to claim sainthood or sage status. Confucius did not regard himself a sage or saint but a junzi (Ã¥ â€ºÃ¥ ­  man/men of great virtue; also called daren Ã¥ ¤ §Ã¤ º º). Junzi are neither perfect nor infallible. They make mistakes. ( Confucian Thought on Heaven and Humanity Chinese Handicrafts and Traditional Chinese Arts, n.d., p. 1) He knew his place in the world was not about title or prestige, but about living a life of kindness, caution, and good faith. It was through this he believed one could attain a state of heaven. As mentioned before, the rich was often a concept Confuscius met with disdain. He believed the rich gained their wealth through dishonrable means. If there were an honorable way to get rich, Id do it, even if it meant being a stooge standing around with a whip. But there isnt an honorable way, so I just do what I like. (Waley, 2013, Chapter 12 Book 7) He did not mind the idea of being rich, but he wanted to pursue it in a way that preseved his integrity.Show MoreRelated Confucius and Confucianism Essay788 Words   |  4 Pagesthe most famous people in ancient China was a wise philosopher named Confucius (circa 551-479 BC). He was a great Chinese philosopher, educator, and political figure. His teachings and proverbs are in the Analects, which is the base of many Chinese theories. He was born in the village of Zou, and the state of Lu. His father died three years after Confucius birth leaving the family in destitution, but Confucius did have a decent education. He was married at the age of 19 and had Read MoreConfucianism: Morality and Confucius1356 Words   |  6 Pagesprescription? Do you think it’s a good one? If you agree, explain why? After studying the Confucius theory of human nature, my outlook and view on many things have either been changed or at least question. I am left to wonder how being born into this belief system may impact an individual views of themselves, and the world around them. Overall Confucianism is a very practical belief system. Confucius seemed very optimistic about humans, in that the species could be improved if they trusted hisRead MoreConfucianism : A Great Chinese Thinker And Philosopher Who Lived During The Warring States1206 Words   |  5 PagesSarthak Singh Mr Rainey 18 June 2015 Ancient China Confucianism Who was Confucius? 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Although Confucianism is no long as prominent as it was during the times of ancient China, it is important to understand these concepts and how later Chinese Dynasty adopted it. Confucianism has developed and influenced modern China and also other East AsianRead MoreConfucianism Is An Eastern Asia Religion / Philosophy1071 Words   |  5 PagesConfucianism is an Eastern Asia religion/philosophy. Described as a way of life. Confucianism is neither said to be a monotheistic nor a polytheistic religion. This religion originated in China during the 6th-5th century BCE, and has since then spread to countries such as Korea, Taiwan and Vietnam. Confucianism which means â€Å"The School of the Scholars† developed from the teachings of Confucius. It is more a system of ethics and social behavior. People who follow the teachings of Confucius also observeRead MoreEssay about Confucianism1154 Words   |  5 Pages Confucianism Confucianism is an East Asian philosophy built up by a Chinese philosopher, Confucius (Kung Fu-tzu), in the 6th - 5th century BCE. It’s a philosophy based on human morals and relationships. To really understand Confucianism, how it originated and what parts of history contributed towards its development, you have to look into: who is Confucius, the history of Confucianism and what happened to Confucianism after the death of Confucius? Who is Confucius? Confucius was of noble descentRead MoreThe Impact of Confucius on the Development of Chinese Thought and Culture1528 Words   |  7 PagesThe impact Confucius had on the development of Chinese thought and culture Confucianism has been a part of Chinese culture for over a thousand years. Many who have studied Confucianism would say that it is not a religion. It is better described as a philosophy or moral code. The philosophy of Confucianism comes mainly from the speeches and writings of Confucius, a great Chinese thinker and educator. He believed that Humanity, Rite, Neutrality, Virtue, Education, and Cultivation were the basisRead MoreConfucianism Is An Ethical And Philosophical System1226 Words   |  5 PagesConfucianism is an ethical and philosophical system, originated from the teachings of the Chinese philosopher Confucius (551–479 BCE). Often viewed as a religion, Confucianism can be understood as a way of thinking or living. Confucius thought the world was made of orderly hierarchies in both of family and government, and that people at the top of the hierarchy should rule with kindness to win the respect of their people. Where Legalism emphasizes law. The law rewards who ob eys and punish who break

Richard Corey Poem Analysis free essay sample

Richard did not seem to have the same feeling, because at the end of the poem he shoots and kills himself. This poem sends the message that the outside appearance of a person might not always reflect what is on the inside. The people of the town envied Richard and thought his life was more than perfect. They held him to a higher set of standards as if he was more than just a human being. This was made clear throughout the whole poem especially in the third stanza, quoted, â€Å"in fine, we thought the he was everything to make us wish that we were in his place† (11, 12). It seems that the envy of the people of the town made Richard more of an outcast of the town rather than a part of it. At the beginning of the poem when it talks about how the people stop and stare when Richard Cory went in town definitely creates the first part of imagery in the poem. We will write a custom essay sample on Richard Corey Poem Analysis or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page The people stop and watch him as if he is something they have not seen before, almost as if he is a god. They seemed to be shocked that when Richard spoke he was in fact human. He seemed to make people nervous just Czinke 2 being in their presence. This was portrayed when the poem states how Richard Cory fluttered pulses when he would say good-morning. This shows just how awkward the town’s people must have made him felt if his pulse is fluttering just to say a simple good-morning. Simply Richard did not find his life anything to be envious of; to him it was nowhere near perfect. The town’s people seem to put Richard above everyone else because of his lifestyle and having so much money. In the third stanza it is said, â€Å"and he was rich, yes richer than a king† (9). This part of the poem shows how there is an over exaggeration of how rich he really was, which is an example of a hyperbole. In the first stanza it says, â€Å"He was a gentleman from sole to crown† (3). This line in the poem is an example of a connotation. There is a higher meaning to this quote, not only does Richard Cory live like a king but maybe his soul and inner being is viewed like a king as well. The people seem to believe living as a king like the see Richard Cory does that he must have true happiness. Even though the people seem to struggle each day they cannot imagine that Richard can possibly have anything to struggle about in his life. The poem shows the peoples struggle in the very last stanza where it says, â€Å"so on we worked, and waited for the light, and went without the meat, and cursed the bread† (13, 14). Money might be able to buy food but for Richard money does not buy his own happiness. Sadly Richard is far from happy and on one summer night Richard puts a bullet through his head. The outer appearance of Richard Cory is in no way a reflection of his inner struggles. This whole poem is about Richard Cory through the eyes of the people of the town. Clearly what they saw and what they viewed Richards Cory’s life as seemed to be the total opposite of how Czinke 3 Richard actually felt. They envied Richard so much and set him to such a high standard that they ended up making him an outcast in the town. It does not make sense how a person could be happy when they were so isolated from everyone else. The people see that Richards Cory’s life is perfect. Richard may have seemed like a king and to have everything, however that is not the case. At the end of the poem it is clear Richard is not happy. While the true reason seems to be a mystery exactly why he shot himself Richard does end up killing himself, showing he is not as he appeared to be. Robinson, Edwin. â€Å"Richard Cory. † Literature: Reading, Reacting, Writing, Karen Mauk, Wadsworth, 2010. 1130-1131. â€Å"Print. †