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Alex Nguyen Mr. Reppenhagen May 28th, 2014 IB Mathematics What is the Central Limit Theorem and what are its applications in statistics?

Table of Contents 1. 2. 3. 4. 5. 6.

Introduction Definition of Terms Explanation History of Central Limit Theorem Demonstration of Proof of Central Limit Theorem Applications a. Sampling b. Polls 7. Weakness 8. Conclusion 9. Works Cited

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Introduction The Central Limit Theorem has many versions, referring to a convergence of means of a probability distribution towards one single distribution: the normal distribution. The Classical Central Limit Theorem states that the arithmetic mean of a large number of independent and identically distributed random variables, each with well-defined mean and variance, will be normally distributed. In simpler terms, if we obtain a large sample, each individual in the sample being independent from one another and random, and calculate the means of these random variables, then the central limit theorem states that this distribution of means will be approximately bell-shaped, or distributed in a normal curve. The Classical Central Limit Theorem allows one to speculate on the probability distribution of the outcome (the mean) of a process (some random variables comprising an event) without knowing much about the nature of the events themselves other than the fact that they are identical and independent. Today, the central limit theorem is made abstract and its hypotheses weakened to allow for some cases in which the variables can be dependent on one another, which widens the scope and applicability of the theorem. The Central Limit Theorem is a surprising result in statistics and probability and is used constantly, which elevates the importance of the normal distribution in statistics and probability.

Definition of Key Terms First, let us define our terms. By a random variable we mean something that changes due to chance. A random variable can be domain of a probability density function. Each value of a random variable, whether discrete or continuous, is assigned a single “probability.” For example, the values of a die roll are values of a random variable. A mathematical function that assigns all possible values of the random variable an associated probability is called a probability distribution. All probability distribution has its area under the curve as one unit, because the chances of something in the sample space happening is certain. The probability distribution for one single die roll is a horizontal straight line. The mean or expected value of a random variable is the value of the variable we would expect if we repeat the random variable an infinite number of times and take the average of all the value. In a way, the mean or expected value is a weighted average of all possible values. The standard deviation is the square root of the root mean square or quadratic mean of the distances between all the values and the expected value.

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The normal distribution is a very commonly occurring continuous probability distribution in which the mean, median, and mode are the exactly the same. A normal distribution with mean deviation

and standard

is given by the equation:

The normal distribution is also called a Gaussian distribution. The value of a normal distribution is practically zero several standard deviations away from the mean. For example, 99.7% of values are present within 3 standard deviations of the mean. Therefore, extreme events are predicted to have very little chances of occurring, to due the exponential decay demonstrated on both sides.

The picture above is a normal distribution with the probability of values lying between each standard deviation from the mean. Then, we need to define what being “independent and identically distributed” means. The phrase is used to describe a collection of random variables. Random variables are independent from each other if one occurring does not alter the probability of the other. Random variables are identically distributed when they have the same probability distribution as the others. For example, the events of rolling the die repeatedly is independent and identically distributed because they have the same probability

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distribution (horizontal straight line) and are independent from one another (one die roll does not affect another). By the mean in this essay we mean the arithmetic mean, and by variance we mean the square of standard deviation.

Explanation The Central Limit Theorem essentially describes the characteristics of a population of means creating from the means of an infinite number of random population samples size N, all drawn from a parent population. The Central Limit Theorem specifically predicts that regardless of the distribution of the parent population, as long as the samples are random and independent from one another, that 1) The mean of a population of means is always equal to the mean of the parent population from which the samples are taken 2) The standard deviation of a population of means is always equal to the standard deviation of the parent population divided by the square root of the sample size. 3) The distribution will increasingly approach a normal distribution as the size N of sample increases. We know that different parts of the distribution converge differently to a normal distribution. The parts close to the mean converges quickly to the normal distribution but the tails converge more slowly to the normal. Therefore, we say that the central limit theorem gives an asymptotic distribution. It requires a large number of observations to stretch the convergence to the tails.

History of the Central Limit Theorem Many natural and social scientists in the 19th century has noticed a pattern in the means of these independent random variables. When the outcome (the means) is affected by a lot of random variables (high sample size) and when each variable only has a slight effect on the outcome as a whole, the mean is distributed in a certain way, regardless of the actual probability distribution of the random variables. Mathematically, however, it is an important and seemingly daunting problem, which requires a mathematician to draw conclusions about the outcome (the mean) from a set of random variables when little is known about the distribution of the various variables. It has been described as "one of the most

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remarkable results in all of mathematics" and "a dominating personality in the world of probability and statistics" (Adams, 1974, p. 2). It is also one of the earliest results of probability theory. The central limit theorem was named by the mathematician Georg Polya, from a paper in 1810 by the French mathematician Laplace, in which Polya recognized a number of theorem that eventually leads to the appearance of the normal distribution. Polya, drawing from Laplace’s foundations, named the theorems “central limit theorems” which is used widely today.

Proof of Central Limit Theorem While the proof of the central limit theorem is too advanced for the scope of this exploration, we can explore the heuristic behind the central limit theorem. The normal distribution satisfies a specific identity about itself

A random variable with a normal distribution of mean random variable distributed normally with mean distributed normally with mean

and standard deviation

and standard deviation

and another

will have a sum that is

and standard deviation √

In essence, normal distributions when added together yield normal distributions up to a degree of scaling. The equation

defines a normal distribution. These properties help us understand how we expect normalized means to converge to a normal distribution. Suppose that the population of means converge to a hypothetical distribution D. We have

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where

√

is simply the normalized sum. To find the mean we simply divide the sum by √ .

We would expect that

so D must be normal.

Applications of Central Limit Theorem Hypothesis testing draws strongly from the central limit theorem. The central limit theorem helps scientists who want to draw claims about a population that they are studying. For example, in areas of knowledge where there are variable behaviors such as in psychology, experts need to formulate hypotheses about a population and need to know the margin of error for their claims. They use statistical experiments to obtain sample data from the population. Information from the data, such as standard deviation, sample size, or the mean can be used to test for the accuracy of a specific hypothesis with regards to a population. Hypothesis testing that assumes data is just from a normal distribution seems unrealistic, because real world data shows outliers, skewing towards one side, multiple peaks and asymmetry. For examples, if we sample the world’s wealth, we have outliers such as Bill Gates or Warren Buffett, which are not taken into by the normal distribution because they are so far from the mean as to be virtually impossible by the normal distribution. We also have a skewed distribution towards poverty (there are more moderately poor people than there are moderately rich people). Therefore, it is unrealistic to treat all data as if they are normal. However, if we take the mean of such data, assuming that in all samples of data are formed of similar composition and assuming the sample size is large enough, we can put the data into a normal distribution.

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Sampling A sampling distribution of a statistic is the probability distribution of a given statistic. For example, if we have one sample we might want to know the mean of that one sample. However, when we repeat the experiment we take the mean of each sample as representative of that sample. This mean is called the sample mean. The sample distribution of the mean is a probability distribution of these means, because the means might differ from one sample to another. For example, realistically we can take samples in one geographical area. Perhaps one mean would be altered because one area is a rich neighborhood and yields a high average income per household. However this is not representative of the actual population. We need to take many samples. The sample distribution of the mean is the distribution of these various means. This is important for statisticians because when they make claims such as 90% of individuals earn an average of 30,000 dollars to to 70,000 dollars 19 out of 20 times, they want to specifically know if current statistical surveys will report the same findings. Sampling distributions of the means are used to generate confidence intervals for survey reports and for significance testing (testing the statistic to see if they actually describe the population). Therefore it is important to know how variable our estimates are. The central limit theorem, by generalizing these sample distributions of the mean into normal distributions, help us figure out specifically what the variability of these statistics are.

Polls An important effect of the central limit theorem affects how we read polls. For example, during election time we usually see polls that are taken to estimate the percentage of a population which supports a certain candidate for presidency. Since it is not possible to survey the entire population the pollsters have to survey only a certain proportion of the population. Suppose the pollsters survey a sample population of size n for their preferences. The preferences of the people in the sample can be represented as a sequence of random variables which are independent and presumably identically distributed. The pollsters sample the mean in the polls. The mean should be distributed normally. As the number of people surveyed increases, the mean of the sample distribution of means should be close to the population mean, which is the number reported in the polls.

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Weakness A weakness of the central limit theorem is the premise of independent and identically distributed variables. However, the theorem still holds if some of its assumptions (independent and identically distributed variables with finite mean and standard deviation) are violated. If the variables are weakly dependent on each other, the sample distribution of the mean converges less quickly to the normal distribution, which means that our estimates are less accurate than it is had the random variables been independent. The main thing to understand is that elementary mathematics is an elegant and simple starting point. Assumptions such as independent and identically distributed random variables are not usually found in the real world. In the same vein, assumptions in physics such as engines completely transforming heat into work do not exist in real life. In the real world, nonstationary processes whose probability distributions, mean, and variance shift as a function of time are commonly found. Therefore, the central limit theorem does not strictly apply to these situations.

Conclusion The central limit theorem is an important theorem in statistics and probability theory. Some mathematicians have called it one of the fundamental theorems of statistics. The theorem tells you how a population of means behave when the sample size approaches infinity. This greatly helps with surveys and help us compute how accurately a survey reflects the population. However, due to the various assumptions that the central limit theorem makes (independent and identically distributed), we can only use the theorem as a starting point. Therefore, the central limit theorem is important for students to know how to use, but we need to know its limitations.

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Works Cited Adams, W.J. (1974). The life and times for the Central Limit Theorem. New York: Kaedmon.

Blacher, René. "Central Limit Theorem by Moments." Statistics & Probability Letters 77.17 (2007): 1647-651. Dartmouth University. Web. 28 May 2014. "The Central Limit Theorem." Intuitor. Web. 28 May 2014. Clauset, Aaron. "Adapted Probability Distributions." Web. 28 May 2014. "Distribution, Normal." Encyclopedia.com. HighBeam Research, 01 Jan. 2008. Web. 28 May 2014. H., Krieger. "Proof of Central Limit Theorem." Harvey Mudd College, 2005. Web. 28 May 2014. Lane, David M. "Sampling Distribution (1 of 3)." Sampling Distribution (1 of 3). Web. 28 May 2014.

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