Standard error, data

Test concentrations of the essential oil were 100 pg/ml. All data are inhibitory percentage of spore germination of fungal pathogens tested and mean of 3 replicates with standard error. Data in a column not followed by the same letter are significantly different, p=0.05, ANVOA with Ducan s multiple range test. 

This sum, when divided by the number of data points minus the number of degrees of freedom, approximates the overall variance of errors. It is a measure of the overall fit of the equation to the data. Thus, two different models with the same number of adjustable parameters yield different values for this variance when fit to the same data with the same estimated standard errors in the measured variables. Similarly, the same model, fit to different sets of data, yields different values for the overall variance. The differences in these variances are the basis for many standard statistical tests for model and data comparison. Such statistical tests are discussed in detail by Crow et al. (1960) and Brownlee (1965). 

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. 

FIGURE 11.2 Paired experimental data. Values of constitutive calcitonin receptor activity [1 -(Tr/Tj) units] in transiently transfected melanophores. Five separate experiments are shown. Points to the left indicate the basal level of constitutive activity before (filled circles) and after (open circles) addition of 100 nM AC512 (calcitonin receptor inverse agonist). Lines join values for each individual experiment. Points to the right are the mean values for constitutive activity in control (filled circles) and after AC512 (open circles) for all five experiments (bars represent standard errors of the mean). Data shown in Table 11.3. 

The Standard Error of Prediction (SEP) is supposed to refer uniquely to those situations when a calibration is generated with one data set and evaluated for its predictive performance with an independent data set. Unfortunately, there are times when the term SEP is wrongly applied to the errors in predicting y variables of the same data set which was used to generate the calibration. Thus, when we encounter the term SEP, it is important to examine the context in order to verify that the term is being used correctly. SEP is simply the square root of the Variance of Prediction, s2. The RMSEP (see below) is sometimes wrongly called the SEP. Fortunately, the difference between the two is usually negligible. 

These two standard errors are to be considered identical, for comparisons of this kind can never be highly refined. The identity of these errors is a welcome indication that the spectrograph system was satisfactorily stable over the rather long period (nearly a working day) required for taking the data in Table 10-1. 

In order to estimate part of this level of error, data from the author s laboratory was used where the student calculated flashpoints of varied substances using the Cleveland apparatus oc and Setaflash cc that work below 70-80°C. Every year about thirty groups of two students analyse flashpoints of about ten substances. Measurements are thus repeated many times by each group, which enables estimations of standard deviations within a group intragroup standard deviation) to be made and also between groups (intergroup standard deviation). 

FIGURE 4.2 Concentrations of MeHg (mean 1 standard error) in zooplankton from Lake 240 of the Experimental Lakes Area (northwestern Ontario, Canada), showing seasonal variation during summer and pronounced rapid increases in mean concentration after the fall overturn. (Source Michael J. Paterson, Fisheries and Oceans Canada, Winnipeg, Manitoba, unpublished data.)... 

Table 4.1 Effect of selected thiols, disulphides, amino acids and antioxidants on the time to the onset and the time to reach maximal ischaemic contracture in isolated perfused rat hearts. Hearts were perfused for a control period of 10 min at the end of which global low-flow (10% of control) ischaemia was initiated. The interventions described above were included in the perfusion fluid 5 min prior to the onset and throughout the ischaemic period. The data are shown as means standard errors of the means (n = 6)...

Figure 4.10 Effect of reduced glutathione (GSH) (0-1.0 mM) on Na/K ATPase inhibition associated with the addition of oxidized glutathione (GSSG) (1 me). Experiments were performed using an isolated bovine ventricular Na/K ATPase preparation. Na/K ATPase activity was quantified by the ouabain-sensitive hydrolysis of ATP to yield inorganic phosphate. The data are presented as means standard errors of the means (n = 6).

TABLE 1 Concentration of K, CP, and Ca in Twigs of Nonstimulated Willow Plants. Emitted X-ray Signals Are Given as Peak Background Ratio. The Data Are Mean Values of 20 Measurements Taken from Three Plants with a Standard Error Lower than 0.5 for K+, 0.2 for Ca + and 0.9 for CP... 

This is equivalent to assuming that the standard error in the i1 1 measurement of the response variable is proportional to its value, again a rather "safe" assumption as it forces least squares to pay equal attention to all data points. 

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). 

Kittrell et al. (1965a) also performed two types of estimation. First the data at each isotherm were used separately and subsequently all data were regressed simultaneously. The regression of the isothermal data was also done with linear least squares by linearizing the model equation. In Tables 16.7 and 16.8 the reported parameter estimates are given together with the reported standard error. Ayen and Peters (1962) have also reported values for the unknown parameters and they are given here in Table 16.9. 

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. 

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