Comparison of standard deviations

Two standard deviations, Si and s2, with corresponding degrees of freedom, and v2, are compared by means of the F-test. The test statistic 

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A comparison of standard deviation s to standard counting error sc is thus a useful criterion for the reliability of analytical results obtained by x-ray emission. To illustrate the simplest possible application of this criterion, consider again the x-ray data plotted in Figure 10-3, which are given in Table 10-1. The individual N s summarized in the table would, in x-ray emission spectrography, appear eventually as analytical results that is, as the Us of Figure 10-1 with x as their mean. For these 393 individual ATs, the standard deviation is... 

Up to now (1971) only a limited number of reaction series have been completely worked out in our laboratories along the lines outlined in Sec. IV. In fact, there are rather few examples in the literature with a sufficient number of data, accuracy, and temperature range to be worth a thorough statistical treatment. Hence, the examples collected in Table III are mostly from recent experimental work and the previous ones (1) have been reexamined. When evaluating the results, the main attention should be paid to the question as to whether or not the isokinetic relationship holds i.e., to the comparison of standard deviations of So and Sqo The isokinetic temperature /J is viewed as a mere formal quantity and is given no confidence interval. Comparison with previous treatments is mostly restricted to this value, which has generally and improperly been given too much atention. 

The other key assumption that we sort of implied was that the comparison of standard deviation is constant. Of course we know that as n changes, the comparison value changes as the square root of n. This is on top of and in addition to the changes caused by the use of the t rather than the Normal (Z) distribution. 

Precision is usually expressed in terms of the standard deviation or relative standard deviation. The latter is useful in case of the comparison of standard deviation values that refer to the dispersion of individual results from samples with different concentrations. 

Table 2.5. Comparison of Standard Deviation for Several Kinetic Models for 11.2 nig of Reagent Beryllium Sulfate Heated at 4.9°/min a=0.1 —0.9 (145).

Comparison of Standard Deviation of BTU for Scott s and Industry Analysis of Natural Gas... 

B. Comparison of Standard Counting Error and Standard Deviation ... 

A T-score is a comparison of the patient s measured BMD to the mean BMD of a healthy, young (20- to 29-year-old), sex-matched, white reference population. The T-score is the number of standard deviations from the mean of the reference population. 

All measured values are normally registered by means of a recorder. In addition, an improved method for data collection and processing is possible today by use of a computer. This has the advantage of automatic, safer data collection in an easy-to-read form. Comparison with standard values, correction of the buoyancy effects, control of linearity, control of standard deviations and peak integrations are thus possible. 

In a series of standard deviations, variances, or ranges the COCHRAN test [ISO 5725] is used for evaluating the largest value. It is primarily used in (planned) method comparisons, cooperative tests, and analysis of variance where the number of the measurements and the levels of the means should be the same. 

At the end of Chapter III on the comparison of means it was mentioned that complications arise in the calculation of standard deviations if the observations are not independent of each other. 

The reports often include extensive data analysis, statistical summaries, and plots. The mean of all results or the mean of results from peer laboratories (those performing the test with similar methods) is taken as the target value and is used for comparison with the individual laboratory s result. Different programs do this in different ways. For example, the statistical significance of any difference between an individual laboratory s observed result and the group mean can be tested by use of the t-test. When the difference is significant, the laboratory is alerted that its results are biased compared with the results of most of the other laboratories. Another approach is to divide the difference by the overall standard deviation of the group, and then to express the difference in terms of the number of standard deviations... 

Using the above equation, the theorehcal sample size needed to represent the bulk composihon of a blend with a given standard deviation can be calculated. Data in the table below are normalized to the relahve standard deviation based on the component concentrahon (P) to allow for easy comparison (relative standard deviation = standard deviation / P x 100). The number of particles is also expressed as grams. It can be seen in the table that the lower the %component in the blend or the greater the confidence in the composihon (lower the acceptable relative standard deviahon), the greater the number of particles are required in the sample. 

Note also that, owing to the erroneous practice mentioned above (which, dimension aside, can he rewritten in the form v sj v for better comparison), the standard deviation will be either over- or underestimated depending on whether the counting time t is shorter or, respectively, longer than the time unit used for the count-rate calculation. In the case of a 1-min measurement, e.g., if the time unit is the second, then the error is overestimated if the unit is the hour, then it is underestimated. The (numerically) correct result is only obtained if the time unit happens to be the minute. 

In the treatment of the theoretically calculated and experimentally obtained residence time functions, the concept of variance is useful, particularly for the comparison of different theories of RTD. Variance is defined as, in mathematical statistics, that is, the quadratic value of standard deviation ... 

As mentioned previously, a pH glass electrode and a calomel reference electrode immersed in pH 7 buffer generate approximately zero millivolts. The standardization or calibration control on the pH meter allows any deviation of these electrodes from zero millivolts to be offset, thus providing comparison of standardization of all electrode pairs. This provides a starting point for a linear function which represents the observed potential versus the pH, and since the slope of this function is predictable, it can be represented by the function illustrated in Figure 1.6. 

A recent paper compares the performances of GC-MS in SIM mode to those of GC coupled to tandem mass spectrometry (GC-MS/MS), either using an ion-trap or a triple quadrupole (Kinani et al, 2006). The ion trap was not suitable for the PASs determination due to a high LOQ (up to 40 mg/kg), lack of linearity and high variability (up to 26% of standard deviation when re-injecting a standard at a concentration of 10 mg/kg). The statistical comparison of the GC-MS method (with a single column) with the triple quadrupole results shows, in the latter case, a lower risk of false positives and negatives. However, the authors conclude that none of the tested methods is fully satisfactory from this point of view. 

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