Normal distribution central limit theorem

The random manner by whieh the inherent inaeeuraeies within the proeess are generated produees a pattern of variation for the dimension that resembles the Normal distribution, as diseussed in Chapter 2. As a first supposition then in the optimization of a toleranee staek with number of eomponents, it is assumed that eaeh eomponent follows a Normal distribution, therefore giving an assembly toleranee with a Normal distribution. It is also a good approximation that if the number of eomponents in the staek is greater than 5, then the final assembly eharae-teristie will form a Normal distribution regardless of the individual eomponent distributions due to the central limit theorem (Misehke, 1980). 

The Central Limit Theorem gives an a priori reason for why things tend to be normally distributed. It says the sum of a large number of independent random distributions having finite means and variances is normally distributed. Furthermore, the mean of the resulting distribution the sum of the individual means the combined variance is the sum of the individual variance.. ... 

Our next result concerns the central limit theorem, which places in evidence the remarkable behavior of the distribution function of when n is a large number. We shall now state and sketch the proof of a version of the central limit theorem that is pertinent to sums of identically distributed [p0i(x) = p01(a ), i — 1,2, ], statistically independent random variables. To simplify the statement of the theorem, we shall introduce the normalized sum s defined by... 

The central limit theorem thus states the remarkable fact that the distribution function of the normalized sum of identically distributed, statistically independent random variables approaches the gaussian distribution function as the number of summands approaches infinity—... 

The physical and conceptual importance of the normal distribution rests on one unique property the sum of n random variables distributed with almost any arbitrary distribution tends to be distributed as a normal variable when n- oo (the Central Limit Theorem). Most processes that result from the addition of numerous elementary processes therefore can be adequately parameterized with normal random variables. On any sort of axis that extends from — oo to + oo, or when density on the negative side is negligible, most physical or chemical random variables can be represented to a good approximation by a normal density function. The normal distribution can be viewed a position distribution. 

Central limit theorem if n independent variates have finite variances, their sum will tend to be normally distributed as n increases. 

From Eq. (A4) it follows that the limiting case a = 2 corresponds to the Gaussian normal distribution governed by the central limit theorem. For / = 0, the distribution is symmetric, y translates the distribution, and c is a scaling factor for X. Thus, y and c are not essential parameters if we disregard them, the characteristic function fulfills... 

Based on central limit theorem [3] the average X has a normal distribution n(p,Ox jn), or... 

One conclusion derived from the central limit theorem is that for large samples the sample mean X is normally distributed about the population mean p with var-iance ax, even if the population is not normally distributed. This means that we can almost always presume that X is normally distributed when we are trying to estimate or make a test on p, providing we have a large sample. 

The mean of the x distribution is k, and the variance is 2k. Because x is the sum of identically distributed variables, its distribution is asymptotically normal, as shown by the central limit theorem. This can be seen in the accompanying figure for large values of k. For large values of k, we can write ... 

There is no a priori reason to doubt that the Central Limit Theorem, and consequently the normal distribution concept, applies to trace element distribution, including Sb and Ba on hands in a human population, because these concentrations are affected by such random variables as location, diet, metabolism, and so on. However, since enough data were at hand (some 120 samples per element), it was of interest to test the normal distribution experimentally by examination of the t-Distribution. The probability density plots of 0.2 and 3 ng increments for Sb and Ba, respectively, had similar appearances. The actual distribution test was carried out for Sb only because of better data due to the more convenient half life of 122Sb. After normalization, a "one tail" test was carried out. 

The exact solutions are not valid if any of the model inputs differ from the distribution type that is the basis for the method. For example, the summation of lognormal distributions is not identically normal, and the product of normal distributions is not identically lognormal. However, the Central Limit Theorem implies that the summation of many independent distributions, each of which contributes only a small amount to the variance of the sum, will asymptotically approach normality. Similarly, the product of many independent distributions, each of which has a small variance relative to that of the product, asymptotically approaches lognormality. 

It is common to measure only a small number of objects or aliquots, and so one has to rely upon the central limit theorem to see that a small set of data will behave in the same manner as a large set of data. The central limit theorem states that as the size of a sample increases (number of objects or aliquots measured), the data will tend towards a normal distribution. If we consider the following case ... 

According to the important theorem known as the central limit theorem, if N samples of size n are obtained from a population with mean, fi, and standard deviation, a, the probability distribution for the means will approach the normal probability distribution as N becomes large even if the underlying distribution is nonnormal. For example, as more samples are selected from a bin of pharmaceutical granules, the distribution of N means, x, will tend toward a normal distribution with mean /j and standard deviation <7- = a/s/n, regardless of the underlying distribution. 

As we saw in Section 3.1.1, the familiar bell-shaped curve describes the sampling distributions of many experiments. Many distributions encountered in chemistry are approximately normal [3], Regardless of the form of the parent population, the central limit theorem tells us that sums and means of samples of random measurements drawn from a population tend to possess approximately bell-shaped distributions in repeated sampling. The functional form of the curve is described by Equation 3.19. 

The great majority of statistical procedures are based on the assumption of normality of variables, and it is well known that the central limit theorem protects against failures of normality of the univariate algorithms. Univariate normality does not guarantee multivariate normality, though the latter is increased if all the variables have normal distributions in any case, it avoids the deleterious consequences of skewness and outliers upon the robustness of many statistical procedures. Numerous transformations are also able to reduce skewness or the influence of outlying objects. 

A normal distribution for experimental errors is frequently encountered in physical chemistry and can be shown to be expected theoretically by means of the central limit theorem of Liapounov, assuming that the errors result from several small factors, which is often the case. 

Given a regular fraction of a 2f experiment and independent response variables, the estimators described above have constant variance even if the individual response variables do not. Also, the estimators are approximately normally distributed by the Central Limit Theorem. However, if the response variables have unequal variances, this unfortunately causes the estimators to be correlated and, therefore, dependent. The use of the data analysis to assess whether the levels of some factors affect the response variability is, itself, a problem of great interest due to its role in robust product design see Chapter 2 for a discussion of available methods and analysis. In the present chapter, we consider situations in which the estimators are independent. 

Closer inspection of Equation A1.4 shows that substances with a high expected risk ratio (nE//iRfD) contribute most to the uncertainty (or variance) in the HI. If 1 or 2 components dominate the mixture, it seems sufficient to base the uncertainty assessment on these dominant components. However, mixtures are often dominated by more than 2 components. Furthermore, the covariance between the individual risk ratios should not be ignored, since exposure estimates (E,) of individual mixture components can be (positively) correlated, as well as their reference values IA>fDr). The uncertainty in the HI may be severely underestimated if these correlations are not accounted for, which is evident from the last part of Equation A1.4. The central limit theorem states that the final HI will approach a normal distribution when the number of substances in the mixture becomes large or if no single risk ratio dominates the sum (De Groot 1986). 

Because it is unlikely that the risk estimates of substances with an independent mode of action are correlated, the covariance can be ignored and the overall variance is simply calculated by adding the variances of the individual components. For a many-compound mixture, the central limit theorem states that the uncertainty of the mixture risk is approximately normally distributed. At high response levels, the uncertainty in the mixture risk depends not only on the variance in the individual responses, but also on the variances of the response products. Calculation procedures are available but are rather complex (Mood et al. 1974). 

The usual assumptions leading to the normal error probability function are those required for the validity of the central limit theorem The assumptions leading to this theorem are sufficient but not always altogether necessary the normal error probability function may arise at least in part from circumstances different from those associated with the theorem. The factors that in fact determine the distribution are seldom known in detail. Thus it is common practice to assume that the normal error probability function is applicable even in the absence of valid a priori reasons. For example, the normal error probability function appears to describe the 376 measurements of Fig. 3 quite well. However, a much larger number of measurements might make it apparent that the true probability function is slightly skewed or flat topped or double peaked (bimodal), etc. 

Actually, least squares is often applied in cases where it is not known with any certainty that measurements of jy conform to a normal distribution or even when it is in fact known that they do not conform to a normal distribution. Does this destroy the applicability of the maximum-likelihood criterion The answer is, not necessarily. The central-limit theorem is discussed briefly in Chapter 11. Simply stated, it says that the sum (or average) of a large number of measurements conforms very nearly to a normal distribution, regardless of the distributions of the individual measurements, provided that no one measurement contributes more than a small fraction to the sum (or average) and that the variations in the widths of the individual distributions are within reasonable bounds. (As we shall see, the average of a group of numbers is a special case of a least-squares determination.)... 

Chapter 4 begins the study of statistical methods and their role in process investigations. Concepts introduced include sample spaces, probability densities and distributions, the central limit theorems of DeMoivre and Lindeberg, the normal distribution, and its extension to multivariate... 

Equations (4.3-4) and (4.3-5) are the first of several important limit theorems that establish conditions for asymptotic convergence to normal distributions as the sample space grows large. Such results are known as central limit theorems, because the convergence is strongest when the random variable is near its central (expectation) value. The following two theorems of Lindeberg (1922) illustrate why normal distributions are so widely useful. 

These central limit theorems are also relevant to physical models based on random processes. These theorems tell us that normal distributions can arise in many ways. Therefore, the occurrence of such a distribution tells very little about the mechanism of a process it indicates only that the number of random events involved is large. 

A very important theorem in statistics, the central limit theorem, states that as sample size increases, the distribution of a series of means from any frequency distribution will become normally distributed. This fact can be used to devise an experimental or sampling strategy that ensures that data are normally distributed, i.e. using means of samples as if they were primary data. 

The standard normal distribution function, with standard deviation equal to 1 and mean equal to zero, is presented in Figure 3.1. The confidence interval defined for an experimental set of data Xi,..., xn by fix ctx means that there is a 68.26 percent probability (see Section 3.1.4) that the correct value lies within the interval. There is a 95.44 percent probability that the correct value lies within the interval confidence interval fix 2ax- The Central-Limit Theorem described in Section 3.1.5 is often invoked to justify using the normal distribution as a basis for interpreting experimental data. 

The Central-Limit Theorem states that the sampling distribution of the mean, for any set of independent and identically distributed random variables, will tend toward the normal distribution, equation (3.17), as the sample size becomes large. ... 

Remember 3.1 The Central-Limit Theorem is often invoked to justify using the normal distribution as a basis for interpreting experimental data. 

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