Central Limit Theorem
The Central Limit Theorem states that the sampling distribution of the sample mean will be approximately normally distributed, regardless of the shape of the original population distribution, as long as the sample size is sufficiently large. This theorem is a fundamental concept in statistics and is widely used in making inferences about population parameters based on sample data.
7 Key excerpts on "Central Limit Theorem"
eBook - ePub- Debbie L. Hahs-Vaughn, Richard Lomax(Authors)
- 2020(Publication Date)
- Routledge(Publisher)
...The Central Limit Theorem states that as sample size n increases, the sampling distribution of the mean from a random sample of size n more closely approximates a normal distribution. If the population distribution is normal in shape, then the sampling distribution of the mean is also normal in shape. If the population distribution is not normal in shape, then the sampling distribution of the mean becomes more nearly normal as sample size increases. This concept is graphically depicted in Figure 5.2. FIGURE 5.2 Central Limit Theorem for normal and positively skewed population distributions. The top row of Figure 5.2 depicts two population distributions, the left one being normal and the right one being positively skewed. The remaining rows are for the various sampling distributions, depending on the sample size. The second row shows the sampling distributions of the mean for n = 1. Note that these sampling distributions look precisely like the population distributions, as each observation is literally a sample mean. The next row gives the sampling distributions for n = 2; here we see for the skewed population that the sampling distribution is slightly less skewed. This is because the more extreme observations are now being averaged in with less extreme observations, yielding less extreme means. For n = 4 the sampling distribution in the skewed case is even less skewed than for n = 2. Eventually we reach the n = 25 sampling distribution, where the sampling distribution for the skewed case is nearly normal and nearly matches the sampling distribution for the normal case. This phenomenon will occur for other nonnormal population distributions as well (e.g., negatively skewed). The moral of the story here is a good one. If the population distribution is nonnormal, then this will have minimal effect on the sampling distribution of the mean except for rather small samples...
eBook - ePubFoundations of Statistics for Data Scientists
With R and Python
- Alan Agresti, Maria Kateri(Authors)
- 2021(Publication Date)
- Chapman and Hall/CRC(Publisher)
...As the sample size n increases, the standard error decreases, so the sample mean tends to be closer to the population mean. The Central Limit Theorem states that for large samples obtained with randomization, the sampling distribution of the sample mean is approximately a normal distribution. This holds no matter what the shape of the population distribution, both for continuous and discrete variables. The Central Limit Theorem applies also to proportions, since the sample proportion is a special case of the sample mean for observations coded as 0 and 1 (such as for two candidates in an election). Many other statistics also have an approximately normal sampling distribution for large n. For instance, the delta method shows this is true for many functions of statistics that themselves have an approximate normal distribution. The bell shape for the sampling distribution of many statistics is the main reason for the importance of the normal distribution. The next two chapters show how the Central Limit Theorem is the basis of methods of statistical inference. Exercises Data Analysis and Applications 3.1 In an exit poll of 2123 voters in the 2018 Senatorial election in Minnesota, 61.0% said they voted for the Democratic candidate Amy Klobuchar in her race against the Republican candidate Jim Newberger. Based on this information, if you could treat this exit poll like a simple random sample, would you be willing to predict the winner of the election? Conduct a simulation to support your reasoning. 3.2 In an exit poll of 1648 voters in the 2020 Senatorial election in Arizona, 51.5% said they voted for Mark Kelly and 48.5% said they voted for Martha McSally. Suppose that actually 50% of the population voted for Kelly...
eBook - ePubInterpreting Statistics for Beginners
A Guide for Behavioural and Social Scientists
- Vladimir Hedrih, Andjelka Hedrih(Authors)
- 2022(Publication Date)
- Routledge(Publisher)
...it is not guaranteed that all of the properties of the sample will be exactly like those of the population. On the contrary, it is quite likely that the properties of the sample might differ somewhat. This is why, when making inferences about the population, this possibility that the sample properties will be more or less different to those of the population has to be accounted for. That is one of the reasons, why we draw a distinction between statistical indicators calculated from the sample and those same indicators in the population, where the former are called statistics and the latter are called parameters. At the moment this book is written, two approaches to addressing the issue of assessing values of parameters based on values of statistics are in most common use: the approach based on the Central Limit Theorem and the approach based on the use of resampling, primarily on the use of bootstrapping. 5.1 The Central Limit Theorem The Central Limit Theorem postulates what happens when we take multiple samples from the same population, which is actually a typical situation in which research findings are verified – one study takes a sample from the population to be studied and reports its findings and, later, another study takes another sample from the same population and reports whether their findings match the findings reported by the first study. The Central Limit Theorem postulates that if we take a large number of random samples from a population and we then calculate the same statistic from each of these samples, the distribution of values of this statistic across these samples will approximate the normal distribution. When postulating this, it is also assumed that either the population in unlimited or very large compared to the sample (to be practically the same as unlimited) or that the sampling is done with replacement (so that probabilities of individual entities for being included in the sample remain constant throughout the sampling process)...
- Bruce B. Frey(Author)
- 2018(Publication Date)
- SAGE Publications, Inc(Publisher)
...The mathematical formulation of the theorem is attributed to the St. Petersburg School of probability, from 1870 until 1910, with Pafnuty Chebyshev, Andrey Markov, and Aleksandr Liapounov. Mathematical Formulation Let X 1,X 2,…,X n be independent random variables that are identically distributed, with mean μ and finite variance σ 2. Let Other denote the empirical average, then from the law of large numbers tends to 0 as n tends to infinity. The Central Limit Theorem establishes that the distribution of tends to a centered normal distribution when n goes to infinity. More specifically, Other We can also write Other or as n → ∞. A Limiting Result as an Approximation This Central Limit Theorem is used to approximate distributions derived from summing, or averaging, identical random variables. Consider for instance a course where 7 students out of 8 pass. What is the probability that (at least) 4 failed in a class of 25 students. Let X be the dichotomous variable that describes failure: 1 if the student failed and 0 if the student passed. That random variable has a Bernoulli distribution with parameter p = 1/8 (with mean 1/8 and variance 7/64). Consequently, if students’ grades are independent, the sum S n = X 1 +…+ X n follows a binomial distribution, with mean np and variance P (1 − p), which can be approximated, by the Central Limit Theorem, by a normal distribution with mean np and variance np (1 − p). Here, μ = 3.125 while σ 2 = 2.734. To compute P (S n ≤ 4), we can use the cumulative probabilities of either the binomial distribution or the Gaussian approximation. In the first case, the probability is 80.47%, Other In the second case, use a continuity correction and compute the probability that S n is less than 4 + 1/2. From the Central Limit Theorem: Other The probability that a standard Gaussian variable is less than this quantity is: Other which can be compared with 80.47% obtained without the approximation (see Figure 1)...
eBook - ePubStatistics for the Behavioural Sciences
An Introduction to Frequentist and Bayesian Approaches
- Riccardo Russo(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...Notice that the sampling distribution of the mean is normal, centred at zero, but now the variance is 0.0625 (i.e., σ 2 n = 0.25 4). In summary, if we want to test hypotheses about means we need to know the characteristics of the distribution of the sample means. As seen above, the Central Limit Theorem tells us that the sampling distribution of the mean is usually normal with μ x ¯ = μ and σ x ¯ = σ n, where µ and σ are the mean and the standard deviation of the parent population of individual observations from which the samples are drawn, and n is the sample size. 7.3 Testing hypotheses about means when σ is known It is usually uncommon to know the standard deviation of a population of scores, so the technique described in this section is of limited application, but it is still worth knowing since there are circumstances in which it can be successfully applied. For example, when a standardised test is applied to a sample of subjects, we then know the population standard deviation and the mean of the individual scores. Let us consider the example described in the Introduction. As stated earlier, we know that the distribution of the population of individual scores in a standardised test measuring reading speed is normal with µ = 200 words per minute and σ = 30. We also know that a random sample of 36 people achieved a mean performance of 218 words per minute in this test after having attended a crash course on how to read fast...
- George V Chilingar, Leonid F. Khilyuk, Herman H. Reike(Authors)
- 2012(Publication Date)
- Gulf Publishing Company(Publisher)
...It had been shown earlier that MI k = p and DI k = pq (q = 1 − p). The number of successes is given by v n = n −1 (I 1 + … + I n). To complete the proof, one can apply corollary 12.3. Central Limit TheoremS The term Central Limit Theorems (CLT) in the probability theory refers to the set of statements about the convergence of the distribution function of a sum of independent small random variables to the normal distribution function, provided that the number of random addenda tends to infinity. More accurately, the theorems of this kind state some conditions that guarantee that the sum of small independent random variables tends to the random variable with N (0, 1). These results are very important not only for probability theory and its applications, but also for general scientific perception of nature and society. They can be considered as a theoretical justification for a wide application of normal law for description of asymptotic behavior of the large assemblies of small independent random variables. The Bell Curve by R. Herrnstein and C. Murray (1994) provides many unexpected examples of manifestation of the above mentioned tendency in modern society. Central Limit Theorems is a generic term for a broad class of theorems. The authors present only the most important interpretations and applications here. Theorem 12.3. Central Limit Theorem for independent random variables with identical distributions Suppose that ξ 1, ξ 2, …, ξ n are mutually independent, identically distributed random variables with finite mathematical expectations M ξ k = a and variances D ξ k = σ 2 (k = 1, 2, …, n). Introduce two derivative random variables: and Then η n converges weakly to the random variable η∈ N (0, 1)...
eBook - ePub- Thomas Pfaff(Author)
- 2019(Publication Date)
- Chapman and Hall/CRC(Publisher)
...11 The Central Limit Theorem and Z-test In this chapter we provide two simulations to illustrate the Central Limit Theorem. We follow that up by demonstrating how to perform a z-test and calculate associated confidence intervals. For both, we show how to perform tests and compute confidence intervals when working from a data set and from summary statistics. The data we use in the chapter for the z-test and interval will be generated randomly from a normal distribution, while the simulations will use data generated from exponential and uniform distributions. The z.test and zsum.test in this chapter use the BSDA package. Please note that you should only load a package once in an R session, although the code below will have library(BSDA) whenever it is needed as a reminder that it is being used. 11.1 A Central Limit Theorem Simulation We demonstrate two Central Limit Theorem simulations, with the differences being the distribution we sample from and how the breaks are set in the histograms. Our first example begins by setting the graph frame, par(mfrow=c(2,2)), to accept four graphs in a 2-by-2 grid. We set trials=1000000, which is the size of the simulation. This isn’t necessary to do here, but it makes it easier to change the simulation size in one place in the code. We begin a for loop for values of i = 1, 10, 50, and 100, which will be the sizes of our samples for the distribution given two lines later. Note that the for loop begins with { and ends with }. Depending on your computer, this simulation size make take a few minutes to complete. The real work in the code is in the next line where simulations stores the output of replicate(trials,mean(rexp(i,0.25))). Unraveling this from the inside out, rexp(i,0.25) selects i random values from the exponential distribution with λ = 0.25, and mean(rexp(i,0.25)) calculates the mean of these values. The replicate(n,f) function repeats an operation f, n times. In our case we repeat mean(rexp(i,0.25)) 1000000 (trials) times...
