Sampling statistical theory

In effect, the standard deviation quantifies the relative magnitude of the deviation numbers, i.e., a special type of average of the distance of points from their center. In statistical theory, it turns out that the corresponding variance quantities s have remarkable properties which make possible broad generalities for sample statistics and therefore also their counterparts, the standard deviations. 

Statistical techniques can be used for a variety of reasons, from sampling product on receipt to market analysis. Any technique that uses statistical theory to reveal information is a statistical technique, but not all applications of statistics are governed by the requirements of this part of the standard. Techniques such as Pareto Analysis and cause and effect diagrams are regarded as statistical techniques in ISO 9000-2 and although numerical data is used, there is no probability theory involved. These techniques are used for problem solving, not for making product acceptance decisions. 

In their seminal work from 1983, Davis and Giddings used a statistical theory to define the number of peaks observable in 1DLC separation upon the injection of a sample of different complexity on a column of a given peak capacity (Davis and Giddings, 1983). The theory was later extended into 2D separation space (Davis, 2005 Shi and Davis, 1993), also discussed in Chapter 2 of this book. The theory implies that when the 1D or 2D separation space is randomly covered with the number of peaks equal to the separation space peak capacity (area), the normalized surface coverage is... 

The subject of statistical researches are the Population (universe, statistical masses, basic universe, completeness) and samples taken from a population. The population must be representative of a collection of a continual chemical process by some features, i.e. properties of the given products. If we are to find a property of a product, we have to take out a sample from a population that, by mathematical statistics theory is usually an infinite gathering of elements-units. 

Statistical estimation uses sample data to obtain the best possible estimate of population parameters. The p value of the Binomial distribution, the p value in Poison s distribution, or the p and a values in the normal distribution are called parameters. Accordingly, to stress it once again, the part of mathematical statistics dealing with parameter distribution estimate of the probabilities of population, based on sample statistics, is called estimation theory. In addition, estimation furnishes a quantitative measure of the probable error involved in the estimate. As a result, the engineer not only has made the best use of this data, but he has a numerical estimate of the accuracy of these results. 

We note from Table 1.19 that the sums of squares between rows and between columns do not add up to the defined total sum of squares. The difference is called the sum of squares for error, since it arises from the experimental error present in each observation. Statistical theory shows that this error term is an unbiased estimate of the population variance, regardless of whether the hypotheses are true or not. Therefore, we construct an F-ratio using the between-rows mean square divided by the mean square for error. Similarly, to test the column effects, the F-ratio is the be-tween-columns mean square divided by the mean square for error. We will reject the hypothesis of no difference in means when these F-ratios become too much greater than 1. The ratios would be 1 if all the means were identical and the assumptions of normality and random sampling hold. Now let us try the following example that illustrates two-way analysis of variance. 

In clinical research it is of particular interest to estimate a population mean on the basis of data collected from a sample of subjects employed in a randomized clinical trial. Sampling and statistical procedures facilitate the estimation of the population mean based on the sample mean and sample SD that are precisely calculated from the data collected in the trial. If we take a sample of 100 numbers from a population of 100,000 numbers and calculate the mean of those 100 numbers, this sample mean, which is precisely known, provides an estimate of the unknown population mean. If we then took another sample of 100 numbers, or indeed many samples, it is extremely unlikely that the numbers in any subsequent sample would be identical to those in the first sample, and it is unlikely that the calculated sample means would be identical to that of the first sample. Therefore, in a randomized clinical trial, a situation in which only one sample is taken from a population, a question that arises is What degree of certainty is there that the mean of that sample represents the mean of the population This question can be answered using statistical theory in conjunction with knowledge of the number of subjects participating in the trial, i.e., the sample size. 

Federer, W.T. (1987). On screening samples in the laboratory and factors in factorial investigations. Communications in Statistics—Theory and Methods, 16, 3033-3049. Kleijnen, J.P.C (1987). Review of random and group-screening designs. Communication in Statistics—Theory and Methods, 16, 2885-2900. 

The presence of filler in the rubber as well as the increase of the surface ability of the Aerosil surface causes an increase in the modulus. The temperature dependence of the modulus is often used to analyze the network density in cured elastomers. According to the simple statistical theory of rubber elasticity, the modulus should increase twice for the double increase of the absolute temperature [35]. This behavior is observed for a cured xmfilled sample as shown in Fig. 15. However, for rubber filled with hydrophilic and hydrophobic Aerosil, the modulus increases by a factor of 1.3 and 1.6, respectively, as a function of temperature in the range of 225-450 K. It appears that less mobile chain units in the adsorption layer do not contribute directly to the rubber modulus, since the fraction of this layer is only a few percent [7, 8, 12, 21]. Since the influence of the secondary structure of fillers and filler-filler interaction is of importance only at moderate strain [43, 47], it is assumed that the change of the modulus with temperature is mainly caused by the properties of the elastomer matrix and the adsorption layer which cause the filler particles to share deformation. Therefore, the moderate decrease of the rubber modulus with increasing temperature, as compared to the value expected from the statistical theory, can be explained by the following reasons a decrease of the density of adsorption junctions as well as their strength, and a decrease of the ability of filler particles to share deformation due to a decrease of elastomer-filler interactions. 

In contrast to the filled samples, the deformation energy for the unfilled ones increases proportionally to the increase in the absolute temperature according to the prediction of the simple statistical theory of rubber elasticity. Thus, it appears that the change of the modulus and the deformation energy with increasing temperature reveals a decrease of the density of adsorption junctions in the elastomer matrix, as well as a decrease of the ability of filler particles to share deformation, resulting from a weakening of elastomer-filler interactions. 

You need not spend much time attempting to master rigorous statistical theory. Because EVOP was developed to be used by nontechnical process operators, it can be applied without any knowledge of statistics. However, be prepared to address the operators tendency to distrust decisions based on statistics. Concepts that you should understand quantitatively include the difference between a population and a sample the mean, variance, and standard deviation of a normal distribution the estimation of the standard deviation from the range standard errors sequential significance tests and variable effects and interactions for factorial designs having two and three variables. Illustrations of statistical concepts (e.g., a normal distribution) will be valuable tools. 

Given two estimates of a statistic, one from a sample of size n and the other from a sample of size 2n, one might expect that the estimate from the larger sample would be more reliable than that from the smaller sample. This is, in fact, supported by statistical theory. If the variance in the population is cr, then the variance of the sample mean for samples of size n is a jn. The square root of this is the standard error of the mean. Consistent with the variance of the sample mean being /n times that of a single determination (cr ), the standard deviation and the CV% of the sample mean are reduced by the square root of n. As a direct consequence, an assay method that relies on the mean of two independent concentration determinations has a CV / /2 that of the same method based on a single determination. This provides an easy way to increase the precision (reduce variability) of a method. An example of this is found in radioimmunoassay in which it is common for a concentration estimate to be calculated from the mean response of two determinations of a specimen. 

The univariate statistical theory is used, for example, for rejecting one extreme value in a set of scattered results in a given sample. For this purpose, the extreme value x is temporarily eliminated from the sample. Then, from the sample Xi, X2...Xn - Xe there are calculated m, s and the value ... 

Another utilization of statistical theory Is to decide if two specimens are the same or different when the measurements performed on each specimen generate the two samples xi, X2... x and xi, X2 ... Xk containing scattered values. For this purpose, the two averages m and m and the two standard deviations s and s are calculated. It can be shown that If the samples are large enough, the variable ... 

Statistical theory teaches that under the assumption that the population means of the two groups are the same (i.e. if Hq is true), the distribution of variable T depends only on the sample size but not on the value of the common mean or on the measurements population variance and thus can be tabulated independently of the particulars of any given experiment. This is the so-called Student s f-distribution. Using tables of the f-distribution, we can calculate the probability that a variable T calculated as above assumes a value greater or equal to 4.7, the value obtained in our example, given that H0 is true. This probability is <0.0001. Thus, if H0 is true, the result obtained in our experiment is extremely unlikely, although not impossible. We are forced to choose between two possible explanations to this. One is that a very unlikely event occurred. The second is that the result of our experiment is not a fluke, rather, the difference Mb — Ma is a positive number, sufficiently large to make the probability of this outcome a likely event. We elect the latter explanation and reject H0 in favor of the alternative hypothesis Hx. 

The guidelines also present the agrument that the predetermined limits for the final blend uniformity should be tighter than the USP requirements because of the increase in process variability as one moves downstream. Let s assume that the geometry of the blender is such that 15 unit dose samples adequately represents the stratification of the blender five samples from the top, the middle, and the bottom of the bin. Reasonable acceptance criteria for the 15 unit dose samples is the stage 1 dose uniformity attributes requirements of no unit outside 85.0-115.0% of label claim. But what do we do about setting requirements for the relative standard deviation (RSD) It is known from statistical theory, refer to Larson (8), that the relationship between the population standard deviation and the sample standard deviation is given by Eq. 1. 

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