Calculator standard deviation

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. 

The uncertainties given are calculated standard deviations. Analysis of the interatomic distances yields a selfconsistent interpretation in which Zni is assumed to be quinquevalent and Znn quadrivalent, while Na may have a valence of unity or one as high as lj, the excess over unity being suggested by the interatomic distances and being, if real, presumably a consequence of electron transfer. A valence electron number of approximately 432 per unit cell is obtained, which is in good agreement with the value 428-48 predicted on the basis of a filled Brillouin polyhedron defined by the forms 444, 640, and 800. ... 

Figure 4.48. The calculated standard deviation and its upper CL. A series of 10 measurements was simulated, bottom panel), with the newest addition at each step given in bold. The corresponding SD is given by the thick line in the top panel, and the 80. .. 97.5% CLy by thin lines. Notice that point 5, which is high, drives the SD up from = 0.9 to = 1.5 (E(a) = 1) the 95% CL is at 2.38 cr, respectively 3.6. The ordinates are both scaled in units of a. This depiction, for just one level of p, is part of the display of program CONVERGE.

In fact, we have talked about the Durbin-Watson statistics in previous chapters, although a long time ago and under a different name. Quite a while ago we published a column titled Alternative Ways to Calculate Standard Deviation [7], One of the alternative ways described was the calculation by Successive Differences. As we shall see, that calculation is very closely related indeed to the Durbin-Watson Statistic. More recently we described this statistic (more directly named) in a sidebar to an article in the American Pharmaceutical Review [8],... 

The other formula is the formula for calculating Standard Deviation by successive Differences ... 

The calculated standard deviation is now regarded as a standard uncertainty and is expressed as ... 

Clearly, both kinetic parameters are well defined with standard errors of less than 5%. Also, all component spectra are clearly resolved, see Figure 4-49. The calculated standard deviation of the residuals matches the noise level of the generated data sets. 

Figure 4-55. Distributions of the calculated standard deviations of the fitted parameters for 1000 experiments. Top panel for n, bottom panel for X2. The white bars represent the %2 fits, the black bars the ssq-fits the arrows represent the standard deviations of the distributions of Figure 4-54.

Parameter considered a standard for the purpose of calculating standard deviations. 

The value of these transformations is that, on the transformed scale, the calculated standard deviation is independent of where we are on the scale. If this were not the case, we would tend to overestimate the variability for small values of y, and underestimate it for large values of y. 

Calculators and spreadsheets include the following function to calculate standard deviation (for an infinite number of measurements, s is replaced by a, x is replaced by p and n — 1 is replaced by n) ... 

For anisometric particles (e.g., needle- or platelet-shaped particles) mathematical statistics may likewise be applied [1.10]. The two-dimensional logarithmic normal distribution of the length L and breadth B of the particles also allows the representation and calculation of the characteristic parameters and mean values. The eccentricity of the calculated standard deviation ellipse (Fig. 2) is a measure of the correlation between the length and breadth of the particle. By using more than two... 

Measure ionized magnesium using selected controls on three instruments. Calculate standard deviation. Report the results as a mean value of four measurements and the results of our measurements are given in Table 1.3. 

Prepare Data Table 1.3. Calculate the percent deviation (+/-) of the experimental volume recorded from the volume value marked on the pipette. See the Radioanalytical Chemistry textbook, Section 10.3 Measurement Uncertainty, for guidance on calculating standard deviation. 

Figure I. Effect of errors in x, T, and ir on the calculated standard deviation of E in Equation 9 (1) 3 simultaneous errors (2) x-error (3) T-error ...

Ett = pressure difference in Equation 8 Pi = vapor pressure of Component i saturated with salt s, s , s, sT = estimate of standard deviation of the error in the experimental variables x, y, tt, and T, respectively Szi, sZ2 = estimate of the standard deviation of the error in the calculated energy parameters SE = calculated standard deviation of E in Equation 9 S == calculated standard deviation of the error in y in Equation 11 T = temperature, °C Vi = molar volume of Component i... 

Fourteen standard copper and brass alloys, the compositions of which have been certified by the National Bureau of Standards, have been used to calculate the concentrations of various elements in the coins (NBS C-1100. 1101, C-1102, 1106, C-1109, C-1111, C-1112, C-lllS, 1116, C-1120, 63C, 62D, 157A, and 158A). All standards were prepared metallographi-cally, ending with a diamond polish to obtain a surface representative of the interior of the standard. Excellent calibration curves were obtained with very good precision for both standards and coins, the calculated standard deviation is about 0.003% for Fe, 0.004% for Ni, 0.005% for Ag, 0.002% for Sn, 0.004% for Sb, and 0.003% for Pb. 

In this expression, as before, 5 is the calculated standard deviation and n is the sample size. The quantity t is known as the r-value, and can be determined from statistical tables. Its precise value depends on the level of accuracy required together with a quantity known as the number of degrees of freedom, which is equal to - 1. Relevant values of t are given in Appendix 5. 

Reactivity profiles obtained from all three chars are evaluated by using the normalisation procedure. The quality of this procedure is indicated by the calculated standard deviation (o) relative to the reference profile a) Wheat straw 11 out of 14 experiments shows coke experiments below 1000 C shows an extremely good fit, a< 2.4%. At 1000 C fiX) changes dramatically for Longyear coke, as the CO2 partial pressure is reduced from 1.0 to 0.03bar. (The calculated standard deviations are based on (0.2< X. 8) and a step size of AX = 0.002.)... 

In addition to allowing fine-tuning of the fitted parameters, the final step of simplex searching offers a convenient means of estimating the error associated with each parameter. This process has been described by Phillips and Eyring [30]. Briefly, one determines a quadratic approximation to the error surface, from which an error matrix is developed. This matrix can then be used to calculate standard deviations of the fitted parameters. These standard deviations are reported as error estimates of the parameters in Table 2. 

The calculated standard deviation s is a good estimator of the population standard deviation cr if the number of measurements is high enough. An equivalent alternative form of Eq. 16.2 is presented in Eq. 16.3 this form is more useful when a nonprogrammable calculator is used. 

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