Standard uncertainties

From the measured properties and their uncertainties (standard deviations), compute Avogadro s number and its uncertainty. To find the uncertainty of c/, use the function y = x in Table 3-1. 

The procedure we use assumes that the errors in the y values are substantially greater than the errors in the x values.7 This condition is usually true in a calibration curve in which the experimental response (y values) is less certain than the quantity of analyte (x values). A second assumption is that uncertainties (standard deviations) in all the y values are similar. 

What is the net sample rate and its uncertainty This raises the general question of calculating the uncertainty in the result of some mathematical operations on an uncertain number. If we consider two independently determined numbers and their uncertainties (standard deviations), A + [Pg.572]

Using the NOAA ARL air-mass trajectory model, the WATOX researchers stratified volume-weighted averages of precipitation composition by compass sector. The results of these stratifications for excess S0 (Figure 6) showed that Bermuda was an ideal sampling platform for air that, at times, was directly impacted by anthropogenic tical uncertainties. standard error of... 

This laboratory reports tritium activities with their total expanded uncertainties (Table 3). The biggest contribution is from the furnace recovery which is 5%. Over 70 standard measurements were used for this. The average value for the furnace recovery is 95 5%. A value of 5.6% expanded uncertainty is used for the final calculation in order to estimate the total expended uncertainty. Standard runs are performed after every five sample runs and the acceptable furnace recovery range is between 80 - 100%. If the furnace recoveries are out of... 

Parameter estimation without an appropriate assessment of reliabihty of the estimates yields no conhdence in such estimates. Estimation of uncertainty enables the use of such parameter estimates in data synthesis. Embarking on data synthesis (e.g., clinical trial simulation) using model parameter estimates without associated uncertainty or poorly dehned uncertainty will produce unreliable outcomes. Sometimes it is impossible to obtain standard errors for population model parameter estimates when small sample sizes are used for population PK/PD modeling. The bootstrap with winsorization has been proposed for the estimation of inestimable uncertainty—standard errors—for population PK/PD parameters that are usually not obtainable using software such as NONMEM because of small sample size... 

Care should be taken to monitor the manufacturer s uncertainty information in relation to the individual components. This can vary considerably, typically between 0.5% and 5% standard uncertainty. This information should influence the choice of manufacturer for calibration and quality control standards, with the lower uncertainty standards being used for calibration. 

As outliers as such are laboratories that must be scrutinized and reasons for their poor performance understood, it is not good practice to remove them from consideration as happens in a method performance study. Unless there is a target mean and target measurement uncertainty (standard deviation) against which the results of all laboratories are assessed, the organizers must make a decision as to the actions to be taken with results that appear to be outlying. There are two approaches. First, outlier tests can be applied such as Cochran s test for laboratory variances (if... 

Continuing from remarks ( 62)-( 63), the following conclusion can be drawn The formula t F = h is often used for the estimation of the natural linewidth. This formula is sometimes interpreted as the time-energy equivalent of the Heisenberg relation, where r is the uncertainty (standard deviation) of the lifetime and F (FWHM) is that of the energy state. It should be stressed, however, that while r can play the assigned role (because the standard deviation is equal to the expected value in the case of the exponential distribution), the quantity F cannot be interpreted as standard deviation, since the Cauchy distribution does not have any. 

Multiscale modeling algorithms, representation and treatment of uncertainties, standard test cases (software, experiments), funding of interdisciplinary research, shared instruments top two issues multiscale modeling algorithms and funding of interdisciplinary research... 

The sampling and analytical variances must be determined from independent measurements. It is well established that once the analytical uncertainty (standard deviation) is reduced to a third or less of the sampling uncertainty, further reduction in analytical uncertainty is of little importance. Therefore, if the uncertainty in sampling is very large, it may be beneficial to opt for an analytical method that is rapid even though it might have lower precision. This will permit more samples to be analyzed, thereby resulting in a better estimate of the mean value. 

As the specially designed common components in the core product, which can be standardized, may increase the cost of materials, companies must carefully assess whether the benefits of standardization outweigh the added costs. Using standardized components decreases the size of the needed buffer inventory. As buffer inventory increases in the uncertainty, standardized components in the core helps reduce cost in a volatile market. For example, products with shorter life cycles increase uncertainty, increasing the benefits of standardization. 

The diagonal elements of this matrix approximate the variances of the corresponding parameters. The square roots of these variances are estimates of the standard errors in the parameters and, in effect, are a measure of the uncertainties of those parameters. 

Below the temperature of the lowest experimental datum, standard-state fugacities were obtained by simple extrapolation. Uncertainties assigned to these fugacities are largest when the fugacities are smallest, for two reasons (1) the extrapolation... 

At temperatures above those corresponding to the highest experimental pressures, data were generated using the Lyckman correlation all of these were assigned an uncertainty of 5% of the standard-state fugacity at zero pressure. Frequently, this uncertainty amounts to one half or more atmosphere for the lowest point, and to 1 to 5 atmospheres for the highest point. 

The Heisenberg uncertainty principle offers a rigorous treatment of the qualitative picture sketched above. If several measurements of andfi are made for a system in a particular quantum state, then quantitative uncertainties are provided by standard deviations in tlie corresponding measurements. Denoting these as and a, respectively, it can be shown that... 

Uncertainty expresses the range of possible values that a measurement or result might reasonably be expected to have. Note that this definition of uncertainty is not the same as that for precision. The precision of an analysis, whether reported as a range or a standard deviation, is calculated from experimental data and provides an estimation of indeterminate error affecting measurements. Uncertainty accounts for all errors, both determinate and indeterminate, that might affect our result. Although we always try to correct determinate errors, the correction itself is subject to random effects or indeterminate errors. 

Suppose that you need to add a reagent to a flask by several successive transfers using a class A 10-mL pipet. By calibrating the pipet (see Table 4.8), you know that it delivers a volume of 9.992 mL with a standard deviation of 0.006 mL. Since the pipet is calibrated, we can use the standard deviation as a measure of uncertainty. This uncertainty tells us that when we use the pipet to repetitively deliver 10 mL of solution, the volumes actually delivered are randomly scattered around the mean of 9.992 mL. 

Propagation of uncertainty allows us to estimate the uncertainty in a calculated result from the uncertainties of the measurements used to calculate the result. In the equations presented in this section the result is represented by the symbol R and the measurements by the symbols A, B, and C. The corresponding uncertainties are sr, sa, sb) and sq. The uncertainties for A, B, and C can be reported in several ways, including calculated standard deviations or estimated ranges, as long as the same form is used for all measurements. 

Using the standard deviation as an estimate of uncertainty, the uncertainty in the total delivered volume is... 

Consider, for example, the data in Table 4.1 for the mass of a penny. Reporting only the mean is insufficient because it fails to indicate the uncertainty in measuring a penny s mass. Including the standard deviation, or other measure of spread, provides the necessary information about the uncertainty in measuring mass. Nevertheless, the central tendency and spread together do not provide a definitive statement about a penny s true mass. If you are not convinced that this is true, ask yourself how obtaining the mass of an additional penny will change the mean and standard deviation. 

A standard solution of Mn + was prepared by dissolving 0.250 g of Mn in 10 ml of concentrated HNO3 (measured with a graduated cylinder). The resulting solution was quantitatively transferred to a 100-mL volumetric flask and diluted to volume with distilled water. A 10-mL aliquot of the solution was pipeted into a 500-mL volumetric flask and diluted to volume, (a) Express the concentration of Mn in parts per million, and estimate uncertainty by a propagation of uncertainty calculation, (b) Would the uncertainty in the solution s concentration be improved... 

Show by a propagation of uncertainty calculation that the standard error of the mean for n determinations is given as s/VTj. 

Determine the density at least five times, (a) Report the mean, the standard deviation, and the 95% confidence interval for your results, (b) Eind the accepted value for the density of your metal, and determine the absolute and relative error for your experimentally determined density, (c) Use the propagation of uncertainty to determine the uncertainty for your chosen method. Are the results of this calculation consistent with your experimental results ff not, suggest some possible reasons for this disagreement. 

A more useful representation of uncertainty is to consider the effect of indeterminate errors on the predicted slope and intercept. The standard deviation of the slope and intercept are given as... 

Standardizations using a single standard are common, but also are subject to greater uncertainty. Whenever possible, a multiple-point standardization is preferred. The results of a multiple-point standardization are graphed as a calibration curve. A linear regression analysis can provide an equation for the standardization. 

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