Error total

Theoiy related to material characteristics states that a minimum quantity of sample is predicated as that amount required to achieve a specified limit of error in the sample-taking process. Theoiy of sampling in its apphcation acknowledges sample preparation and testing as additional contributions to total error, but these error sources are placed outside consideration of sampling accuracy in theoiy of sample extraction. 

The problem with plant data becomes more significant when sampling, instrument, and cahbration errors are accounted for. These errors result in a systematic deviation in the measurements from the actual values. Descriptively, the total error (mean square error) in the measurements is... 

Any main CT that is itndeiioaded will also add to the error in the measurement. Similarly, if provision is tnade in the primary of the summation CT to accommodate fntnre circuits but is not being utilized it tnust be left open, otherwise it will also add to the error. The impedance of the shorting terminals will add to the impedance of the circuit and will increase the total error. 

Assume Z and R to be constant for the expander. This assumption can lead to some minor errors in the calculation. However, if Pj is assumed constant then the variation for Z can be rolled into the new function of R, and the total error becomes even smaller. Now, Equation 7-4 can be simplified to ... 

K is the Kalman gain and the total error varianee expeeted is... 

I he origins of the above two errors are chfferent in cause and nature. A sim ple example is, when the mass of a weight is less than its nominal value, a systematic error occurs, which is constant in absolute value and sign. This is a pure systematic error. A ventilation-related example is, when the instrument faaor of a Pitot-static tube, which defines the relationship between the measured pressure difference and the velocity, is incorrect, a systematic error occurs. On the other hand, if a Pitot-static tube is positioned manually in a duct in such a way that the tube tip is randomly on either side of the intended measurement point, a random error occurs. This way, different phenomena create different ty pes of error. I he (total) error of measurement usually is a combination of the above two types. 

The bias error is a quantity that gives the total systematic error of a measuring instrument under defined conditions. As mentioned earlier, the bias should be minimized by calibration. The repeatability error consists of the confidence limits of a single measurement under certain conditions. The mac-curacy or error of indication is the total error of the instrument, including the... 

The errors arising in sampling, particularly in the case of heterogeneous solids, may be the most important source of uncertainty in the subsequent analysis of the material. If we represent the standard deviation of the sampling operation (the sampling error) by ss and the standard deviation of the analytical procedures (the analytical error) by sA, then the overall standard deviation sT (the total error) is given by... 

Example 1. If the sampling error is +3 per cent and the analytical error is 1 per cent, from equation (1) we can see that the total error sT is given by... 

If, in the above example, the analytical error was 0.2 per cent then the total error sT would be equal to 3.006 per cent. Hence the contribution of the analytical error to the total error is virtually insignificant. Youden7 has stated that once the analytical uncertainty is reduced to one-third of the sampling uncertainty, further reduction of the former is not necessary. It is most important to realise that if the sampling error is large, then a rapid analytical method with relatively low precision may suffice. 

So, we can discard the third eigenvector and, along with it, that portion of the variance in our spectra that displaced the data out of the plane of the noise-free data. We are in fact, discarding a portion of the noise without significantly distorting the spectra The portion of the noise we discard is called the extracted error or the residuals. Remember that the noise we added also displaced the points to some extent within the plane of the noise-free data. This portion of the noise remains in the data because it is spanned by the eigenvectors that we must retain. The noise that remains is called the imbedded error. The total error is sometimes called the real error. The relationship among the real error (RE), the extracted error (XE), and the imbedded error (IE) is... 

Since the machine performs only arithmetic operations (and these only approximately), iff is anything but a rational function it must be approximated by a rational function, e.g., by a finite number of terms in a Taylor expansion. If this rational approximation is denoted by fat this gives rise to an error fix ) — fa(x ), generally called the truncation error. Finally, since even the arithmetic operations are carried out only approximately in the machine, not even fjx ) can usually be found exactly, and still a third type of error results, fa(x ) — / ( ) called generated error, where / ( ) is the number actually produced by the machine. Thus, the total error is the sum of these... 

The true standard deviation Ox is expected inside the confidence interval CI(5 , ) = /Vi. .. /V with a total error probability 2 p (in connection with F and x P taken to be one-sided). 

Total error variance may not be constant with conversion ... 

We want to make the overall error minimal for fixed n. We express the total error in terms of h and n... 

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In the most general case, if both variables are subject to error and, therefore, neither the condition sx sy/b nor sy/b sx is fulfilled, the following total error results ... 

The total error has to be calculated here in different way compared with robustness. Whereas in (i) of-A is the variance within a laboratory, o kA is the variance between laboratories plus that within the laboratories, °ijkA = °k + °ijA- The interpretation is analogous to (i), if F < F( a>Vl>V2, then the null hypothesis cannot be rejected and the procedure can be considered as to be rugged. Advantageously, the test can be carried out by schemes of ANOVA (analysis of variance) as given in Sect. 5.1.1. 

The more sophisticated treatment of Ingle and Crouch [7] comes very close but also misses the mark for an unexplained reason they insert the condition ... it is assumed there is no uncertainty in measuring Ert and Eot... . Now in fact this could happen (or at least there could be no variation in AEr) for example, if one reference spectrum was used in conjunction with multiple sample spectra using an FTIR spectrometer. However, that would not be a true indication of the total error of the measurement, since the effect of the noise in the reference reading would have been removed from the calculated SD, whereas the true total error of the reading would in... 

The total error for the measurements with the present method is made up of various errors as follows ... 

As there are few data on the ratio of thoron concentration to radon, we have assumed that the value is constant and is equal to 0.14, Error(4) and Error(5) are assumed to be negligible for the error calculation. The error due to ambiguities in classification and discrimination of tracks according to the shape or the size was assessed by experiment and the relative error (1 S.D.) was less than 10%. The relative error due to the variations of F, f, and (V a) is less than 13% as described in the previous section. The combined error due to these factors is estimated to be 16%. The total error can be obtained by combining this error and Error(l). If we use the definition that the lowest detectable limit is the radon concentration at 50 % relative S.D., the lowest detectable... 

Of the 17 kcal mol-1 total error, about half is estimated to arise from the single hp— h[ j sigma-type interaction shown in Fig. 2.8, while the remainder arises from weaker pi-type interactions (2-3 kcal mol-1 each). For example, we can carry out a partially localized variational calculation, similar to that described above but with only h prevented from delocalizing into tip (hisleads to a stabilization energy (at 7 = 1.6 A)... 

Thus, the overall standard deviation, sT, i.e., the total error is given by the following expression ... 

Hoffmann, D., Kringle, R. A Total Error Approach for the Validation of Quantitative Analytical Methods. Pharm. Res., 24, 2007, 1157. 

The value of this work is to illustrate the importance of including the estimated amount interval with every calculated amount estimate in written reports. These can be calculated at any response level. As an illustration of this process we can use the data of Table XIII. If an analysis of fenvalerate were being performed and the standards were those of Dataset A, an amount estimated to be 1.5 ng would be reported as having a total error range of 1.16 to 1.97 ng or in rounded figures 1.2 to 2.0 ng. 

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