Fractional errors

The fractional error is logarithmically unbiased that is, an M which is k times produces the same magnitude fractional error (but of opposite sign) as an M which is l/k times E. 

The quantity occurring in braces is very nearly equal to (1/6) l—V2 — V provided v 2 is small. The fractional error in the composition calculated using this approximation is of the order of v x/b. Substituting this result in Eq. (A-3)... 

Mass spectrometric measurements on corals typically result in errors in and °Th of 2 per mil or better (2a), with the exception that fractional error in °Th typically increases progressively from this value for samples progressively younger than several ka. This results from the low concentrations of °Th in very young corals. Errors in Pa are typically somewhat larger than those of the other isotopes, with errors of several per mil, except for corals younger than a few ka. 

In principle, FCS can also measure very slow processes. In this limit the measurements are constrained by the stability of the system and the patience of the investigator. Because FCS requires the statistical analysis of many fluctuations to yield an accurate estimation of rate parameters, the slower the typical fluctuation, the longer the time required for the measurement. The fractional error of an FCS measurement, expressed as the root mean square of fluorescence fluctuations divided by the mean fluorescence, varies as 1V-1/2, where N is the number of fluctuations that are measured. If the characteristic lifetime of a fluctuation is r, the duration of a measurement to achieve a fractional error of E = N l,/- is T = Nr. Suppose, for example, that r = 1 s. If 1% accuracy is desired, N = 104 and so T = 104 s. 

Figure 12. Plot of I/FeO versus 5 Fe values of lunar regolith samples from the Lunar Soil Characterization Consortium. The sub-scripted numbers after the sample numbers are the I,/FeO values measured for the <250 pm sized fraction. All analyses are for bulk samples of the different sized fractions error bars are 2a as calculated from 2 or more complete Fe isotope analyses. Modified from Wiesli et al. (2003a).

Figure 10.19 Fractional error associated with equation 10.132. (A) Rayleigh s crystallization and fractional melting. (B) Equilibrium melting.

Next, the number of components, number of columns, and all physical data are read, followed by the two preset error-limits used by the program. prderr is the highest acceptable sum of component unbalances, and is expressed in moles, bdferr is the accuracy to which bubble-points, dew-points, and flashes are calculated, expressed as a fractional error. The authors have found a tight limit on this error is beneficial, and have commonly used 1 x 10 . 

Thus, the error in the ordinate of the calibration curve, y, is actually a value proportional to the error in M divided by M. In other words, the fractional error in M is involved rather than the absolute error. 

For a measurement of a Rayleigh velocity in the vicinity of 3000 m s 1, if the fractional errors in / and Az are Sf/f and SAz/Az and the error in the temperature measurement is ST (measured in degrees kelvin), then the overall fractional measurement error is... 

The algorithm used is attributed to J. B. J. Read. For many manipulations on large matrices it is only practical for use with a fairly large computer. The data are arranged in two matrices by sample i and nuclide j one matrix, V, contains the amount of each nuclide in each sample the other matrix, E, contains the variances of these numbers, as estimated from counting statistics, agreement between replicate analyses, and known analytical errors. It is also possible to add an arbitrary term Fik to each variance to account for random effects between samples not considered in the model this is usually done in terms of an additional fractional error. Zeroes are inserted for missing data in cases in which not all nuclides were measured in every sample. 

An estimate of the probable errors in the correction factors and cutoff values follows. From Equation 3 one sees that the fractional errors in both are of the same order of magnitude as the fractional error in the velocity, Av/v, averaged over the region of motion. There are three main contributions to this error. One comes from the approximation to the Davies equations (8 and 10). The average fractional error is of the order of Av/v —5%, the minus sign occurring since Equations 9 and 10 underestimate the true values of Re and v. The other error contributions come from the approximations for air density and viscosity. One sees from Equations 7-9 that the first-order term in v is independent of p and has a 1/rj dependence. The second-order term is directly proportional to P. Since this term contributes a maximum of 30% to the velocity and the maximum error in p is 8%, this contribution to Av/v should be... 

The summative-fractioruition method was extended to apply to narrow-distribution polymers with polydispersity (Mw/ Mn) less than 1.12. A fractionation parameter H, previously defined and calculated for theoretical molecular weight distributions for normal polymers, was computed for narrow-distribution polymers. The calculations were made both with and without correction for fractionation errors, using the Flory-Huggins treatment. The method was applied to a well-characterized anionic polystyrene with Mw = 97,000, for which the polydispersity was estimated by this technique to be 1.02 (in the range 1.014-1.027, 95% confidence limits). 

Fractionation Error. The effect of imperfect fractionation was studied by integrating Equation 3 numerically with f(M) given by the Flory-Huggins theory... 

This rule is an approximation to a more exact statement that the fractional error of a product or quotient is the square root of the sum of the squares of the fractional errors in the numbers being multiplied or divided. 

Notice that it is the fractional error which counts in multiplication or division. For example, if A = 100 3 (3% error) and B = 20.0 0.8 (4% error), the product AB will have a 5% error using this rule above (AB = 2000 100). The number of moles of dissolved silver chloride is calculated to be... 

Often the data analysis requires multiplication by some number B such as Avo-gadro s number, which is typically known to much higher accuracy than any of the measured data. In that case, the rule for multiplication above simplifies immediately to AC/C = A A/A—the fractional error after multiplication by a constant is unchanged. 

Notice that the fractional error in A2 is twice the fractional error in A, not V2 times the fractional error as would happen if you multiplied two independent numbers with the same error. The solubility product is the square of the concentration, so it is (3.83 0.12) x 10 n M2. By the way, notice that each individual measurement was made with much higher accuracy than the final result—correct error propagation is quite important. 

Explain why the fractional error bars for A2 are larger than the fractional error bars for AB, even if A and B each have the same fractional random error. 

Fig. 1. Approximate behavior of the fractional error as a function of the mean size parameter a, = ir D,/ and the fractional standard deviation o. Where D, is the mean particle diameter.

Rule 2. When multiplying or dividing measurements with associated errors, the percentage errors (or the fractional errors) are summed. 

In this result, the fractional error at a given value of z, that is, of kn, is proportional in the first approximation to fc2, instead of to fc, as in Eq. (5-13). This means that the approximation to the derivative given... 

The fractional error is simply the absolute error divided by the actual value. In the first example, this is 0.09/4.87 or 0.018, in the second, -0.35/4.87 or -0.072. The fractional error is converted to a percentage error by multiplying by 100, so the respective percentage errors for these examples are 1.8% and-7.2%. 

Note that changing from the true to the measured value makes very little difference to the fractional and percentage errors. In the first case above we would have a fractional error of 0.09/4.96 or 0.018 and a percentage error of 1.8%, in the second a fractional error of —0.077 and a percentage error of -7.7%. 

If a burette reading is given as 27.35 cm, the fractional error will be 0.05 cm /27.35 cm or 0.0018 and the percentage error 100 times this or 0.18%. Note that the fractional and percentage errors never have units since they are calculated by taking the ratio of two quantities having the same units, i.e. the units cancel. 

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