Standard deviation, determination

As indicated in Section 1.7.2, the standard deviations determined for the small sets of observations typical for analytical chemistry are trustworthy only to one or two significant digits. 

The mean determines the location of the curve with respect to the x axis, and the standard deviation determines the shape. Figure 2-3 shows the effect of the standard deviation on the shape. As the standard deviation decreases, the distribution curve becomes more pronounced around the mean value. 

Figures. Dose-response results for each of the six chemicals. Cell cultures were exposed to the chemicals for 30 min. Each chemical was tested for a dose range of 0-300 pM. Each data point is representative of the results for a set of chemically treated triplicates reported as an average % survival standard deviation determined at the 95%...

An established external quality control (QC) scheme is not currently available. Pooled disease control CSF retained from other analyses is used. Aliquots of the pooled CSF are made and stored at -70°C. This CSF is analysed on five separate occasions and the mean and standard deviation determined. For an analytical/diagnostic run to proceed, analysis of QC material must provide concentration values that are within two standard deviations (plus and minus) of the calculated mean for that particular QC. Construction of Levy-Jennings type control charts provide historical information of overall performance and highlight potential deterioration in the performance of the system. 

Standard deviation determined from the volume-pressure relationship upon distension with isotonic saline independent of age. 

The mean determines the location of the peak of the distribution in relation to the y-axis. If the mean is zero, the central line of the profile overlaps with the y-axis. The standard deviation determines the width of the distribution profile. A large a implies a wide variation featuring a short and fat profile, while a small a indicates a narrow variation corresponding to a long and thin profile. 

The liposomes formed under such conditions were mostly multilamellar vesicles (MLV) as can be seen in Fig. 2. They were characterized by the mean vesicle diameter and their standard deviation, determined by SEM and calculated according to the Gauss distribution equation. From results shown in Table 1 it can be observed that the increase in PRO concentration causes an increase in the mean vesicle diameter. It could be assumed that PRO was mostly located in the aqueous phase of the liposomes, between the lamellas, because of their distinctive polar character and the position of the hydroxyl groups seen in Fig. 1. All the dispersions examined have marked polydispersity of the liposomes. 

Since the benches in the quarry are usually horizontal, the computer can, via the standard deviation, determine coefficients of variation and limiting concentrations for selected areas of the deposit. From this information the bench height and bench sections can then in turn be obtained. 

Identify standard deviation, determine shifted target property ranges and introduce degree of satirfaction for property superiority and robustness... 

In Equation (24), a is the estimated standard deviation for each of the measured variables, i.e. pressure, temperature, and liquid-phase and vapor-phase compositions. The values assigned to a determine the relative weighting between the tieline data and the vapor-liquid equilibrium data this weighting determines how well the ternary system is represented. This weighting depends first, on the estimated accuracy of the ternary data, relative to that of the binary vapor-liquid data and second, on how remote the temperature of the binary data is from that of the ternary data and finally, on how important in a design the liquid-liquid equilibria are relative to the vapor-liquid equilibria. Typical values which we use in data reduction are Op = 1 mm Hg, = 0.05°C, = 0.001, and = 0.003... 

The combined result of two such determinations yielded a leak size figure of 8.8% of the feed flow (with a relative standard deviation of less than 5%). This figure could sufficiently explain the product quality problems experienced, whose alternative explanation in turn was catalyst poisoning. 

The shear viscosity is a tensor quantity, with components T] y, t],cz, T)yx> Vyz> Vzx> Vzy If property of the whole sample rather than of individual atoms and so cannot be calculat< with the same accuracy as the self-diffusion coefficient. For a homogeneous fluid the cor ponents of the shear viscosity should all be equal and so the statistical error can be reducf by averaging over the six components. An estimate of the precision of the calculation c then be determined by evaluating the standard deviation of these components from tl average. Unfortunately, Equation (7.89) cannot be directly used in periodic systems, evi if the positions have been unfolded, because the unfolded distance between two particl may not correspond to the distance of the minimum image that is used to calculate the fore For this reason alternative approaches are required. 

The first application of the Gaussian distribution is in medical decision making or diagnosis. We wish to determine whether a patient is at risk because of the high cholesterol content of his blood. We need several pieces of input information an expected or normal blood cholesterol, the standard deviation associated with the normal blood cholesterol count, and the blood cholesterol count of the patient. When we apply our analysis, we shall anive at a diagnosis, either yes or no, the patient is at risk or is not at risk. 

The breadth or spread of the curve indicates the precision of the measurements and is determined by and related to the standard deviation, a relationship that is expressed in the equation for the normal curve (which is continuous and infinite in extent) ... 

An analytical procedure is often tested on materials of known composition. These materials may be pure substances, standard samples, or materials analyzed by some other more accurate method. Repeated determinations on a known material furnish data for both an estimate of the precision and a test for the presence of a constant error in the results. The standard deviation is found from Equation 12 (with the known composition replacing /x). A calculated value for t (Eq. 14) in excess of the appropriate value in Table 2.27 is interpreted as evidence of the presence of a constant error at the indicated level of significance. 

To calculate the standard deviation, we obtain the difference between the mean value (3.117 see Example 4.1) and each measurement, square the resulting differences, and add them to determine the sum of the squares (the numerator of equation 4.1)... 

It is much easier to determine the standard deviation using a scientific calculator with built-in statistical functions. ... 

Many scientific calculators include two keys for calculating the standard deviation, only one of which corresponds to equation 4.3. Your calculator s manual will help you determine the appropriate key to use. 

To evaluate the effect of indeterminate error on the data in Table 4.1, ten replicate determinations of the mass of a single penny were made, with results shown in Table 4.7. The standard deviation for the data in Table 4.1 is 0.051, and it is 0.0024 for the data in Table 4.7. The significantly better precision when determining the mass of a single penny suggests that the precision of this analysis is not limited by the balance used to measure mass, but is due to a significant variability in the masses of individual pennies. 

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. 

How we report the result of an experiment is further complicated by the need to compare the results of different experiments. For example. Table 4.10 shows results for a second, independent experiment to determine the mass of a U.S. penny in circulation. Although the results shown in Tables 4.1 and 4.10 are similar, they are not identical thus, we are justified in asking whether the results are in agreement. Unfortunately, a definitive comparison between these two sets of data is not possible based solely on their respective means and standard deviations. 

To begin with, we must determine whether the variances for the two analyses are significantly different. This is done using an T-test as outlined in Example 4.18. Since no significant difference was found, a pooled standard deviation with 10 degrees of freedom is calculated... 

The data we collect are characterized by their central tendency (where the values are clustered), and their spread (the variation of individual values around the central value). Central tendency is reported by stating the mean or median. The range, standard deviation, or variance may be used to report the data s spread. Data also are characterized by their errors, which include determinate errors... 

Vitha, M. F. Carr, P. W. A Laboratory Exercise in Statistical Analysis of Data, /. Chem. Educ. 1997, 74, 998-1000. Students determine the average weight of vitamin E pills using several different methods (one at a time, in sets of ten pills, and in sets of 100 pills). The data collected by the class are pooled together, plotted as histograms, and compared with results predicted by a normal distribution. The histograms and standard deviations for the pooled data also show the effect of sample size on the standard error of the mean. 

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. 

There is an obvious similarity between equation 5.15 and the standard deviation introduced in Chapter 4, except that the sum of squares term for Sr is determined relative toy instead of y, and the denominator is - 2 instead of - 1 - 2 indicates that the linear regression analysis has only - 2 degrees of freedom since two parameters, the slope and the intercept, are used to calculate the values ofy . 

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