Statistical analysis standard error

RMS = root mean square noise of the instrument. SEC = standard error of calibration for each constituent. SEA = standard error of analysis or the difference between analysis values from the same samples analyzed by NIR and the reference laboratory. SED = standard error of a difference. Factory SED = standard error of the difference between the same samples analyzed by the master and slave instrument at different times. H = standardized H statistic. SEE = standard error of the laboratory reference values. (This statistic can be either the difference between duplicates in one laboratory or the difference between the same samples analyzed by two different laboratories.) M SEE = standard error of blind duplicates in the master reference laboratory. 

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 precision stated in Table 10 is given by the standard deviations obtained from a statistical analysis of the experimental data of one run and of a number of runs. These parameters give an indication of the internal consistency of the data of one run of measurements and of the reproducibility between runs. The systematic error is far more difficult to discern and to evaluate, which causes an uncertainty in the resulting values. Such an estimate of systematic errors or uncertainties can be obtained if the measuring method can also be applied under circumstances where a more exact or a true value of the property to be determined is known from other sources. 

Scale bar 200 pm. Red arrows indicate the avascular zones. Quantification of digital analysis of the fluorescence angiography images number of branching points (mm2) (B) and mean mesh size (102 pm2) (C) as markers of vessel density for CAM. P < 0.05 was considered to be statistically significant. Error bars represent standard error of the mean. 

They include simple statistics (e.g., sums, means, standard deviations, coefficient of variation), error analysis terms (e.g., average error, relative error, standard error of estimate), linear regression analysis, and correlation coefficients. 

The primary goal of this series of chapters is to describe the statistical tests required to determine the magnitude of the random (i.e., precision and accuracy) and systematic (i.e., bias) error contributions due to choosing Analytical METHODS A or B, and/or the location/operator where each standard method is performed. The statistical analysis for this series of articles consists of five main parts as ... 

Statistical analysis In each treatment, 10 microspores were used to measure the maximal fluorescence. The means and their standard errors are determined, if the investigator has microspectrofluorimeter or microspectrofluorimeter having special statictic t- test programme. 

Statistical analysis Statistical analysis consists of determination of Standard error of means (SEM) or the Student- t-testing. The measurement error was about 1-2% (10 spectra per one variant, n=10). Counting was performed in four or five replicates (the number of Petri dishes per a treatment). SEM for the fluorescence spectra was 2 %. 

Equation 2.67 indicates that the standard enthalpy and entropy of reaction 2.64 derived from Kc data may be close to the values obtained with molality equilibrium constants. Because Ar// is calculated from the slope of In AT versus l/T, it will be similar to the value derived with Km data provided that the density of the solution remains approximately constant in the experimental temperature range. On the other hand, the error in ArSj calculated with Kc data can be roughly estimated as R In p (from equations 2.57 and 2.67). In the case of water, this is about zero for most solvents, which have p in the range of 0.7-2 kg dm-3, the corrections are smaller (from —3 to 6 J K-1 mol-1) than the usual experimental uncertainties associated with the statistical analysis of the data. 

In any book, there are relevant issues that are not covered. The most obvious in this book is probably a lack of in-depth statistical analysis of the results of model-based and model-free analyses. Data fitting does produce standard deviations for the fitted parameters, but translation into confidence limits is much more difficult for reasonably complex models. Also, the effects of the separation of linear and non-linear parameters are, to our knowledge, not well investigated. Very little is known about errors and confidence limits in the area of model-free analysis. 

The statistical analysis is problematic for this example as the residuals are obviously not normally distributed. Nonetheless, the high errors in the parameters and the large standard deviation of the residuals indicate a bad fit. 

Whereas precision (Section 6.5) measures the reproducibility of data from replicate analyses, the accuracy (Section 6.4) of a test estimates how accurate the data are, that is, how close the data would represent probable true values or how accurate the analytical procedure is to giving results that may be close to true values. Precision and accuracy are both measured on one or more samples selected at random for analysis from a given batch of samples. The precision of analysis is usually determined by running duplicate or replicate tests on one of the samples in a given batch of samples. It is expressed statistically as standard deviation, relative standard deviation (RSD), coefficient of variance (CV), standard error of the mean (M), and relative percent difference (RPD). 

In obtaining Monte Carlo data such as shown in Figs. 2, 3, 5, it is also necessary to understand the statistical errors that are present because the number of states M — Mq over which we average (Eq. (24)) is finite. If the averages m, E, i/ are calculated from a subset of n uncorrelated observations m(Xy), E(Xy), ilf Xy), Standard error analysis applies and yields estimates for the expected mean square deviations, for n- cx),... 

All samples were dried for 72 hours at 80°C, and dry weights were calculated. Dried samples were milled to spectrometrically measure the specific activity of i Cs. The standard error of specific activity was in the range 10-20%. Statistical analysis used the software package MS Excel. 

Calibration curve quality. Calibration curve quality is usually evaluated by statistical parameters, such as the correlation coefficient and standard error of estimate, and by empirical indexes, such as the length of the linear range. Using confidence band statistics, curve quality can be better described in terms of confidence band widths at several key concentrations. Other semi-quantitative indexes become redundant. Alternatively, the effects of curve quality can be incorporated into statements of sample analysis data quality. 

After data analysis, the statistically significant effects induced by certain factors are identified. This is typically achieved using standardized Pareto plots (Figure 14). The plot shows the standard values of the effects in descending order of magnitude. The length of each bar is proportional to the standardized effect, which is equal to the effect divided by the standard error. This is identical to the calculation of a lvalue for... 

It is generally the case that when more complex statistical analysis strategies and designs are under consideration, standard sample size calculations are inadequate to cover them. In such circumstances simulation is often used to determine the t)rpe-I and type-II errors of the proposed studies for a given sample size. 

Methods of statistical meta-analysis may be useful for combining information across studies. There are 2 principal varieties of meta-analytic estimation (Normand 1995). In a hxed-effects analysis the observed variation among estimates is attributable to the statistical error associated with the individual estimates. An important step is to compute a weighted average of unbiased estimates, where the weight for an estimate is computed by means of its standard error estimate. In a random-effects analysis one allows for additional variation, beyond statistical error, making use of a htted random-effects model. 

Finally, it must unfortunately be concluded that the analytical literature about the determination of ruggedness contains many errors and bad practices, the main ones being inapt choice of levels (too large differences), inclusion of factors that should not be included (because they should stay constant in properly standardized analytical procedures) and errors in the statistical analysis. The reader is therefore urged not to base his methodology on examples from the literature without having verified that they are correct. 

Using meta-analytic techniques based on the means and the standard errors presented graphically in the poster, we estimated pooled data of the four effective dosages of quetiapine both for the BPRS and the CGI severity of illness change scores from baseline to endpoint. Quetiapine produced an improvement of 0.43 effect-size units in comparison with placebo, a difference that was highly statistically significant and about the same improvement as haloperidol. Thus, based on the BPRS or PANSS, quetiapine was similar to neuroleptics in efficacy (i.e., differences were nonsignificant). Based on our meta-analysis, quetiapine is clearly superior to... 

Quantitative methodology uses large or relatively large samples of subjects (as a rule students) and tests or questionnaires to which the subjects answer. Results are treated by statistical analysis, by means of a variety of parametric methods (when we have continuous data at the interval or at the ratio scale) or nonparametric methods (when we have categorical data at the nominal or at the ordinal scale) (30). Data are usually treated by standard commercial statistical packages. Tests and questionnaires have to satisfy the criteria for content and construct validity (this is analogous to lack of systematic errors in measurement), and for reliability (this controls for random errors) (31). 

A more useful statistical term for error analysis is standard deviation, a measure of the spread of the observed values. Standard deviation, s, for a sample of data consisting of n observations may be estimated by Equation 1.3. 

AQ/Qo = (Q - Qo)/Qo, where Q0 is charge consumption without analyte and Q is that at certain thrombin, concentration. An example of calibration curve for two independently prepared electrodes is shown in Fig. 47.3. It is seen, that results are well reproducible. Statistical analysis, performed earlier [4] revealed that standard error is approximately 11%. Interferences of this aptasensor with other compounds, human serum albumine (HSA) and human IgG are relatively low. An example is shown in Fig. 47.4, where calibration curve for thrombin is compared with those for HSA and IgG. Please note, that concentrations of HSA and IgG are much higher in comparison with that of thrombin. 

The statistical uncertainty arising from the analytical measurement is derived from the automatic data collection procedure noted before. The AGAS computer performs a standard error analysis and produces both a mean and the standard error of the mean associated with that value. A computer program is used to combine the uncertainties from the primary gravimetric process with the uncertainties produced from the standard deviation of the instrument s response for each of the gas mixtures. 

Calibration curves were constructed with the NIST albumin (5 concentrations in triplicate) and with the FLUKA albumin (5 concentrations in duplicate) in the concentration range of 50 250 mg/1. The measured values of individual concentrations fluctuated around the fitted lines, with a standard error of 0.007 of the measured absorbance. The difference between FLUKA and NIST albumin calibration lines was statistically insignificant, as evaluated by the t-test P=0.14 > 0.05. The calibration lines differed only in the range of a random error. The FLUKA albumin was, thus, equivalent to that of NIST. Statistical evaluation was carried out using the regression analysis module of the statistical package SPSS, version 4.0. 

Where is the error on the ith reading and the expectation value of 1 is 0 and expectation value of e is a2. The values of x are taken to be Normally distributed with the mean p and the standard deviation cr. The values of p, and a are estimated from the actual readings. Thus although the analysis is carried out in terms of the random errors the data provides an estimate of <7 which is the uncertainty arising from random effects. This confusion between error and uncertainty is often added to by referring to cr as the standard error. In addition the statistical analysis is very rarely extended to include systematic errors. 

The time to reach the maximum lipase activity did not show any satisfactory correlation with the studied variables. This behavior is probably owing to the high error associated with this variable or with other factors, such as inoculum age and concentration, which were not considered in the statistical analysis and may be influencing this response, although inoculum standardization procedures were used in all the experiments. Fermentation time was not found to be a determinant factor. However, this maybe important on an industrial scale, since it can directly compromise enzyme productivity and production costs. 

Dry matter yield averaged over two harvests each year. b Yield data in parentheses were not included in statistical analysis because an insufficient number of plants survived the preceding winter. c Standard error of the mean. 

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