Mean errors and standard deviations

AACMM evaluation test From the above data sets mean error and standard deviation for diameter are obtained and represented for each position. Total standard deviations of 0.0136 and 0.0181 mm and ranges of 0.0709 and 0.0828 mm (laboratory 1, circle 1 and 2 respectively) have been found. In laboratory 2, test measurements results on standard deviations of 0.0107 and 0.0116 mm and ranges of 0.05601 and 0.0610 mm. Fig. 2 shows the variation of the center error and its standard deviation per gauge position. 

Each measure characterizes a specific aspect of the performance of the model and the basis s The standard deviation Astd characterizes the distribution of errors A,- — A about a mean value A and Astd thus quantify systematic and nonsystematic errors. The mean absolute error Aabs represents the typical magnitude of the error in the calculations and Amax gives the largest error. Graphically, the mean error and standard deviation can be represented in terms of the normal distiibutUm... 

In the present subsection, we consider the mean errors and standard deviations of bond distances calculated using the Hartree-Fock, MP2, MP3. MP4, CCSD, CCSD(T) and CISD models in the cc-pVDZ, cc-pVTZ and cc-pVQZ sets with all electrons correlated. The mean errors and standard deviations are listed in Table 15.4 and plotted in Figure 15.3. As discussed in Section 15.3.1, the sample comprises all bonds in Table 15.3 except those of HNO. 

In the next several sections, the theoretical distributions and tests of significance will be examined beginning with Student s distribution or t test. If the data contained only random (or chance) errors, the cumulative estimates x and 5- would gradually approach the limits p and cr. The distribution of results would be normally distributed with mean p and standard deviation cr. Were the true mean of the infinite population known, it would also have some symmetrical type of distribution centered around p. However, it would be expected that the dispersion or spread of this dispersion about the mean would depend on the sample size. 

You need not spend much time attempting to master rigorous statistical theory. Because EVOP was developed to be used by nontechnical process operators, it can be applied without any knowledge of statistics. However, be prepared to address the operators tendency to distrust decisions based on statistics. Concepts that you should understand quantitatively include the difference between a population and a sample the mean, variance, and standard deviation of a normal distribution the estimation of the standard deviation from the range standard errors sequential significance tests and variable effects and interactions for factorial designs having two and three variables. Illustrations of statistical concepts (e.g., a normal distribution) will be valuable tools. 

Fifty data sets containing absolute errors were then generated by adding random numbers of mean zero and standard deviation 0.05 to each v value. Another 50 data sets were generated to contain relative error, by multiplying each v value by a random number of mean one and standard deviation 0.10. These 100 data sets were then analyzed individually by each of the four methods, to generate 100 Km values for each method. The average Km value and its uncertainty for each method and each error type are summarized in Table 2.4. 

At this point, it is worth emphasizing the difference between the terms "standard error" and "standard deviation," which, despite the same initial word, represent very different aspects of a data set. Standard error is a measure of how certain we are that the sample mean represents the population mean. Standard deviation is a measure of the dispersion of the original random variable. There is a standard error associated with any statistical estimator, including a sample proportion, the difference in two means, the difference in two proportions, and the ratio of two proportions. When presented with the term "standard error" in these applications the concept is the same. The standard error quantifies the extent to which an estimator varies over samples of the same size. As the sample size increases (for the same standard deviation) there... 

GraphPad Prism software (San Diego, CA) to a sigmoidal dose-response curve with variable slope model. Data points are the mean values and standard deviations are indicated with error bars (data from reference [89]). Reproduced with permission from (2006) Analytical Chemistry, 78, 3659-66. Copyright 2006 American Chemical Society. 

Figure 1.5 Sample residual plot. Paired (x, Y) data were simulated using the model Y = 13 + 1.25x + 0.265x2. To each Y value was added random error from a normal distribution with mean zero and standard deviation 25. The top plot is a plot ordinary residuals versus predicted values when the fitted model was a second-order polynomial, the same model as the data-generating model. The bottom plot is the same plot when the fitted model was linear model (no quadratic term). Residual plots should appear as a shotgun blast (like the top plot) with no systematic trend (like the bottom plot).

Parametric methods for estimating the Cl of a statistic require the assumption of a sampling distribution for the statistic and then some way to calculate the parameters of that distribution. For example, the sampling distribution for the sample mean is a normal distribution having mean p, which is estimated by the sample mean x, and standard deviation equal to the standard error of the mean SE (x), which is calculated using... 

Consider, for example, the function f(x) = exp(x) on foe interval [1,2], whose slope is also exp(x). This graph is shown in as a solid line in Figure 7.1. To it has been added random error that we will call noise. In our example foe noise is normally distributed with mean 0 (i.e. no systematic error) and standard deviation 0.25. We examine how (me might set about identifying and eliminating this noise in a way appropriate to TS-PFRdata. 

To obtain an expression for k p) we will assume that the process execution times for both algorithms (a) and (b) form a normal distribution, which is a reasonable assumption (according to the Central Limit Theorem from probability theory) when there is a large number of tasks per process. Assuming a normal distribution with mean /r and standard deviation a, the probability of a process execution time being below fj. + ka can be computed as 5 + jerf(k/V2), where erf denotes the error function. If there are p processes, the probability that all process execution times are below fi+ka is then given as [ -F jerf k/V2)]P. We need Eqs. 7.5 and 7.6 to be fairly accurate estimates for the maximum execution time, and we must therefore choose k such that... 

To apply the Bayesian model updating procedure, prior distributions with large standard deviations are selected for the model parameters. Lognormal (LN) distributions with mean 1 and standard deviation of the logarithm equal to 1 are adopted for the stiffness parameters LN distributions with mean 0.02 and standard deviation of the logarithm equal to 1 are adopted for the damping ratios LN distributions with mean —1.651 and standard deviation of the logarithm equal to 1 are adopted for the prediction-error variances for the structural accelerations. 

Figures 1 and 2 show our results for samples that were charred for two hours, in air and in evacuated tubes. We show mean values and standard deviations of the parameters. The standard deviations are shown as error bars in the figures.

The mean absolute error and standard deviation of the maximal propene yield represent relative errors of approximately 0.7% and 0.8% with respect to the global maximum. 

The mean absolute error and standard deviation of the diversity among the retained catalysts, i.e., the mean absolute error and standard deviation of the difference between the highest propene yield and the mean propene yield of all the retained catalysts, are 0.07% and 0.09%, respectively. It is worth noticing that these values are smaller than the t5 ical experimental accuracy of catalytic mesurements, which is 0.1%. 

In Figure 15.8 we have, for the calculated dipole moments in the aug-cc-pVXZ basis sets, plotted the mean errors, the standard deviations, the mean absolute errors and the maximum absolute errors. The plots indicate that the calculated dipole moments depend in a systematic mann - on the cardinal number and the correlation treatment In general, the dipole moments arc reduced as we improve the correlation treatment. Indeed, with the exceptions of CO and HNC, the dipole moment is always reduced as we go from Hartree-Fock to CCSD and then on to CCSD(T) - see Table 15.11. The MP2 dipole moments are less systematic but are usually slightly smaller than the CCSD numbers. At the aug-cc-pVQZ level, the mean absolute errors are 0.17 D for the Hartree-Fock model, 0.05 D for the MP2 model, 0.04 D for the CCSD model and 0.01 D for... 

Figure 7.3 Mean values and standard deviation of errors in bond energies for svrp)...

The F statistic, along with the z, t, and statistics, constitute the group that are thought of as fundamental statistics. Collectively they describe all the relationships that can exist between means and standard deviations. To perform an F test, we must first verify the randomness and independence of the errors. If erf = cr, then s ls2 will be distributed properly as the F statistic. If the calculated F is outside the confidence interval chosen for that statistic, then this is evidence that a F 2. 

Vitha, M. F. Carr, P. W. A Laboratory Exercise in Statistical Analysis of Data, /. Chem. Educ. 1997, 74, 998-1000. Students determine the average weight of vitamin E pills using several different methods (one at a time, in sets of ten pills, and in sets of 100 pills). The data collected by the class are pooled together, plotted as histograms, and compared with results predicted by a normal distribution. The histograms and standard deviations for the pooled data also show the effect of sample size on the standard error of the mean. 

This same data is plotted in the chart on the following page. The mean absolute deviation and standard deviation are plotted as points with error bars, and the shaded blocks plot the largest positive and negative-magnitude errors. 

For the usual accurate analytical method, the mean f is assumed identical with the true value, and observed errors are attributed to an indefinitely large number of small causes operating at random. The standard deviation, s, depends upon these small causes and may assume any value mean and standard deviation are wholly independent, so that an infinite number of distribution curves is conceivable. As we have seen, x-ray emission spectrography considered as a random process differs sharply from such a usual case. Under ideal conditions, the individual counts must lie upon the unique Gaussian curve for which the standard deviation is the square root of the mean. This unique Gaussian is a fluctuation curve, not an error curve in the strictest sense there is no true value of N such as that presumably corresponding to a of Section 10.1—there is only a most probable value N. 

Mean and standard deviations, in % of the nominal concentration, found for simulations under various combinations of (a) random variation of EO, (b) volumetric (reading) error in Yl, and (c) use of a pH/lon-meter with a resolution of 0.1 or 0.001 mV. For the last line the exact volume V2 added was varied in the range 2... 3 ml to simulate actual working conditions, and 100 repetitions were run. 

Features The allowed range is defined by two specification limits, the experimental mean and standard deviation are assumed, and the probability of error a is preset. The calculation is done using Eq. (1.37) in Section 1.6. 

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