Statistical terms defined

Thus, when a property of the sample (which exists as a large volume of material) is to be measured, there usually will be differences between the analytical data derived from application of the test methods to a gross lot or gross consignment and the data from the sample lot. This difference (the sampling error) has a frequency distribution with a mean value and a variance. Variance is a statistical term defined as the mean square of errors the square root of the variance is more generally known as the standard deviation or the standard error of sampling. 

These methods of test require a knowledge of what is known as the number of degrees of freedom. In statistical terms this is the number of independent values necessary to determine the statistical quantity. Thus a sample of n values has n degrees of freedom, whilst the sum (x — x)2 is considered to have n — 1 degrees of freedom, as for any defined value of x only n — 1 values can be freely assigned, the nth being automatically defined from the other values. 

In order to determine how many samples we require, it is necessary to consider the sources of uncertainty in the final result. Uncertainty is dealt with in more detail in Chapter 6. In this section, we are mainly concerned with the uncertainty arising from sampling. It is necessary to use a few statistical terms namely, sample standard deviation (s) and variance (.v2). These terms are defined in Chapter 6, Section 6.1.3. 

The sums of squares of the individual items discussed above divided by its degrees of freedom are termed mean squares. Regardless of the validity of the model, a pure-error mean square is a measure of the experimental error variance. A test of whether a model is grossly adequate, then, can be made by acertaining the ratio of the lack-of-fit mean square to the pure-error mean square if this ratio is very large, it suggests that the model inadequately fits the data. Since an F statistic is defined as the ratio of sum of squares of independent normal deviates, the test of inadequacy can frequently be stated... 

In order to take into account the spontaneity and irreversibility of real processes (heat always goes from a hot substance to a cold one, but not the reverse), thermodynamics invokes the notion of the entropy S. In statistical terms entropy is defined as the probability of accessible states for each molecule in the system ... 

Two statistical terms involving error analysis that are often used and misused are accuracy and precision. Precision refers to the extent of agreement among repeated measurements of an experimental value. Accuracy is defined as the difference between the experimental value and the true value for the quantity. Since the true value is seldom known, accuracy is better defined as the difference between the experimental value and the accepted true value. Several experimental measurements may be precise (that is, in close agreement with each other) without being accurate. 

While the word confidence in the previous sentence occurs in its everyday use, the term is also used in Statistics in a precise manner, analogously to the statistical terms Normal and significant. Confidence intervals constitute a range of values that are defined by the lower limit and the upper limit of the interval. These limits are symmetrically placed on either side of the sample mean. A commonly used Cl is the 95% Cl. A commonly expressed view of a 95% Cl is that one can be 95% certain that... 

CM 18.1 to CM 18.3 were assessed in terms of their Cooper statistics, which define an upper limit to predictive performance. In addition, cross-validated Cooper statistics, which provide a more realistic indication of a model s capacity to predict the classifications of independent data, were obtained by applying the threefold cross-validation procedure to the best-sized CTs. In the threefold cross-validation procedure, the data set is randomly divided into three approximately equal parts, the CT is re-parameterized using two thirds of the data, and predicted classifications are made for the remaining third of the data. The cross-validated Cooper statistics are the mean values of the usual Cooper statistics, taken over the three iterations of the cross-validation procedure. The Cooper statistics for CM 18.1 to CM 18.3 are summarized in Table 18.6. 

For a system consisting of the total number of particles N and maintaining its total energy U and volume V constant, statistical thermodynamics defines the entropy, S, in terms of the logarithm of the total number of microscopic energy distribution states Q N,V,U) in the system as shown in Eq. 3.6 ... 

The aim of this chapter is to demonstrate suitable protocols for the planning of experiments and the analysis of the data. The important question to keep in mind is, What is the purpose of the experiment and what do I propose as the outcome Usually, defining the question takes greater effort than performing any analysis. Defining the question is more technically termed defining the research hypothesis, following which the statistical tools can be used to determine whether the stated hypothesis is found to be true. 

A third type of error is the systematic error that cannot be defined in statistical terms. Systematic error is treated by analyzing reference samples to establish corrections. The calibration procedure can produce calendar dates that have large error ranges even when the conventional 14C age is known with very high precision (13). 

Any obtained set of measurements (xi, X2... Xn is defined in statistics as a sample of this continuous range of values. Only all possible measurements (which must be infinite in number to cover the whole range) would generate the real set, which is called population. (The statistical term sample can easily be confused in analytical chemistry with the term sample = specimen, as the term population may also have a different meaning. To avoid this confusion, the statistical terms sample and population will always be italicized.)... 

This has been defined (B5) as the smallest single result which, with some assurance, can be distinguished from zero, or, in statistical terms, the smallest single result whose fiducial limits for, say, P = 0.05 do not include zero. This review will be primarily concerned with clinical chemical methods that have acceptable levels of sensitivity, and to which therefore statistical methods of quality control can be applied throughout the range of concentrations which may be encountered in physiological and pathological conditions. To take an extreme example, therefore, the statistical methods of quality control discussed in this review would not be fully applicable to determinations of plasma epinephrine or... 

In Chapter 10 the use of this method was discussed in terms of estimating the median survival time for participants in a clinical trial. The median survival time can be helpful as a single summary statistic that defines a typical survival time. However, survival distributions may deviate at various points in time. In this section we present the logrank test, which can be used to test the equality of two or more survival distributions. This is not the only test that can be used for this purpose, but it is a natural extension of a method that we have already described and so we have chosen to discuss it. 

Measure for comparing approximations In order to conveniently eompare the approximations, let / be a target FCD and let g be an approximate function to /. The cross-entropy distance between/ and g with respect to /, also known as Kullback-Leibler (KL) information of g at /, has been considered to assess the performance in statistical terms of g when approximating /. In fact, a number of authors have adopted the KL distance for measuring the quahty of proposal ftmetions in infering over their target densities Neil et al. (2007) and Keith et al. (2008) are some recent examples. The KL distance is defined as the expected value... 

The F statistic can also be useful in recognizing suspected outliers within a calibration sample set if the F value decreases when a sample is deleted, the sample was not an outlier. This situation is the result of the sample not affecting the overall fit of the calibration line to the data while at the same time decreasing the number of samples (N). Conversely, if deleting a single sample increases the overall F for regression, the sample is considered a suspected outlier. F is defined as the mean square for regression divided by the mean square for residual (see statistical terms in this table). Statistic Coefficient of Multiple Determination Abbreviations R or r ... 

The nonwoven fabric uniformity is originally defined as the fabric mass per unit area (or fabric density) distribution in the fabric structure. The basic statistic terms of fabric mass uniformity in nonwoven industries are the standard deviation (measured parameters (eg, fabric weight, fabric thickness, fabric density, optical levels, rays absorption amounts, grey level intensity of images, etc.) as follows ... 

We have formulated three equivalent relations, Eqs. (A.13), (A.17) and (A.21), which may all be employed in the evaluation of scattering data. All three equations express a Fourier-relation between S q) and functions which describe properties of the microscopic structure in statistical terms. To put special emphasis on this well defined structural background, S q), firstly introduced as the scattering function , is often also addressed as the structure function or structure factor . We will use all these different names, chosen freely. 

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