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# Topic: Central limit theorem

###### In the News (Thu 25 Apr 19)

 Central limit theorem in Statistics The central limit theorem is one of the most remarkable results of the theory of probability. In its simplest form, the theorem states that the sum of a large number of independent observations from the same distribution has, under certain general conditions, an approximate normal distribution. By the central limit theorem, the distribution of the total number of heads will be, to a very high degree of approximation, normal. www.stattucino.com /berrie/clt.html   (350 words)

 PlanetMath: Lindeberg's central limit theorem Gauss derived the normal distribution, not as a limit of sums of independent random variables, but from the consideration of certain “natural” hypotheses for the distribution of errors; e.g. Nowadays, the central limit theorem supports the use of normal distribution as a distribution of errors, since in many real situations it is possible to consider the error of an observation as the result of many independent small errors. This is version 16 of Lindeberg's central limit theorem, born on 2002-12-10, modified 2006-06-28. planetmath.org /encyclopedia/LindebergsCentralLimitTheorem.html   (358 words)

 Central limit theorem Summary But, this limit is just the characteristic function of a standard normal distribution, N(0,1), and the central limit theorem follows from the Lévy continuity theorem, which confirms that the convergence of characteristic functions implies convergence in distribution. Thus the central limit theorem can be interpreted as a statement about the properties of density functions under convolution: the convolution of a number of density functions tends to the normal density as the number of density functions increases without bound, under the conditions stated above. There are some theorems which treat the case of sums of non-independent variables, for instance the m-dependent central limit theorem, the martingale central limit theorem and the central limit theorem for mixing processes. www.bookrags.com /Central_limit_theorem   (1852 words)

 Central Limit Theorem The Central Limit Theorem states that whenever a random sample of size n is taken from any distribution with mean µ; and variance, then the sample mean will be approximately normally distributed with mean µ; and variance /n. The central limit theorem demonstrates that in large enough samples, the distribution of a sample mean approximates a normal curve, amazingly, regardless of the shape of the distribution from which it is sampled. Abstract: A general central limit theorem is proved for estimators defined by minimization of the length of a vector-valued, random criterion function. www.lycos.com /info/central-limit-theorem.html   (362 words)

 Sampling Theory - SAMPLING DISTRIBUTIONS - CENTRAL LIMIT THEOREM   (Site not responding. Last check: 2007-09-06) The central limit theorem states that the sampling distribution of the mean, for any set of independent and identically distributed random variables, will tend towards the normal distribution as the sample size gets larger. The central limit theorem is one of the most important theorems in the field of probability as well as statistical inference, as it justifies the use of the normal curve in a wide range of statistical applications, both theoretical and practical. The normal approximation to the binomial distribution is a special case of the central limit theorem, where the independent random variables are Bernoulli variables with parameter p. library.thinkquest.org /10030/8stsdclt.htm   (188 words)

 Sampling Theory - SAMPLING DISTRIBUTIONS - CENTRAL LIMIT THEOREM   (Site not responding. Last check: 2007-09-06) The central limit theorem states that the sampling distribution of the mean, for any set of independent and identically distributed random variables, will tend towards the normal distribution as the sample size gets larger. The central limit theorem is one of the most important theorems in the field of probability as well as statistical inference, as it justifies the use of the normal curve in a wide range of statistical applications, both theoretical and practical. The normal approximation to the binomial distribution is a special case of the central limit theorem, where the independent random variables are Bernoulli variables with parameter p. library.advanced.org /10030/8stsdclt.htm   (188 words)

 The Central Limit Theorem   (Site not responding. Last check: 2007-09-06) Example #1 illustrates the central limit theorem when the underlying distribution is normal or chi-square. Example #2 illustrates the central limit theorem when the underlying distribution is uniform, bowtie, right wedge, left wedge, and triangular. Exercise #1 requires you to discuss the central limit theorem when the underlying distribution is uniform, bowtie, right wedge, left wedge, and triangular. www.stat.wvu.edu /SRS/Modules/CLT/clt.html   (171 words)

 Lecture Notes 7 The real advantage of the central limit theorem is that sample data drawn from populations not normally distributed or from populations of unknown shape also can be analysised by using the normal distribution, because the sample means are normally distributed for sample sizes of n>=30. Since the central limit theorem states that sample means are normally distributed regardless of the shape of the population for large samples and for any sample size with normally distributed population, thus sample means can be analysised by using Z scores. The central limit theorem also applies to sample proportions in that the normal distribution approximates the shape of the distribution of sample proportion if (n x p) > 5 and [n (1 - p)] > 5, where p is the population proportion. business.clayton.edu /arjomand/business/l7.html   (1543 words)

 The Central Limit Theorem Roughly, the central limit theorem states that the distribution of the sum of a large number of independent, identically distributed variables will be approximately normal, regardless of the underlying distribution. The importance of the central limit theorem is hard to overstate; indeed it is the reason that many statistical procedures work. This is a slightly less general version of the central limit theorem, because it requires that the moment generating function of the underlying distribution be finite on an interval about 0. www.ds.unifi.it /VL/VL_EN/sample/sample5.html   (1031 words)

 The Central Limit Theorem Although the figures at right are for a population with a uniform distribution, the central limit theorem works well regardless of the population's distribution. The central limit theorem tells us that a sampling distribution always has significantly less wildness than the population it’s drawn from. Thanks to the central limit theorem, we can be sure that a mean or x-bar based on a reasonably large randomly chosen sample will be remarkably close to the true mean of the population. www.intuitor.com /statistics/CentralLim.html   (1999 words)

 Central Limit Theorem: Miscellaneous Wilson saw that, because of the central limit theorem, the structure factor distribution for a crystal structure with a sufficient number of atoms would have to be Gaussian. The conditions given improve on existing results in the literature in terms of the amount of dependence and heterogeneity permitted, and in particular, these appear to be the first such theorems in which virtually the same assumptions are sufficient for both modes of convergence. This article illustrates the central limit theorem via an example for which the computation can be done quickly by hand on paper, unlike the more computing-intensive example in the article titled illustration of the central limit theorem. www.lycos.com /info/central-limit-theorem--miscellaneous.html   (350 words)

 Central limit theorem For a theorem of such fundamental importance to statistics and applied probability, the Central limit theorem has a remarkably simple Proof using characteristic functions. But, this limit is just the characteristic function of a standard normal distribution, N(0,1), and the Central limit theorem follows from the Lévy continuity theorem, which confirms that the convergence of characteristic functions implies convergence in distribution. There are some theorems which treat the case of sums of non-independent variables, for instance the m-dependent Central limit theorem, the martingale Central limit theorem and the central limit theorem for mixing processes. central-limit-theorem.mindbit.com   (1001 words)

 The Central Limit Theorem In Exercise 1, you should have been struck by the fact that the density of the sample mean becomes increasingly bell-shaped, as the sample size increases, regardless of the shape of the density from which we are sampling. This theorem, known as the central limit theorem, is one of the fundamental theorems of probability. The central limit theorem implies that if the sample size n is "large," then the distribution of the sample mean is approximately normal, with the same mean and standard deviation as the underlying basic distribution. www.fmi.uni-sofia.bg /vesta/Virtual_Labs/sample/sample5.html   (767 words)

 Central Limit Theorem (1 of 2) The central limit theorem states that given a distribution with a mean μ and variance σ²;, the sampling distribution of the mean approaches a normal distribution with a mean (μ) and a variance σ²/N as N, the sample size, increases. The amazing and counter-intuitive thing about the central limit theorem is that no matter what the shape of the original distribution, the sampling distribution of the mean approaches a normal distribution. On the next page are shown the results of a simulation exercise to demonstrate the central limit theorem. davidmlane.com /hyperstat/A14043.html   (190 words)

 central limit theorem - HighBeam Encyclopedia In statistics, a theorem showing (roughly) that the sum of any sufficiently large number of unrelated variables tends to be distributed according to the normal distribution, irrespective of how the individual variables are distributed, except in certain special cases. Demonstrating the Central Limit Theorem using a TI-83 calculator. Sample sizes and the central limit theorem: the Poisson distribution as an illustration. www.encyclopedia.com /doc/1O87-centrallimittheorem.html   (590 words)

 Illustration of the Central Limit Theorem It was Lyapunov's analysis that led to the modern characteristic function approach to the Central Limit Theorem. the limiting log-characteristic function of the normalized sum of variables with the distribution function p(z) as n→∞ is that of a normal distribution with mean of zero and variance of 1/12. This is an instance of the Central Limit Theorem. www2.sjsu.edu /faculty/watkins/randovar.htm   (930 words)

 Central Limit Theorem For increasing sample size, n, the distribution of sample means approaches a normal distribution centered on the population mean with a decreasing variance (proportional to 1/n). This is true regardless of how values are distributed within a population and is the essence of the central limit theorem (more info). The Central Limit Theorem link found at Statistical JAVA under Statistical Theory has several interactive applets that can help your students visualize this. serc.carleton.edu /introgeo/teachingwdata/Statcentral.html   (215 words)

 Learning by Simulations: Central Limit Theorem The central limit theorem is considered to be one of the most important results in statistical theory. It states that means of an arbitrary finite distribution are always distributed according to a normal distribution, provided that the number of observations for calculating the mean is large enough. The central limit theorem is the reason why normal distributions are so frequent in nature. www.vias.org /simulations/simusoft_cenlimit.html   (148 words)

 Illustrations of the Central Limit Theorem   (Site not responding. Last check: 2007-09-06) After a certain point, you can't tell the difference between the curve that shows the sampling distribution and the curve that shows a normal distribution with that same mean and standard deviation. At that point, you could definitely use the Central Limit Theorem to approximate the sampling distribution of the sample mean by a normal distribution. It is stated in the conclusion of the Central Limit Theorem and can be proved mathematically. www2.austin.cc.tx.us /mparker/1342/cltdemos0.html   (480 words)

 CM: NetLogo ProbLab: Central Limit Theorem Central Limit Theorem is authored in the NetLogo modeling-and-simulation environment. Central Limit Theorem demonstrates relations between population distributions and their sample mean distributions as well as the effect of sample size on this relation. In this model, a population is distributed by some variable, for instance by their total assets in thousands of dollars. ccl.northwestern.edu /curriculum/ProbLab/CentralLimitTheorem.html   (846 words)

 Illustration of the Central Limit Theorem It was Lyapunov's analysis that led to the modern characteristic function approach to the Central Limit Theorem. the limiting log-characteristic function of the normalized sum of variables with the distribution function p(z) as n→∞ is that of a normal distribution with mean of zero and variance of 1/12. This is an instance of the Central Limit Theorem. www.sjsu.edu /faculty/watkins/randovar.htm   (930 words)

 Quality Practice Tips: Central Limit Theorem to the Rescue!   (Site not responding. Last check: 2007-09-06) A: Control chart construction and theory for x-bar, R and S charts are based on the assumption of normality of the points being plotted, not necessarily the underlying data. Fortunately, the Central Limit Theorem comes into play to make it possible to use an X-bar chart (of subgroup averages) with data from non-normal distributions. The Central Limit Theorem states that the distribution of subgroup means (averages) from any distribution will approach a normal distribution as the subgroup size increases. www.statit.com /support/quality_practice_tips/central_limit_theorem.htm   (426 words)

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