Normalized standard error

During the training session the performance of the network must be monitored. Different performance criteria are possible, but usually the normalized standard error, NSE, is used. 

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. 

The ability of the sequential design to discriminate among the rival models should be examined as a function of the standard error in the measurements (oe). For this reason, artificial data were generated by integrating the governing ODEs for Model 1 with "true" parameter values kt=0.31, k2=0.18, k3=0.55 and k4=0.03 and by adding noise to the noise free data. The error terms are taken from independent normal distributions with zero mean and constant standard deviation (oE). 

Figure 43-6 Family of curves of multiplication factor as a function of Er, for different values of the parameter sigma (the noise standard deviation), for Normally distributed error. Values of sigma range from 0.1 to 1.0 for the ten curves shown, (see Color Plate 5)...

Fig. 4.6. The integrated signals of harmonics from an H2 gas (lO.OTorr), a D2 gas (lO.OTorr), and a gas of their mixture (lO.OTorr) measured as a function of the harmonic order q. The signals are normalized to those from the H2 gas and vertical errors represent SEM (standard error of the mean) for 800 laser shots...

Fig. 6A, B Average relative profile and standard deviation of FMs in A influent and B primary effluent in the U.S. and Europe [11]. The highest concentration FM was normalized to 1. The highest concentration FM (in pg/L) is in parentheses. The error bars represent the normalized standard deviation of the mean...

Fig. 9. Viscoelastic properties of ultrasound-treated ex vivo porcine muscle specimen. Muscle samples were coagulated with focused ultrasounds in selected regions. MRE using the method of Ref. 23 was carried out and shear moduli were calculated in normal and heated regions at different shear wave frequencies. Upper curve FUS-treated tissue. Lower curve normal tissue. Error bars are for standard deviations. (From Ref 47, reprinted by permission of Wiley-Liss, Inc., a subsidiary of John Wiley Sons, Inc.)...

Figure 1. Top portion shows a plot of the observed Fenvalerate response vs. the mass (ng). Lower plot gives ordered, normalized residuals from the fit of model-3 to the data (Table IV) using the weights given in Table III. (Symbols indicate the five replicates, and the plotted residuals are normalized by the standard deviations for these individual replicates. The "goodness of fit residuals of the model to the means of the replicates are larger by 1/5T because they are normalized by the standard errors at each concentration.)...

It can be shown that if the uncertainties associated with the measurements of the response are approximately normally distributed (see Equation 3.8), then parameter estimates obtained from these measurements are also normally distributed. The standard deviation of the estimate of a parameter will be called the standard uncertainty, s, of the parameter estimate (it is usually called the standard error ) and can be calculated from the matrix if an estimate of is available. 

A statistical method for plotting the relative frequency (dN/N) of a probable error in a single measured value X versus the deviation (z) from fi, the mean of the data, in units of standard deviation (o-), such that z = (x -fji)/a. The standard error curve (shown below) does not depend on either the magnitude of the mean or the standard deviation of the data set. The maximum of the normal error curve is poised at zero, indicating that the mean is the most frequently observed value. 

An approach that is sometimes helpful, particularly for recent pesticide risk assessments, is to use the parameter values that result in best fit (in the sense of LS), comparing the fitted cdf to the cdf of the empirical distribution. In some cases, such as when fitting a log-normal distribution, formulae from linear regression can be used after transformations are applied to linearize the cdf. In other cases, the residual SS is minimized using numerical optimization, i.e., one uses nonlinear regression. This approach seems reasonable for point estimation. However, the statistical assumptions that would often be invoked to justify LS regression will not be met in this application. Therefore the use of any additional regression results (beyond the point estimates) is questionable. If there is a need to provide standard errors or confidence intervals for the estimates, bootstrap procedures are recommended. 

Concentrations in yg/m all data dispersion normalized. Values of F and r are given for the overall equation and the standard error of each coefficient is given. 

The first stage in deciding how to treat the results from a ruggedness test is to select a range of parameters to measure which will provide both qualitative and quantitative information on the method s performance. The second stage is to decide how best to evaluate the main effects, standard errors and interaction effects provided by the selected experimental design. For this discussion we will consider only the application of HPLC, normally one of the most complex analytical methods to evaluate. 

This expression of results allows us to immediately interpret the main effects and standard errors in terms of a percentage deviation from the normal observed values for that method. 

When we are dealing with samples rather than populations, we cannot use the standard normal deviate, Z, to make predictions since this requires knowledge of the population mean and variance or standard deviation. In general, we do not know the value of these parameters. However, provided the sample is a random one, its mean 5 is a reliable estimate of the population mean p, and we can use the central limit theorem to provide an estimate of o. This esti mate, known as the standard error of the mean, is given by ... 

Any experimental design that is intended to determine the effect of a parameter on a response must be able to differentiate a real effect from normal experimental error. One usual means of doing this determination is to run replicate experiments. The variations observed between the replicates can then be used to estimate the standard deviation of a single observation and hence the standard deviation of the effects. However, in the absence of replicates, other methods are available for ascertaining, at least in a qualitative way, whether an observed effect may be statistically significant. One very useful technique used with the data presented here involves the analysis of the factorial by using half-normal probability paper (19). 

The joint solution is p = 3.2301 and a = 2.9354. It might not seem obvious, but we can also derive asymptotic standard errors for these estimates by constructing them as method of moments estimators. Observe, first, that the two estimates are based on moment estimators of the probabilities. Let x, denote one of the 500 observations drawn from the normal distribution. Then, the two proportions are obtained as follows Let z,(2.1) = l[x, < 2.1] and z,(3.6) = l[x, < 3.6] be indicator functions. Then, the proportion of 35% has been... 

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