Statistical variability

Virtually all design parameters such as tolerances, material properties and service loads exhibit some statistical variability and uncertainty that influence the adequacy of the design. A key requirement in the probabilistic approach is detailed knowledge... 

Random copolymers are similar to PEO but when the regular helical structure of the chains is demolished, the crystallinity is also destroyed. One of the simplest and most successful amorphous host polymers is an oxyethylene- oxymethylene structure in which medium length but statistically variable EO units are interspersed with methylene oxide groups. First described in 1990 [37], aPEO has the general structure... 

Another indicator is also largely used for time var5dng statistical variables, the autocorrelation function... 

H. Hotelling, Analysis of a complex of statistical variables into principal components. J. Educ. 

PROC MEANS, PROC SUMMARY, and PROC TABULATE are other SAS procedures that you can use to get descriptive statistics and place them into output data sets. However, those procedures do not offer any descriptive statistical variables that you cannot get from PROC FREQ or PROC UNIVARIATE. 

Contain a stochastic climate generator capable of simulating daily precipitation and other weather parameters that are similar in amount and statistical variability to historical weather records for the site. 

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 ). 

Principal Components (i. e, PCs) are linear combinations of random or statistical variables, which have special properties in terms of variances. 

In most problems involving boundary conditions, the boundary is assigned a specific empirical or deterministic behavior, such as the no-slip case or an empirically determined slip value. The condition is defined based on an averaged value that assumes a mean flow profile. This is convenient and simple for a macroscopic system, where random fluctuations in the interfacial properties are small enough so as to produce little noise in the system. However, random fluctuations in the interfacial conditions of microscopic systems may not be so simple to average out, due to the size of the fluctuations with respect to the size of the signal itself. To address this problem, we consider the use of stochastic boundary conditions that account for random fluctuations and focus on the statistical variability of the system. Also, this may allow for better predictions of interfacial properties and boundary conditions. 

Because of the first of these uncertainties (the extrapolation across species), assessments of risks to human health apply an uncertainty or safety factor of 100 to the experimentally derived no observed adverse effect concentration (NOAEC), in other words the NOAEC is divided by 100 to derive a no-effect level for human toxicity. This factor has been used since 1961, when it was chosen on an essentially arbitrary basis (RCEP, 2003, p22). In the assessment of risks to the environment, application factors of 10, 50, 100 or 1000 are applied to the results of tests carried out on specific species,2 depending on the species used and whether the tests were long term or short term. Evidence to the Royal Commission on Environmental Pollution (RCEP) for their report Chemicals in products indicated that these are merely extrapolation factors — they express the statistical variability of test results but do not effectively take into account inter-species variability, the vulnerability of threatened species, lifetime exposures or the complexity of biological systems... 

Hotelling, H., Analysis of a Complex of Statistical Variables into Principal Components /. Educ. 

Additional factors which must be taken into account are environmental effects (thermal as well as chemical), effects of defects, statistical variability of the material, long-term behavior, and cyclic versus static loading effects. Assessment of these effects requires the end user to conduct a large series of tests using multiple specimens. A typical series will examine a unidirectional material in tension in the 0, 90, and cross-ply directions 0, 90, and cross-ply in compression and 1-2, 1-3, and 2-3 shear at different temperatures ranging from —54°C to the expected service temperature creep rupture at temperatures up to the expected service temperature and fatigue at room and elevated temperature. This series of tests, shown in Table 12.1, may require over 400 specimens. 

Figure 5.7 Graphic introduction of the correlation between statistical variables, (a) independent variables, (b) dependent variables.

Finally, it should be mentioned that the equilibrium statistical mechanics is thermodynamically self-consistent if the statistical variables (x1,. .., xn), the potentials (/, g,. ..), and the variables (u1,. ..,un) are homogeneous variables of the first- or zero-order satisfying Eqs. (21)-(25). 

The inhibition rate for all inhibitor assays was measured from the initial velocities within the linear time-activity relationships of the control and inhibited samples. The standard deviations were calculated for the Iso values of the compounds against the three enzymes used and the maximal values were used to compare the statistical variability within each assay test. 

Pavlik, H. E Runnells, D. D. "Evaluation of the Validity and Statistical Variability of Lithium and Fluoride Distribution Coefficient Values in a Complex Geochemical System" Research Highlights, DOE/ER-0261, U.S. Department of Energy Washington, DC, 1986 ... 

The electrons in the island can then be regarded as being localized and they behave classically, although they undergo the same thermodynamic fluctuations as every statistical variable. 

Figure 8. Evolution of several statistical variables as a function of parameter aid a) n co b) cross 5 c) P(Icross 0)...

The reason for the ambiguity of results of assessment of the CHG full-body shower wash reported in the literature may be attributed to statistical variability. A 1- or 2-day presurgical application period is simply too brief to establish the necessary levels of residual antimicrobial activity to be of value in reducing postsurgical infection rates. It will be interesting to see what data other investigations produce, when the full-body shower wash procedure is employed over a period of 3-5 consecutive days. 

Consider two statistic variables, say x andy, for each of which a distribution function, say f x) and g x), and for both of which a joint distribution function, F x,y), is defined. We call the variables independent if and only if... 

Having a value for the mean tells you what value the data points "cluster" around but it does not tell you how spread out they are from the center. A second statistical variable called the standard deviation is used for that. The standard deviation is essentially the average distance between the data points and their mean. The distance is expressed as a squared distance in order to prevent negative deviations from lessoning the effect of positive deviations. 

The theory and application of statistical moment rest on the tenet that the movement of individual drug molecules through a body compartment is governed by probability. The residence time of a molecule of drug in the body, therefore, can be regarded as a random statistical variable. The mean and the variance of the retention times of a mass of drug molecules reflect the overall behavior of these drug molecules in the... 

---