Absolute error, mass

The relative error is the absolute error divided by the true value it is usually expressed in terms of percentage or in parts per thousand. The true or absolute value of a quantity cannot be established experimentally, so that the observed result must be compared with the most probable value. With pure substances the quantity will ultimately depend upon the relative atomic mass of the constituent elements. Determinations of the relative atomic mass have been made with the utmost care, and the accuracy obtained usually far exceeds that attained in ordinary quantitative analysis the analyst must accordingly accept their reliability. With natural or industrial products, we must accept provisionally the results obtained by analysts of repute using carefully tested methods. If several analysts determine the same constituent in the same sample by different methods, the most probable value, which is usually the average, can be deduced from their results. In both cases, the establishment of the most probable value involves the application of statistical methods and the concept of precision. 

Figure 2.11. For various combinations of n (5 10 resp. 20) and m (1, 2, resp. 3) the estimated CI(X) is plotted versus absorbance y. The left figure shows the absolute values t ixl, while the right figure depicts the relative ones, namely 100 t Sx/X in %, It is obvious that it would be inopportune to operate in the region below about 90% of nominal (in this particular case below y = 0.36 the absolute error for y = 0.36 is smaller than that for y = 0.6, but the inverse is true for the relative error, see arrows). There are three remedies increase n ox m (and costs), or reduce the calibration concentrations to shift the center of mass (x ean, ymean) below 100/0.42. At y = 0.6 and m - 1 (no replicates ) one finds X = 141.4 with a Cl of 3.39 (+2.4%, circle).

Example The [M-Cl]" ion, [CHCl2], represents the base peak in the El spectrum of chloroform. The results of three subsequent determinations for the major peaks of the isotopic pattern are listed below (Fig. 3.15). The typical printout of a mass spectrometer data system provides experimental accurate mass and relative intensity of the signal and an error as compared to the calculated exact mass of possible compositions. For the [ CH Cl2] ion, the experimental accurate mass values yield an average of 82.9442 0.0006 u. The comparatively small absolute error of 0.6 mmu corresponds to a relative error of 7.5 ppm. 

The scale we use to measure mass accuracy is parts per million (ppm), which is a relative unit obtained by dividing the absolute error in the mass measurement by the relative molecular mass according to equation (5.3) ... 

The mass accuracy is usually described as either the absolute mass error in milli-daltons (or thousandths of a mass unit) or as a part-per-million error which is the ratio of the absolute error and the calculated mass multiplied by one million. For instance, if an ion has a calculated or theoretical mass of 400.000 and the measured mass is 400.002, then the error is 2 mDa. The ppm error is calculated by dividing the 2-mDa error by the calculated mass of 400.000 and multiplying by one million to give 5 ppm. Note that if the same absolute error of 2 mDa was found for a compound of theoretical mass of 800, the ppm error would be 2.5 ppm. For the same absolute error, the ppm error varies with mass. 

Drug A formulation lots were then examined for evidence of iron as well as other transition metal ions. Inductively coupled plasma-mass spectroscopy (ICP-MS) was used initially in a semiquantitative scanning mode. This mode of detection allows for determination of the elements sodium through mercury with detection limits of about Ippb with absolute errors typically about 30%. The three manufactured batches shown in Table 3 were examined. With the exception of iron, no first-, second-or third-row transition metal ion was found in any lot at greater than 2 ppb. Most transition metals were undetectable. Iron, in contrast, was detected at between 10 and 30 ppb. Iron levels were quantitated more accurately by using the method of standard... 

Yanson et al. [41] using field-ionization mass spectrometry studied the formation of gas-phase GC, CC, AT and TT pairs. From measurements of temperature dependence of equilibrium constants, an interaction enthalpy for the base pair formation was derived. This technique was sometimes questioned because the determination of enthalpy from the slopes of appropriate van t Hoff curves might not be unambiguous. From Table 6 it is evident that the agreement with the present theoretical values is good, and concerns not only the relative interaction enthalpies but even the absolute values the average absolute error is less than 1.5 kcal/mol. 

Mass accuracy Difference between the measured m/z and the calculated m/z. The mass accmacy can be expressed as an absolute error (in mDa), or as a relative error (given in ppm) Relative Error = Measured - Calculated Calculated... 

Note that in Eq. 14.48, for the sake of simplicity, the absolute error Arjg — At]Q i) is referred here to the approximate value (Atjg i), not to the true value (Arje). In practice, the result is the same as long as the relative error is small. The relative error given by Eq. 14.48 is obviously small if the second term of the RHS of Eq. 14.45 is small compared to the first one, i.e., if the mass transfer resistances are small, and the column is highly efficient. If this is not true, however, and the second term in the RHS of Eq. 14.45 is dominant, Eq. 14.48 becomes... 

Although the isotope ratios of samples may be compared with high precision, the absolute errors may be quite large, usually of the order of 1.0%. These errors may be attributed to the method of introducing the sample into the mass spectrometer or to discriminating effects within the ion source, analyzer, or detector regions of the instrument (26). 

What is the absolute minimum mass of phosphorus that can be detected using the reaction (n, a) under irradiation with 14-MeV neutrons Assume 4> = 10 neutrons/(m s), o- = 0.150 b, counting system background = 15 0.5 counts/min, counting time = 2 min, and maximum acceptable error is 30 percent. 

The true mass of a glass bead is 0.1026 g. A student takes four measurements of the mass of the bead on an analytical balance and obtains the following results 0.1021 g, 0.1025 g, 0.1019 g, and 0.1023 g. Calculate the mean, the average deviation, the standard deviation, the percentage relative standard deviation, the absolute error of the mean, and the relative error of the mean. 

Note that absolute values of estimated errors are converted to relative errors in the second step by dividing the absolute estimated errors by the original data. Moreover, an absolute value of the maximum error is obtained in the fourth step by multiplying the sum of the relative errors (with a subscripted nonsignificant figure) by the calculated result. For clarification, absolute errors are boldfaced, whereas relative errors are not. As a second example, the density of a sample and its maximum error can be calculated as mass divided by volume. 

Equation 12-11 was developed to deal with random relative error (noise). Develop an appropriate equation suitable to generate random absolute errors at various levels. Use both to examine the absolute standard deviation of a measurement of mass of 0.2000 g of a substance whose purity is known to 1%, assuming a weighing error of + 0.0002 g. Obtain 20 values using randomization of each factor and 20 values with both factors employed. What are the standard errors of the resultant means ... 

A random absolute error can be achieved by multiplying the expected random absolute error, e.g., 0.0002g in mass measurement by the noise function using P = 1. The expected relative error, in this case 1%, arising from uncertainties in purity, is obtained by multiplying the mass by the noise function where P=0.02. (See Figure 12.4) Hence W, the randomized mass is... 

The design and operation of a highly accurate large flow cryogenic calibration stand has proven quite successful. The test system has been used to calibrate flowmeters using liquid oxygen, liquid nitrogen and distilled water. The maximum absolute error of the test system as obtained from the error analysis is 0.28 for the calibration of volumetric-type flowmeters, and 0.17 for the calibration of mass type flowmeters. 

The precision of quantitative mass spectral measurements by the procedure just described usually ranges between 2% and 10% relative. The analytical accuracy varies considerably depending on the complexity of the mixture being analyzed and the nature of its components. For gaseous hydrocarbon mixtures containing five to ten components, absolute errors of 0.2 to 0.8 mole percent are typical. 

In summary, the determination of the coupling efficiency on the polymer support has severe difficulties due to the fact that the resin comprises the vast majority of the materials mass. Though the absolute error remains constant, the percent error increases for each iteration along the sequence. Therefore, yield determinations based on elemental composition changes along... 

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. 

Taylor PDF, Maeck R, De Bievre P (1992) Determination of the absolute isotopic composition and Atomic Weight of a reference sample of natural iron. Int J Mass Spectrom Ion Processes 121 111-125 Taylor PDF, Maeck R, Hendrickx F, De Bievre P (1993) The gravimetric preparation of synthetic mixtures of iron isotopes. Int J Mass Spectrom Ion Processes 128 91-97 Thirlwall MF (2002) Multicollector ICP-MS analysis of Pb isotopes using a Pb- Pb double spike demonstrates up to 4000 ppm/amu systematic errors in Tl-normalization. Chem Geol 184 255-279... 

Example A magnetic sector mass spectrometer allows for an absolute mass accuracy of 2-5 mmu in scanning mode over a range of about m/z 50-1500. At m/z 1200 an error of 3.5 mmu corresponds to inconspicuous 2.9 ppm, whereas the same error yields 70 ppm at m/z 50 which seems to be unacceptably large. 

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