Statistics measures

Unlike the solid state, the liquid state cannot be characterized by a static description. In a liquid, bonds break and refomi continuously as a fiinction of time. The quantum states in the liquid are similar to those in amorphous solids in the sense that the system is also disordered. The liquid state can be quantified only by considering some ensemble averaging and using statistical measures. For example, consider an elemental liquid. Just as for amorphous solids, one can ask what is the distribution of atoms at a given distance from a reference atom on average, i.e. the radial distribution function or the pair correlation function can also be defined for a liquid. In scattering experiments on liquids, a structure factor is measured. The radial distribution fiinction, g r), is related to the stnicture factor, S q), by... 

Correlation analysis reveals the interdependence between variables. The statistical measure for the interdependence is the correlation coefficient. 

The model building step deals with the development of mathematical models to relate the optimized set of descriptors with the target property. Two statistical measures indicate the quality of a model, the regression coefficient, r, or its square, r, and the standard deviation, a (see Chapter 9). 

A statistical measure of the average deviation of data from the data s mean value (s). 

Statistical Criteria. Sensitivity analysis does not consider the probabiUty of various levels of uncertainty or the risk involved (28). In order to treat probabiUty, statistical measures are employed to characterize the probabiUty distributions. Because most distributions in profitabiUty analysis are not accurately known, the common assumption is that normal distributions are adequate. The distribution of a quantity then can be characterized by two parameters, the expected value and the variance. These usually have to be estimated from meager data. 

In isotropic turbulence, statistical measures of fluctuations are equal in all directions. 

The statistical measures can be calculated using most scientific calculators, but confusion can arise if the calculator offers the choice between dividing the sum of squares by N or by W — 1 . If the object is to simply calculate the variance of a set of data, divide by N . If, on the other hand, a sample set of data is being used to estimate the properties of a supposed population, division of the sum of squares by W — r gives a better estimate of the population variance. The reason is that the sample mean is unlikely to coincide exactly with the (unknown) true population mean and so the sum of squares about the sample mean will be less than the true sum of squares about the population mean. This is compensated for by using the divisor W — 1 . Obviously, this becomes important with smaller samples. 

Standard deviation A statistical measure of the scatter of a series of numbers or measurements about their mean value. 

You will need to produce more than one part to verify that the process is stable. You need to form a sample large enough to take statistical measurement. If the measurements taken on the product fall within the central third of the control limits then the set-up can be approved - if not, then adjustments should be made and further samples produced until this condition is achieved. The Note in clause 4.9.4 indicates that regardless of the number of parts in the sample, it is the comparisons made on the last part that establish the conditions for commencement of production. 

Equipment failure rate data points carry varying degrees of uncertainty expressed by two measures, confidence and tolerance. Confidence, the statistical measurement of uncertainty, expresses how well the experimentally measured parameter represents the actual parameter. Confidence in the data increases as the sample size is increased. 

The Hammett equation is said to be followed when a plot of log k against a is linear. Most workers take as the criterion of linearity the correlation coefficient r, which is required to be at least 0.95 and preferably above 0.98. A weakness of r as a statistical measure of goodness of fit is that r is a function of the slope p if the slope is zero, the correlation coefficient is zero. A slope of zero in an LEER is a chemically informative result, for it demonstrates an absence of a substituent... 

The statistical measure of homogeneity is expressed as a function of the element geometry and the length of the unit (i.e., the number of elements in the mixer assembly). 

Density The simplest statistical measure that can be used to characterize a configuration is the density pi t), defined to be the average fraction of sites with value a = I at time t. For a disordered configuration, for example, pi = p. 

In order to quantitatively trace behavior as a function of A, it is clear that we need to look at statistical measures that distinguish between ordered and random behavior. To this end, consider the spreading rates of differenc e patterns and entropy. 

Except for using the statistical measure p to characterize the temporal evolution, the evolution itself has so far been entirely deterministic. We now take explicit account of temperature, as introduced via equation 7.111 ... 

The evolution of. systems starting from random initial value states is generally difficult to follow vi.sually, particularly for Fs that induce many structural changes, and must therefore be studied indirectly. The simplest way is to chart the time-development by recording selected statistical measures. A more detailed accound is given in [ilachSS]. 

Mean absolute deviation MAD is a statistical measure of the mean (average) difference between a product s forecast and actual usage (demand). The deviations (differences) are included without regard to whether the forecast was higher than actual or lower. 

Statistical measures of changes in line with the mentoring objectives (e.g. alterations in productivity or staff turnover, cost-benefit analyses). 

There is currently debate on the best methods to define the applicability domain for a model in predictive toxicology. The ultimate solution is likely to be lacking for a number of years. However, there are some initiatives that are beginning to address the issue of applicability domain, which include the use of statistical measures and also mechanistic appreciation. 

It is usual to have the coefficient of determination, r, and the standard deviation or RMSE, reported for such QSPR models, where the latter two are essentially identical. The value indicates how well the model fits the data. Given an r value close to 1, most of the variahon in the original data is accounted for. However, even an of 1 provides no indication of the predictive properties of the model. Therefore, leave-one-out tests of the predictivity are often reported with a QSAR, where sequentially all but one descriptor are used to generate a model and the remaining one is predicted. The analogous statistical measures resulting from such leave-one-out cross-validation often are denoted as and SpR ss- Nevertheless, care must be taken even with respect to such predictivity measures, because they can be considerably misleading if clusters of similar compounds are in the dataset. 

If the probability distribution of the data is or assumed Gaussian, several statistical measures are available for interpreting the data. These measures can be used to interpret the latent variables determined by a selected data analysis method. Those described here are a combination of statistical measures and graphical analysis. Taken together they provide an assessment of the statistical significance of the analysis. 

The greatest need in model performance testing and validation is clearly the use of quantitative measures to describe comparisons of observed and predicted values. As noted above, although a rigorous statistical theory for model performance assessments has yet to be developed, a variety of statistical measures has been used in various combinations and the frequency of use has been increasing in recent years. The FAT workshop (3.) identified three general types of comparisons that are often made in model performance testing ... 

Statistical measures for the paired-data, and integrated paired-data performance tests noted above are essentially identical. 

Young and Alward (5), and Lorber and Mulkey (7) demonstrate the use of these statistical measures as quantitative assessments of model performance for a wide range of models. 

Models of the polymer coil are based on the end-to-end distance, which is generally not directly available as a quantitative feature. Coils in dilute solution can be characterized in terms of the radius of gyration, Rg, which is a statistical measure of the distribution of mass about the center of gravity or in terms of the hydrodynamic radius, Rh, that is usually determined through the use of Stokes law and a measurement of a drag coefficient or friction factor, /drag/ for the coil,... 

---