Standard error calculations

The observed difference Xj — X2 = 5.4 mmHg is the signal and suppose also by applying the formula the standard error calculation has given se = 2.57. 

For each parameter, means and standard error calculated, an analysis of variance is performed and the appropriate tests of significance are applied. The mean values of each parameter of the treated groups are compared with the values of the vehicle control group. It is important to compare the results with reference to their use for the rat strain commonly-used in the Laboratory. For many studies it may be advisable to include groups treated with reference compounds of established endocrine activity. 

Figure 2. Iron absorption by adults from a range of foods. The bars represent the mean absorptions and standard errors, calculated from the logarithms of the percentage absorptions. (Reproduced, with permission, from Ref. 15. Copyright 1971, Grune and Stratton, Inc.)...

In general, all the estimation methods produced roughly equivalent parameter estimates with none of the methods considered to be significantly different from the others. Having the standard errors calculated using only the gradient or Hessian had no effect what-... 

The standard error calculated from the data in our example is 4.665, resulting in the interval... 

Typical accuracy values reported in the literature for integrating the various functional groups by NMR range from 3-5% 102), In these experiments, the error analysis was based on reproducing the LMS biopretreatment utilizing VA as the mediator three separate times. The black liquor lignin from each was isolated and analyzed. The standard error calculated from these three treatments for the NMR experiments was 3%. 

The actual values found at the pressure of the experiment, p, are simply Dab/P- AU errors given in this table are standard errors calculated by regressions analysis. This is defined by Eq. 16. 

Figure 5. Effect of free energy standard error (in calories) on temperature standard error for the brucite (+), muscovite (o) and chlorite (A) equilibria. In each case the large square indicates the temperature standard error calculated from the entropy of reaction and the actual standard error of the free energy of reaction. Other points on each curve were calculated by the Monte Carlo Method using arbitrarily assigned errors of the individual minerals.

Fig. 7.5 Yield of eS using a scavenger concentration of 0.1 M. An initial ion-pair separation of 20 A was used. Standard error calculated on the final yield to one standard deviation is 0.0012 (IRT) and 0.00114 (MC). Here MC refers to random flights simulation...

Fig. 7.16 Yield of eS at five different ion pair encounter distances a 3 A, b 5 A, c 7 A, d 10 A and e 13 A, using a scavenger concentration of 0.5M and an Onsager distance of 190A. An initial distance of 20 A between the radical ion pair was used, together with a scavenging encounter radius of 4 A. Standard error calculated on the final yield to one standard deviation is a 0.00216 (MC and IRT) and 0.00213 [IRT (uncorrected)] b 0.00216 (MC and IRT) and 0.00211 [IRT (uncorrected)] c 0.00213 (MC and IRT) and 0.0021 [IRT (uncorrected)] d 0.002 (MC and IRT) e 0.002 (MC and IRT). Here Monte Carlo (MC) refers to random flights simulation. Units of x-axis are in A ...

In Eq. (14) "est stands for the calculated (estimated) response, exp" for the experimental one, and n and m are the numbers of objects in the training set and the test set respectively. CoSE stands for Compound Standard Error. As an option, one can employ several test sets, if needed. 

Show by a propagation of uncertainty calculation that the standard error of the mean for n determinations is given as s/VTj. 

Example 1 Sample Quantity for Composition Quality Control Testing An example is sampling for quality control of a 1,000 metric ton (VFg) trainload of-Ks in (9.4 mm) nominal top-size bentonite. The specification requires silica to be determined with an accuracy of plus or minus three percent for two standard errors (s.e.). With one s.e. of 1.5 percent, V is 0.000225 (one s.e. weight fraction of 0.015 squared). The problem to be solved is thus calculating weight of sample to determine sihca with the specified error variance. 

Example 2 Calculation of Error with Doubled Sample Weight Repeated measurements from a lot of anhydrous alumina for loss on ignition established test standard error of 0.15 percent for sample weight of 500 grams, noting V is the square of s.e. Calculation of variance V and s.e. for a 1000 gram sample is... 

Example 3 Calculating Sample Weight for Screen-Size Measurement Weight W of bulk sample for screen analysis is calculated by the Gayle model for percent retained on a specified screen with relative standard error s.e. in percent... 

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... 

Alternatively, the experimental error can be given a particular value for each reaction of the series, or for each temperature, based on statistical evaluation of the respective kinetic experiment. The rate constants are then taken with different weights in further calculations (205,206). Although this procedure seems to be more exact and more profoundly based, it cannot be quite generally recommended. It should first be statistically proven by the F test (204) that the standard errors in fact differ because of the small number of measurements, it can seldom be done on a significant level. In addition, all reactions of the series are a priori of the same importance, and it is always a... 

The whole calculation can also be extended easily to measurements of different weights, i.e., of different accuracy. Let us denote the standard error in Yij by 6y. The weight wy of a measurement can be defined as... 

Two models of practical interest using quantum chemical parameters were developed by Clark et al. [26, 27]. Both studies were based on 1085 molecules and 36 descriptors calculated with the AMI method following structure optimization and electron density calculation. An initial set of descriptors was selected with a multiple linear regression model and further optimized by trial-and-error variation. The second study calculated a standard error of 0.56 for 1085 compounds and it also estimated the reliability of neural network prediction by analysis of the standard deviation error for an ensemble of 11 networks trained on different randomly selected subsets of the initial training set [27]. 

LOQs for each of the three parent herbicides in surface water were determined using all the analytical results (not corrected for background) of samples fortified at the lowest fortification level, 0.05 pgL , during the analysis in years 1995-2001. The calculated LOQs were below 0.05 pgL for acetochlor and metolachlor and approximately 0.05 pgL for alachlor. If the true concentration of an analyte is at the LOQ or greater, the standard error of individual measured concentration values relative to the true concentration is at most 10%. 

Once we have estimated the unknown parameters that appear in an algebraic or ODE model, it is quite important to perform a few additional calculations to establish estimates of the standard error in the parameters and in the expected response variables. These additional computational steps are very valuable as they provide us with a quantitative measure of the quality of the overall fit and inform us how trustworthy the parameter estimates are. 

If on the other hand we wish to compute the(l-a) 100% confidence interval of the response of y, at t=to, we must include the error term (e0) in the calculation of the standard error, namely we have... 

Subsequently, Watts performed a parameter estimation by using the data from all temperatures simultaneously and by employing the formulation of the rate constants as in Equation 16.19. The parameter values that they found as well as their standard errors are reported in Table 16.18. It is noted that they found that the residuals from the fit were well behaved except for two at 375°C. These residuals were found to account for 40% of the residual sum of squares of deviations between experimental data and calculated values. 

Once the standard error of estimate of the mean forecasted response has been estimated, i.e., the uncertainty in the total production rate, one can compute the probability level, a, for which the minimum total production rate is below some pre-determined value based on a previously conducted economic analysis. Such calculations can be performed as part of the post-processing calculations. 

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