Statistical methods standard deviation

Table 18. Statistical comparison (F-test [125]) of the methods. Standard deviation Sxo of the calibration curves for diethylstilbestrol and ethinylestradiol [114].

The next step is to arrange the seven differences, Aa to AG, in numerical order (ignoring the sign). To calculate if any of the differences are statistically significant, a statistical test (t-test) is applied. Equation (4.17) is used to compare the difference A with the expected precision of the method, s. The value of t used corresponds to the value obtained from statistical tables for the degrees of freedom appropriate for the estimation of s and the level of confidence used. For example, if the method standard deviation was obtained from ten results, i.e. nine degrees of freedom, t(95%) = 2.262. 

In the maximum-likelihood method used here, the "true" value of each measured variable is also found in the course of parameter estimation. The differences between these "true" values and the corresponding experimentally measured values are the residuals (also called deviations). When there are many data points, the residuals can be analyzed by standard statistical methods (Draper and Smith, 1966). If, however, there are only a few data points, examination of the residuals for trends, when plotted versus other system variables, may provide valuable information. Often these plots can indicate at a glance excessive experimental error, systematic error, or "lack of fit." Data points which are obviously bad can also be readily detected. If the model is suitable and if there are no systematic errors, such a plot shows the residuals randomly distributed with zero means. This behavior is shown in Figure 3 for the ethyl-acetate-n-propanol data of Murti and Van Winkle (1958), fitted with the van Laar equation. 

The method allows variables to be added or multiplied using basic statistical rules, and can be applied to dependent as well as independent variables. If input distributions can be represented by a mean, and standard deviation then the following rules are applicable for independent variables ... 

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. 

Sulcer and Denson (Ref 19) used the gas chromatographic—B .T. procedure for the analysis of Class I A1 powder (45 u max dia) which cannot be tested satisfactorily by sedimentation methods because of the presence of aggregates. A rough statistical evaluation of this procedure was made by running twelve determinations and calculating the standard deviation as shown in Table 14 ... 

The flowsheet shown in the introduction and that used in connection with a simulation (Section 1.4) provide insights into the pervasiveness of errors at the source, random errors are experienced as an inherent feature of every measurement process. The standard deviation is commonly substituted for a more detailed description of the error distribution (see also Section 1.2), as this suffices in most cases. Systematic errors due to interference or faulty interpretation cannot be detected by statistical methods alone control experiments are necessary. One or more such primary results must usually be inserted into a more or less complex system of equations to obtain the final result (for examples, see Refs. 23, 91-94, 104, 105, 142. The question that imposes itself at this point is how reliable is the final result Two different mechanisms of action must be discussed ... 

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. 

The Student s (W.S. Gossett) /-lest is useful for comparisons of the means and standard deviations of different analytical test methods. Descriptions of the theory and use of this statistic are readily available in standard statistical texts including those in the references [1-6]. Use of this test will indicate whether the differences between a set of measurement and the true (known) value for those measurements is statistically meaningful. For Table 36-1 a comparison of METHOD B test results for each of the locations is compared to the known spiked analyte value for each sample. This statistical test indicates that METHOD B results are lower than the known analyte values for Sample No. 5 (Lab 1 and Lab 2), and Sample No. 6 (Lab 1). METHOD B reported value is higher for Sample No. 6 (Lab 2). Average results for this test indicate that METHOD B may result in analytical values trending lower than actual values. 

The test to determine whether the bias is significant incorporates the Student s /-test. The method for calculating the t-test statistic is shown in equation 38-10 using MathCad symbolic notation. Equations 38-8 and 38-9 are used to calculate the standard deviation of the differences between the sums of X and Y for both analytical methods A and B, whereas equation 38-10 is used to calculate the standard deviation of the mean. The /-table statistic for comparison of the test statistic is given in equations 38-11 and 38-12. The F-statistic and f-statistic tables can be found in standard statistical texts such as references [1-3]. The null hypothesis (H0) states that there is no systematic difference between the two methods, whereas the alternate hypothesis (Hf) states that there is a significant systematic difference between the methods. It can be seen from these results that the bias is significant between these two methods and that METHOD B has results biased by 0.084 above the results obtained by METHOD A. The estimated bias is given by the Mean Difference calculation. 

You may be surprised that for our example data from Miller and Miller ([2], p. 106), the correlation coefficient calculated using any of these methods of computation for the r-value is 0.99887956534852. When we evaluate the correlation computation we see that given a relatively equivalent prediction error represented as (X - X), J2 (X - X), or SEP, the standard deviation of the data set (X) determines the magnitude of the correlation coefficient. This is illustrated using Graphics 59-la and 59-lb. These graphics allow the correlation coefficient to be displayed for any specified Standard error of prediction, also occasionally denoted as the standard error of estimate (SEE). It should be obvious that for any statistical study one must compare the actual computational recipes used to make a calculation, rather than to rely on the more or less non-standard terminology and assume that the computations are what one expected. 

Phosphoric Acid. The 2nd-order rate method for analyzing the TGA data was statistically best (Table IV) for the cellulose/H PO samples. This suggests that the conclusions from a prior study which assumed a lst-order reaction (29) may need to be reexamined. While Wilkinson s approximation method gave high r values, the rate constant is determined by the intercept rather than the slope in this method. Thus, the standard deviation of the rates determined by Wilkinson s approximation method is still relatively high when compared to the other methods. In addition, the reaction order as determined by the Wilkinson approximation method was unrealistically high, ranging from 2.6 to 5.8. 

If a large number of readings of the same quantity are taken, then the mean (average) value is likely to be close to the true value if there is no systematic bias (i.e., no systematic errors). Clearly, if we repeat a particular measurement several times, the random error associated with each measurement will mean that the value is sometimes above and sometimes below the true result, in a random way. Thus, these errors will cancel out, and the average or mean value should be a better estimate of the true value than is any single result. However, we still need to know how good an estimate our mean value is of the true result. Statistical methods lead to the concept of standard error (or standard deviation) around the mean value. 

A given device, procedure, process, or method is usually said to be in statistical control if numerical values derived from it on a regular basis (such as daily) are consistently within 2 standard deviations from the established mean, or the most desirable value. As we learned in Section 1.7.3, such numerical values occur statistically 95.5% of the time. Thus if, say, two or more consecutive values differ from the established value by more than 2 standard deviations, a problem is indicated because this should happen only 4.5% of the time, or once in roughly every 20 events, and is not expected two or more times consecutively. The device, procedure, process, or method would be considered out of statistical control, indicating that an evaluation is in order. 

Analytical laboratories, especially quality assurance laboratories, will often maintain graphical records of statistical control so that scientists and technicians can note the history of the device, procedure, process, or method at a glance. The graphical record is called a control chart and is maintained on a regular basis, such as daily. It is a graph of the numerical value on the y-axis vs. the date on the x-axis. The chart is characterized by five horizontal lines designating the five numerical values that are important for statistical control. One is the value that is 3 standard deviations from the most desirable value on the positive side. Another is the value that is 3 standard deviations from the most desirable value on the negative side. These represent those values that are expected to occur only less than 0.3% of the time. These two numerical values are called the action limits because one point outside these limits is cause for action to be taken. 

While the value for coefficient of variation is a general statement about the imprecision of a method, only the value for standard deviation can be used in any statistical comparison of two methods. The use of coefficient of variation assumes a constant relationship between standard deviation and the mean value and this is not always true (Table 1.5). 

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