Standard deviation calculation

Finally, it is worth investigating how deterministic values of material strength are calculated as commonly found in engineering data books. Equation 4.14 states that the minimum material strength, as used in deterministic calculations, equals the mean value determined from test, minus three standard deviations, calculated for the Normal distribution (Cable and Virene, 1967) ... 

The uncertainties are standard deviations, calculated in the appropriate manner (Whittaker Robinson, 1937). 

Means and standard deviations calculated according to Eqs. (4.9) and (4.12) or the following, respectively, are only characteristic for sample subsets if certain pre-requisites are satisfied, the fulfilment of these can be verified by tests. The observed values of measurement series have to... 

The precision limits r and R are given by equations (4.3a) and (4.3b), respectively, where tv>a is the Student /-value for v degrees of freedom and a corresponds to the stated probability, sr is the repeatability standard deviation and Sr is the reproducibility standard deviation calculated from (v + 1) results ... 

Most of the observations are likely to be within the range x 2 s, and practically all within the range x 3. . In common with the sample mean, the value of the standard deviation calculated from a set of observations is only an estimate of the true or population value of the standard deviation s becomes a better estimate of population value) as N increases and... 

Mean and standard deviation calculated from data from 10 rats per diet group. Corrected by adjusting test diets to 2.50 for casein (AOAC). 

If we don t have such an ideal control sample, but only one with a matrix different from the routine sample (e.g. a standard solution) than we have to consider also the uncertainty component arising from changes in the matrix. For this purpose we use the (repeatability) standard deviation calculated from repeated measurements of our routine samples (performed e.g. for a range control chart). When we estimate the reproducibility within laboratory we now have to combine both contributions by calculating the square root of the sum of squares. 

To measure the thermal expansion coefficients, a 0.125 in. thick sample was taken from the cured plates. The Increase in length with temperatures was measured by use of a Dupont 941 Thermomechanical Analyzer (TMA), with a heating rate of 5°C/min. The instrument was calibrated with an aluminum standard. Three runs were made for each sample and standard deviations calculated. 

Dpdft is the mean pore diameter calculated from the NLDFT pore size distribution ctpdft is the standard deviation calculated from the NLDFT pore size distribution dwaii = ao - DPDFT- the pore wall thickness assuming cylindrical pores... 

Suppose one takes n measurements of a quantity and the measurements are subject to random errors. The value of the mean so measured follows a normal distribution about the true mean. Most calculators give a standard deviation calculated from the formula... 

Figure 3. Relative standard deviations calculated from 10 measurements...

Initial concentration 5.00 x 10-4 M. b Standard deviations calculated by the author. 

Concentration of spinach powder in slurry, % m/V Mean absorbance Relative standard deviation, % Calculated detection limit, xg/g... 

Standard deviation calculated from experimental data. 

Chromatograph five injections of equal volume, up to 25 pL, of the Standard Solution, and measure the peak response. The relative standard deviation, calculated by the formula... 

Standard Solutions Run through the chromatograph a commercially available standard containing a mixture of fatty-acid methyl esters. Fatty acids and methyl esters with a wide range of carbon numbers and double-bond configurations can be purchased. The calculated concentration should compare to that claimed within 2(7, where cr is the standard deviation calculated from at least 10 replicate determinations, preferably made over a period of several days. 

The computerized aqueous chemical model of Truesdell and Jones (, 3), WATEQ, has been greatly revised and expanded to include consideration of ion association and solubility equilibria for several trace metals, Ag, As, Cd, Cu, Mn, Ni, Pb and Zn, solubility equilibria for various metastable and(or) sparingly soluble equilibrium solids, calculation of propagated standard deviation, calculation of redox potential from various couples, polysulfides, and a mass balance section for sulfide solutes. Revisions include expansion and revision of the redox, sulfate, iron, boron, and fluoride solute sections, changes in the possible operations with Fe (II, III, and II + HI), and updating the model s thermodynamic data base using critically evaluated values (81, 50, 58) and new compilations (51, 26 R. M. Siebert and... 

Cetin et al. [29] proposed a simple, precise, fast, and low-cost differential pulse (DP) polarographic method for fhe deferminafion of lornoxicam in pharmaceutical formulations. Results have been compared with those of high-performance liquid chromatographic (HPLC) method. Mean values and standard deviations calculated by 10 determinations were 8.10 0.12 mg for DP polarography and six determinations were 8.02 0.08 mg for HPLC mefhod. The sfafisfical evaluations indicated that there was no significant difference between the mean values and precessions of fhe fwo mefhods af 95% confidence level (f = 2.64, F = 2.25). 

The term system stability is a measure of the bias in assay results generated during a preselected time interval, e.g., every hour up to 10 h, using a single solution. Stability should be determined by replicate analysis of the sample solution and the relative standard deviation calculated. The assay results obtained in different time intervals should not exceed more than 20% of the corresponding value of the system precision. If the value is higher on plotting the assay results as a function of time, the maximum duration of the usability of the sample solution can be calculated. 

Each value is the average concentration (and standard deviation) calculated from... 

Note-. The coefficient of variation following multiple runs using the indicated positive control test substances in one laboratory is shown. The overall average of the intralaboratory CVs listed is 17%. n is the number of times the assay has been conducted with the indicated positive control material units indicates the measurement units obtained from the alternative method mean indicates the average value obtained for all the indicated runs SD is the standard deviations calculated associated with the mean of the alternative method sources CV is the coefficient of variation (mean/SD) SLS is sodium lauryl sulfate. 

Note in Example 6-1 that the difference between and (Sx,) /iV is very small. If we had rounded these numbers before subtracting them, a serious error would have appeared in the computed value of s. To avoid this source of error, never round a standard deviation calculation until the very end. Furthermore, and for the same reason, never use Equation 6-5 to calculate the standard deviation of numbers containing five or more digits. Use Equation 6-4 instead. Many calculators and computers with a standard deviation function use a version of Equation 6-5 internally in the calculation. You should always be alert for roundoff errors when calculating the standard deviation of values that have five or more significant figures. 

Some analysts, such as US EPA (2002) for the analysis of 1,3-butadiene carcinogenicity examined in more detail below, assumed a distribution shape (normal in that case, with a standard deviation calculated from the estimated MLE and confidence limits of the qi estimates). However, as will be shown below, such assumptions are approximate at best even in favorable cases. Zeise et al. (1991) demonstrated a procedure to generate the required distribution by tracing the likelihood profile of the linear term ( i) for a linearized multistage fit to dichotomous tumor data. This used a modified version (Zeise and Salmon 1991) of the MSTAGE program developed by Crouch (1985). More recently. Crouch (2006) developed a similar program that provides the likelihood distribution for estimates of qi when... 

Q = R/r, where R = observed standard deviation, and r = theoretical standard deviation, calculated under the assumption of complete randomness. 

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