Proportional standard deviations

Figure 14-28 Distances from data points to the line in the vertical direction in WLR assuming proportional standard deviations for random errors in x2 and no random error in xl. (From Linnet K. Necessary sample size for method comparison studies based on regression analysis. Clin Chem 1999 45 882-94.)...

Another way to improve the error in a simulation, at least for properties such as the energy and the heat capacity that depend on the size of the system (the extensive properties), is to increase the number of atoms or molecules in the calculation. The standard deviation of the average of such a property is proportional to l/ /N. Thus, more accurate values can be obtained by running longer simulations on larger systems. In computer simulation it is unfortunately the case that the more effort that is expended the better the results that are obtained. Such is life ... 

For example, when the activity is determined by counting 10,000 radioactive particles, the relative standard deviation is 1%. The analytical sensitivity of a radiochemical method is inversely proportional to the standard deviation of the measured ac-... 

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. 

The following set of data represents the outeome of a tensile test experiment to determine the yield strength in MPa of a metal. There are 50 individual results and they are displayed in the order they were reeorded. It is required to find the mean and standard deviation when the data is represented by a histogram. It is also required to find the strength at —3cr from the mean for the metal and the proportion of individuals that eould be expeeted to have a strength greater than 500 MPa. 

Then as the retention volume is n(vm + Kvs) and twice the peak standard deviation at the points of inflexion is 2 Vn (vm + Kvs), then, by simple proportion,... 

The relative standard deviation RSD (or c.o.v. = coefficient of variation ) is constant over the whole range, such as in many GC methods, that is, the standard deviation Sy is proportional to y. 

One performs so many repeat measurements at each concentration point that standard deviations can be reasonably calculated, e.g., as in validation work the statistical weights w, are then taken to be inversely proportional to the local variance. The proportionality constant k is estimated from the data. 

The sampling variance of the material determined at a certain mass and the number of repetitive analyses can be used for the calculation of a sampling constant, K, a homogeneity factor, Hg or a statistical tolerance interval (m A) which will cover at least a 95 % probability at a probability level of r - a = 0.95 to obtain the expected result in the certified range (Pauwels et al. 1994). The value of A is computed as A = k 2R-s, a multiple of Rj, where is the standard deviation of the homogeneity determination,. The value of fe 2 depends on the number of measurements, n, the proportion, P, of the total population to be covered (95 %) and the probability level i - a (0.95). These factors for two-sided tolerance limits for normal distribution fe 2 can be found in various statistical textbooks (Owen 1962). The overall standard deviation S = (s/s/n) as determined from a series of replicate samples of approximately equal masses is composed of the analytical error, R , and an error due to sample inhomogeneity, Rj. As the variances are additive, one can write (Equation 4.2) ... 

Table 2. Flipping ratios of strong and weak reflections measured starting from different time proportions (three steps of 60s). When using the optimised method, the standard deviation c(R) does not depend on the initial conditions. 

The only problem with this method is observed for weak reflections where (+)/(-) count-rates are similar (i.e. R 1). The (+)/(-) optimised counting-time proportions must be 50%/50%, but with low count-rates, we have observed that the lack of precision may lead to proportions which are not optimum (e.g. 47%/53%). The same behaviour has been observed for peak to background proportions. In fact, when measuring a flipping ratio in many steps, we observe oscillations of the time proportions which slow the decrease of the standard deviation. Of course, these time variations have no sense, and one should calculate the variances of the optimised counting-times (Equations (16)) to avoid such spurious fluctuations ... 

Poisson-distributed noise, however, has an interesting characteristic for Poisson-distributed noise, the expected standard deviation of the data is equal to the square root of the expected mean of the data ([11], p. 714), and therefore the variance of the data is equal (and note, that is equal, not merely proportional) to the mean of the data. Therefore we can replace Var(A s) with Es in equation 47-17 and Var(A r) with Et ... 

The variations in the background, the sensitivity to moisture, the alpha activity of the chamber itself and the influence of recombination were discussed by Hultqvist. The standard deviation due to counting statistics was estimated to be about 3 % (in a few measurements 6 %). The calibration was made by counting each alpha particle by a proportional counter specially designed at the Department for this purpose. The statistical uncertainty of the calibration of the equivalent radon concentration was estimated to be 12 %. 

Note that the mean /i (8) has a proportional influence on the standard deviation of the total ordered amount. The tendency is intuitive The formula means that only a few large orders require more safety stock to cover random demand variations than many small orders. It is therefore a good idea to measure the mean and the variance of demands twice. First in the usual way as mean and variance of a sequence of, say weekly, figures and then by analyzing the orders of a historic period and applying Eq. (6.6). The comparison of the two results obtained often provides insight in the independence and the randomness of the historic demand. If the deviation of the two mean values and/or two variances is large then the demand can not be considered as a random sum. A reason could be, for example, that the demand... 

At any one instant, only a very small proportion of the total number of unstable nuclei in a radioactive source undergo decay. A Poisson distribution which expresses the result of a large number of experiments in which only a small number are successful, can thus be used to describe the results obtained from measurements on a source of constant activity. In practical terms this means that random fluctuations will always occur, and that the estimated standard deviation, 5, of a measurement can be related to the total measurement by ... 

A sample of metamorphic carbonate contains calcite CaC03, dolomite Cao.5Mgo.5CO3, and diopside CaMgSi206. A chemical analysis on the calcinated (C02-free) rock indicates the following molar proportions 0.525 (0.03) CaO, 0.225 (0.01) MgO, and 0.25 (0.02) Si02 with standard deviations given in parentheses. Find the molar proportions of each mineral in the rock and their standard deviation. 

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