Normal distribution properties

When a small number of replicate analyses are performed their mean value, x is used to estimate the population mean, systematic errors (i.e., with unbiased measurements), a would be the true value of the analyte concentration. But because random errors occur x will not be exactly equal to a. We thus wish to define an interval within which a lies with a given degree of probability, and thanks to the central limit theorem we can apply normal distribution properties to the sampling distribution of the mean to do this. Remembering that the standard error of x is ajy/n (see above), the normal distribution shows that 95% of the values of x will lie within 1.96cr/v of the mean. That is, the 95% confidence limits of the mean are given by... 

It is important to verify that the simulation describes the chemical system correctly. Any given property of the system should show a normal (Gaussian) distribution around the average value. If a normal distribution is not obtained, then a systematic error in the calculation is indicated. Comparing computed values to the experimental results will indicate the reasonableness of the force field, number of solvent molecules, and other aspects of the model system. 

To predict the properties of a population on the basis of a sample, it is necessary to know something about the population s expected distribution around its central value. The distribution of a population can be represented by plotting the frequency of occurrence of individual values as a function of the values themselves. Such plots are called prohahility distrihutions. Unfortunately, we are rarely able to calculate the exact probability distribution for a chemical system. In fact, the probability distribution can take any shape, depending on the nature of the chemical system being investigated. Fortunately many chemical systems display one of several common probability distributions. Two of these distributions, the binomial distribution and the normal distribution, are discussed next. 

Interpreting Control Charts The purpose of a control chart is to determine if a system is in statistical control. This determination is made by examining the location of individual points in relation to the warning limits and the control limits, and the distribution of the points around the central line. If we assume that the data are normally distributed, then the probability of finding a point at any distance from the mean value can be determined from the normal distribution curve. The upper and lower control limits for a property control chart, for example, are set to +3S, which, if S is a good approximation for O, includes 99.74% of the data. The probability that a point will fall outside the UCL or LCL, therefore, is only 0.26%. The... 

A remarkable property of the normal distribution is that, almost regardless of the distribution of x, sample averages x will approach the gaussian distribution as n gets large. Even for relatively small values of n, of about 10, the approximation in most cases is quite close. For example, sample averages of size 10 from the uniform distribution will have essentially a gaussian distribution. 

Normal Distribution of Observations Many types of data follow what is called the gaussian, or bell-shaped, curve this is especially true of averages. Basically, the gaussian curve is a purely mathematical function which has very specif properties. However, owing to some mathematically intractable aspects primary use of the function is restricted to tabulated values. 

Many distributions occurring in business situations are not symmetric but skewed, and the normal distribution cui ve is not a good fit. However, when data are based on estimates of future trends, the accuracy of the normal approximation is usually acceptable. This is particularly the case as the number of component variables Xi, Xo, etc., in Eq. (9-74) increases. Although distributions of the individual variables (xi, Xo, etc.) may be skewed, the distribution of the property or variable c tends to approach the normal distribution. 

The data used to generate the maps is taken from a simple statistical analysis of the manufacturing process and is based on an assumption that the result will follow a Normal distribution. A number of component characteristics (for example, a length or diameter) are measured and the achievable tolerance at different conformance levels is calculated. This is repeated at different characteristic sizes to build up a relationship between the characteristic dimension and achievable tolerance for the manufacture process. Both the material and geometry of the component to be manufactured are considered to be ideal, that is, the material properties are in specification, and there are no geometric features that create excessive variability or which are on the limit of processing feasibility. Standard practices should be used when manufacturing the test components and it is recommended that a number of different operators contribute to the results. 

It has been argued that material properties sueh as the ultimate tensile strength have only positive values and so the Normal distribution eannot be the true distribution. 

Theoretically, the effects of the manufacturing process on the material property distribution can be determined, shown here for the case when Normal distribution applies. For an additive case of a residual stress, it follows that from the algebra of random variables (Carter, 1997) ... 

There is no data available on the endurance strength in shear for the material chosen for the pin. An approximate method for determining the parameters of this material property for low carbon steels is given next. The pin steel for the approximate section size has the following Normal distribution parameters for the ultimate tensile strength, Su ... 

If we plot a Normal distribution for an arbitrary mean and standard deviation, as shown in Figure 4, it ean be shown that at lcr about the mean value, the area under the frequeney eurve is approximately 68.27% of the total, and at 2cr, the area is 95.45% of the total under the eurve, and so on. This property of the Normal distribution then beeomes useful in estimating the proportion of individuals within preseribed limits. 

Property 1 indicates tliat tlie pdf of a discrete random variable generates probability by substitution. Properties 2 and 3 restrict the values of f(x) to nonnegative real niunbers whose sum is 1. An example of a discrete probability distribution function (approaching a normal distribution - to be discussed in tlie next chapter) is provided in Figure 19.8.1. 

The least squares estimator has several desirable properties. Namely, the parameter estimates are normally distributed, unbiased (i.e., (k )=k) and their covariance matrix is given by... 

When the Gauss-Newton method is used to estimate the unknown parameters, we linearize the model equations and at each iteration we solve the corresponding linear least squares problem. As a result, the estimated parameter values have linear least squares properties. Namely, the parameter estimates are normally distributed, unbiased (i.e., (k )=k) and their covariance matrix is given by... 

MINITAB readily produces many useful manipulations of data such as were obtained in this experiment. Figure 2 shows histograms of the responses, indicating that, for the limited number of data points, the experimental values for each response approach a normal distribution. Thus, the statistical analysis was considered valid. Table III shows a copy of the computer printout of a correlation table with all the responses. Clearly, Property A and Property B are negatively correlated, as predicted, but Property B and Property E are not well correlated. 

Each point in the MND can be projected onto the planes defined by each pair of the axes of the coordinate system. For example, Figure 1-2 shows the projection of the data onto the plane at the bottom of the coordinate system. There it forms a two-dimensional MND, which is characterized by several parameters, the two-dimensional MND being the prototype for all MNDs of higher dimension and the properties of this MND are the characteristics of the MND that are the key defining properties of it. First of all, the data contributing to an MND itself has a Normal distribution along any of the... 

Now we ask ourselves the question If we calculate the standard deviation for a set of data (or errors) from these two formulas, will they give us the same answer And the answer to that question is that they will, IF (that s a very big if , you see) the data and the errors have the characteristics that statisticians consider good statistical properties random, independent (uncorrelated), constant variance, and in this case, a Normal distribution, and for errors, a mean (fi) of zero, as well. For a set of data that meets all these criteria, we can expect the two computations to produce the same answer (within the limits of what is sometimes loosely called Statistical variability ). 

Statistical properties of a data set can be preserved only if the statistical distribution of the data is assumed. PCA assumes the multivariate data are described by a Gaussian distribution, and then PCA is calculated considering only the second moment of the probability distribution of the data (covariance matrix). Indeed, for normally distributed data the covariance matrix (XTX) completely describes the data, once they are zero-centered. From a geometric point of view, any covariance matrix, since it is a symmetric matrix, is associated with a hyper-ellipsoid in N dimensional space. PCA corresponds to a coordinate rotation from the natural sensor space axis to a novel axis basis formed by the principal... 

Wu, Ruff and Faethl249 made an extensive review of previous theories and correlations for droplet size after primary breakup, and performed an experimental study of primary breakup in the nearnozzle region for various relative velocities and various liquid properties. Their experimental measurements revealed that the droplet size distribution after primary breakup and prior to any secondary breakup satisfies Simmons universal root-normal distribution 264]. In this distribution, a straight line can be generated by plotting (Z)/MMD)°5 vs. cumulative volume of droplets on a normal-probability scale, where MMD is the mass median diameter of droplets. The slope of the straight line is specified by the ratio... 

In this case the summation is the sum of the squares of all the differences between the individual values and the mean. The standard deviation is the square root of this sum divided by n — 1 (although some definitions of standard deviation divide by n, n — 1 is preferred for small sample numbers as it gives a less biased estimate). The standard deviation is a property of the normal distribution, and is an expression of the dispersion (spread) of this distribution. Mathematically, (roughly) 65% of the area beneath the normal distribution curve lies within 1 standard deviation of the mean. An area of 95% is encompassed by 2 standard deviations. This means that there is a 65% probability (or about a two in three chance) that the true value will lie within x Is, and a 95% chance (19 out of 20) that it will lie within x 2s. It follows that the standard deviation of a set of observations is a good measure of the likely error associated with the mean value. A quoted error of 2s around the mean is likely to capture the true value on 19 out of 20 occasions. 

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. 

The sum Z of two normal variables X and Y is a normal variable. Its mean is the sum of individual means, its variance the sum of individual variances. If X is normal variable with mean n and variance a2, 2X is normal with mean 2/i and variance 2a2. This property is easily extended to the sum of any number of normal variables. An important consequence on sampling distributions is that, from equation (4.1.40), the mean x of m observations Xj from the same normal distribution with mean n and variance a2 is a normal variable with mean mfi/m=fi and variance ma2/m2 - a2/ /m. 

Parallel to the case of a single random variable, the mean vector and covariance matrix of random variables involved in a measurement are usually unknown, suggesting the use of their sampling distributions instead. Let us assume that x is a vector of n normally distributed variables with mean n-column vector ft and covariance matrix L. A sample of m observations has a mean vector x and annxn covariance matrix S. The properties of the t-distribution are extended to n variables by stating that the scalar m(x—p)TS ( —p) is distributed as the Hotelling s-T2 distribution. The matrix S/m is simply the covariance matrix of the estimate x. There is no need to tabulate the T2 distribution since the statistic... 

The least-square solution itself does not depend on the probability distribution of y it is simply a minimum-distance estimate. Later in this Chapter, it will be shown, however, that its sampling properties are most easily described when the measurements are normally distributed. 

The data of this examples have been simulated as follows xi and x2 have been systematically varied as shown in Table 4.2 x3 contains random numbers from a normal distribution N(0, 5). Property y is calculated as a theoretical value 5x, + 4x2 with noise added from a normal distribution 7/(0, 3). 

There are several properties of linear regression that should be noted. First, it is assumed that the model errors are normally distributed. Second, the relationship between the x and y variables is assumed to be linear. In analytical chemistry, the first assumption is generally a reasonable one. However, the second assumption might not be sufficiently accurate in many situations, especially if a strong nonlinear relationship is suspected between x and y. There are some nonlinear remedies to deal with such situations, and these will be discussed later. 

The normal distribution has some properties that are very important in understanding statistical results. The curve is symmetrical about the central value p. 68,27% of the values he widiin p + la, 95,45% within p + 2a and 99,73% within p + 3a. 

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