Standard deviation measurement

The random nature of most physieal properties, sueh as dimensions, strength and loads, is well known to statistieians. Engineers too are familiar with the typieal appearanee of sets of tensile strength data in whieh most of the individuals eongregate around mid-range and fewer further out to either side. Statistieians use the mean to identify the loeation of a set of data on the seale of measurement and the variance (or standard deviation) to measure the dispersion about the mean. In a variable x , the symbols used to represent the mean are /i and i for a population and sample respeetively. The symbol for varianee is V. The symbols for standard deviation are cr and. V respeetively, although a is often used for both. In this book we will always use the notation /i for mean and cr for the standard deviation. 

Geometrical standard deviation A measure of the range of particulate sizes present in a collection of particles. 

There are instances where deviations, as measured by the standard error, are scaled to the magnitude of the mean to yield the coefficient of variation. This is calculated by... 

Finally, a sample of 21 substances and the analysis of the standard deviations of measurements of LEL show that these standard deviations are not equal and therefore reflect different causes of variation. Without making the statistical analysis worse, the experimental values are very unstable eind therefore heirdly useful. 

Moving ahead to describe the details of this claim, we need here to develop a few basic review concepts. The standard deviation of measurements is determined by first calculating the mean, then taking the difference of each control result from the mean, squaring that difference, dividing by n — 1 then taking the square root. All these operations are implied in the following equation 71-3 ... 

Lee and Chau [66] have discussed the development and certification of a sediment reference material for total polychlorobiphenyls. Alford Stevens et al. [49] in an inter-laboratory study on the determination of polychlorobiphenyls in environmentally contaminated sediments showed the mean relative standard deviation of measured polychlorobiphenyl concentrations was 34%, despite efforts to eliminate procedural variations. Eganhouse and Gosset [67] have discussed the sources and magnitude of bias associated with the determination of polychlorobiphenyls in environmental sediments. Heilman [30] studied the adsorption and desorption of polychlorobiphenyl on sediments. 

The standard deviation in measurements, however, can vary with the analyte concentrations. On the other hand, RSD, which is expressed as the ratio of standard deviation to the arithmetic mean of replicate analyses and is given as a percentage, does not have this problem and is a more rational way of expressing precision ... 

FIG. 14. Plot of nanotubule diameter versus plating time for the Au nanotubule membranes. The standard deviation reflects measurements on at least three different membranes prepared under identical conditions. The circles are for membranes that have the Au surface layers still present on the membrane faces. The triangles are after removal of both surface layers. 

Weights o) = 1/s = standard deviation at concentrations i. n = number of standard measurements... 

Bergman and Agren (1984) used a similar cell to study the properties of MnO-NiO and also conducted a detailed analysis of the standard deviation of measured EMFs as a function of composition, showing that this could vary substantially across the system. 

Ri-measured PO2. Global Ischemia showed complete hypoxia for both groups with or without KCI arrest. Error bars represent one standard deviation of measurements from multiple hearts (data adapted from Ph.D. thesis of HImu Shukla, UT Southwestern 1994) [405]. 

A popular approach to comparing the tolerance and standard deviation of measurements is to define the process capability index, Cp or CI, as... 

There are several terms used in measurement uncertainty that must be defined. An uncertainty arising from a particular source, expressed as a standard deviation, is known as the standard measurement uncertainty (u). When several of these are combined to give an overall uncertainty for a particular measurement result, the uncertainty is known as the combined standard measurement uncertainty (uc), and when this figure is multiplied by a coverage factor ( ) to give an interval containing a specified fraction of the distribution attributable to the measurand (e.g., 95%) it is called an expanded measurement uncertainty [U). I discuss these types of uncertainties later in the chapter. 

Quantitation Limit. When the quantity of heavy metals is determined from the calibration curve, it is recommended to estimate the lowest value of heavy metals concentration as the quantitation limit. The methods to estimate the quantitation limit are described in JP and ICH Q2B Guidelines [2], and an appropriate method should be selected from among these methods. An estimation of the quantitation limit can be obtained from the standard deviation of measured values of the low-concentration test solution. The standard deviation of background noise will be used to estimate a value for the standard signal-to-noise ratio (10 1). 

The function f(x) has its maximum value at x = x and drops off exponentially with the square of the deviation of x from the mean, when such deviations are measured as fractions or multiples of the standard deviation. 

The preexponential factor accomplishes the normalization of the function that is, the integral of the function over all possible values of x (— oo to oo) equals unity. In a broad distribution a is large, and the exponential does not drop off as rapidly as in a narrow distribution (recall that all deviations are measured relative to the standard deviation). 

The standard deviation, s, measures how closely the data are clustered about the mean. The smaller the standard deviation, the more closely the data are clustered about the mean (Figure 4-2). 

For quality assessment of an analytical process, a control chart could show the relative deviation of measured values of calibration check samples or quality control samples from their known values. Another control chart could display the precision of replicate analyses of unknowns or standards as a function of time. 

If m and ri2 are unity, r2/cA is plotted versus ca/cr. Then ki is obtained from the intersection of the resulting straight line and the ordinate, whereas ki is its slope. Standard mathematical methods, such as linear- and multiple regression, or search techniques based on least-squares-methods to minimize the deviation of measured and calculated reaction rates, must be applied to determine the rate constants when m and m are different from unity. 

Standard Deviation Terms Measurements Plate Number n=... 

A more useful statistical term for error analysis is standard deviation, a measure of the spread of the observed values. Standard deviation, s, for a sample of data consisting of n observations may be estimated by Equation 1.3. 

Although Euclidean and Mahalanobis distances are the ones most commonly used in analytical chemistry applications, there are other distance measures that might be more appropriate for specific applications. For example, there are standardized Euclidean distances, where each of the dimensions is inversely weighted by the standard deviation of that dimension in the calibration data (standard deviation-standardized), or the range of that dimension in the calibration data (range-standardized). 

Standard deviation usual measure of the dispersion of observed values or results expressed as the positive square root of the variance (ASTM D-2013 ASTM D-2234). 

As in any measurement, there are error sources whose effects must be quantified. There are two ways to assess error in our analysis. The first, a simple comparison of analytical results with actual elevations (Fig. 9), results in a standard deviation (between measured and actual) of a = 372 m, which is small relative to the elevation changes we consider in major tectonic events. This simple empirical approach to the error depends on the number of samples analyzed and is thus not intrinsic to the technique. The error can be reduced simply by taking more samples... 

Element True value Standard deviation of replicates Deviation of measured value from true value Deviation of modeled value from true value... 

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