Standard deviation with calculator

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 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.

Wanek, P. M., et al., Inaccuracies in the Calculation of Standard Deviation with Electronic Calculators, Anal. Chem. 54, 1982, 1877-1878. 

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. 

To calculate the standard deviations of the average actual data and use these in future runs as a trend warning when the standard deviation with new data added changes measurably. 

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. 

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. 

To illustrate the coefficient of variation, consider the following (extremely simple and artificial) example. Imagine tliat there are two random variables in an early therapeutic exploratory clinical trial. One random variable is pulse (ranging from 50 to 80) and the other is age, which in this case is pulse minus 20. We can see that, from this example, values of pulse and age are just as disperse, but what differs between them is the mean. Hence, when we calculate the standard deviation, one random variable will appear to have more or less dispersion, but, after re-scaling the standard deviation with the sample mean, the measure of dispersion is the same. 

It has been possible to use the statistics from the least squares fit of the microwave spectrum through propagation of errors formulas (including the correlation of errors between the determined rotational and distortion constants) to calculate standard deviations for the force constants calculated from the distortion constants. These standard deviations, which represent measurement errors, can be compared with the lack of internal consistency between force constants calculated from different distortion constants. This comparison gives the relative magnitude of measurement vs. model errors. 

Highlight cells B9 C13, and click on From the Statistical function, scroll down to LINEST and chck OK. For BCnown y s, enter the array B3 B7, and for Known x s, enter A3 A7. Then in each of the boxes labeled Const and Stats, type true . Now we have to use the keyboard to execute the calculations. Depress Shift, Control, and Enter, and release. The statistical data are entered into the highlighted cells. This keystroke combination must be used whenever performing a function on an array of cells, like here. The slope is in cell B9 and its standard deviation in cell BIO. The intercept is in cell C9 and its standard deviation in cell CIO. The coefficient of determination is in cell Bll. Compare the standard deviations with those calculated in Example 3.22, and the slope, intercept, and with Example 3.21 or Figure 3.8. 

The 8-run design gives us a twofold decrease in the standard deviation, with respect to that of a direct comparison. The calculation demonstrates clearly that small effects may be determined with a considerably better precision than by changing one factor at a time. Any error in one of the experiments is "shared" equally between the estimates of the different parameters. The design is robust. 

We take the example of 3 variables, for a manufacturing or factory scale process whose performance and reproducibility is well known. The factors effects may be investigated by a 2 design (4 experiments) plus centre point. A cycle of the 5 experiments is performed and the effect of each variable determined as described in chapter 2. Since its standard deviation a is already known, each measured effect may be compared with the calculated standard deviation for each effect, c/2. 

Figure 1.56. Salt form of a phenyl-end-capped tetramer of PANI. Bond lengths are in A. The calculated standard deviation in the least significant figure is in parentheses. (Reproduced from ref 339 with kind permission. Copyright (1988) American Institute of Physics.)...

In our example, since all information comes from a single sample, the value of n under the square root sign is the same as that appearing in tn-i-As we have said, the value of n in tn-i arises from the number of observations used to estimate the standard deviation whereas the n in /n is the number of observations used to calculate the sample average. These values, however, can be different. Later we will see examples in which we combine information from several different samples to estimate a standard deviation. With this procedure, the value of s — and therefore the value of — will have more degrees of freedom than that for a single sample. As a consequence, the confidence intervals will become narrower, and our predictions more precise. 

Kragten, J. (1994) Calculating standard deviations and confidence intervals with a universally applicable spreadsheet technique. Analyst, 119, 2161-2165. 

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