Subtraction, uncertainties

In the Tables, values may be found for the calculated (subtracted) uncertainties for the X tristimulus value, the Y tristimulus and the Z, as well as the L, a, and b " uncertainty values. These are given in columns from left to right under the following uncertainty components calculated instrumait uncertainty (CIU), calculated operator uncertainty (COU), and calculated levelness uncertainty (CLU). The total uncertainty is the vector length (the square root of the sum of the squares) of the three components. Because these components add under quadratme, the total uncertainty is the same as the largest component in the case of Table 1, but this need not always be the case. 

It is easy to see that combining uncertainties in this way overestimates the total uncertainty. Adding the uncertainty for the first delivery to that of the second delivery assumes that both volumes are either greater than 9.992 mL or less than 9.992 mL. At the other extreme, we might assume that the two deliveries will always be on opposite sides of the pipet s mean volume. In this case we subtract the uncertainties for the two deliveries,... 

Many chemical calculations involve a combination of adding and subtracting, and multiply and dividing. As shown in the following example, the propagation of uncertainty is easily calculated by treating each operation separately using equations 4.6 and 4.7 as needed. 

When measured quantities are added or subtracted, the uncertainty in the result is found in a quite different way than when they are multiplied and divided. It is determined by counting the number of decimal places, that is, the number of digits to the right of the decimal point for each measured quantity. When measured quantities are added or subtracted, the number of decimal places in the result is the same as that in the quantity with the greatest uncertainty and hence the smallest number of decimal places. 

We see that the uncertainty in a derived quantity is fixed by the uncertainties in the measurements that must be combined. For an addition or a subtraction, the maximum uncertainty is simply the sum of the uncertainties in the components 0.2 + 0.2 = 0.4. 

The magnitude of t0 can be measured from the intercept of a f(a)—time plot. The existence of the induction period can introduce uncertainty into a reduced time analysis if the temperature coefficient of t0 differs from that later applicable, and it is necessary to plot (t — t0)/(tb — t0) against a where tb is the time at which the selected common value of a is attained. The occurrence of a slow initial process can be reflected in deviations from linearity in the f(a) time plot, though in favourable systems the contribution may be subtracted before analysis [40]. 

The uncertainties of forecast have been calculated from the standard deviations of forecast awarding five points. The uncertainty subtracted and added to the estimated value gives a confidence interval of 95% for the LEL. 

In contrast, a systematic error remains constant or varies in a predictable way over a series of measurements. This type of error differs from random error in that it cannot be reduced by making multiple measurements. Systematic error can be corrected for if it is detected, but the correction would not be exact since there would inevitably be some uncertainty about the exact value of the systematic error. As an example, in analytical chemistry we very often run a blank determination to assess the contribution of the reagents to the measured response, in the known absence of the analyte. The value of this blank measurement is subtracted from the values of the sample and standard measurements before the final result is calculated. If we did not subtract the blank reading (assuming it to be non-zero) from our measurements, then this would introduce a systematic error into our final result. 

The tracer-subtraction procedure adds negligible uncertainty to the measured CaJ Ca ratios. However, it is in fact essentially impossible to entirely eliminate the effects of instrumental mass discrimination for the measurements of either the Ca- Ca mixed tracer or for the standard Ca isotope ratios. Hence, it is necessary to have a standard material with an agreed-upon value of 5 Ca. At the time of writing of this article there is no such standard. 

In a sense, calculating the mean replicate response removes the effect of purely experimental uncertainty from the data. It is not unreasonable, then, to expect that the deviation of these mean replicate responses from the estimated responses is due to a lack of fit of the model to the data. The matrix of lack-of-fit deviations, L, is obtained by subtracting f from J... 

Fig. 3. Plot of logio normalized ion-exchange rate at amorphous silica saturation vs. the amount of excess alkalis (Na, K), denoted by the molar ratio XAlk/(Al + IVB + FeT). All boron is treated as four-fold coordinated (IVB) and total iron (FeT) is regarded as ferric. The ion-exchange rate subtracts out the contribution of alkalis to solution from matrix dissolution. As the amount of excess alkali increases, the ion-exchange rate increases. This increase in rate reflects the increasing amount of alkalis in non-bridging oxygen (NBO) configurations. Error bars represent 2-

Answer Subtract 0.169 mV. Note that this is less than the typical uncertainty of 1 mV in E°.]... 

Uncertainties always combine according to equations like 6.28 even if the quantities are subtracted (i.e., the squares of the uncertainties are always added). This is why a calculation that eventuates in the subtraction of two large numbers of nearly equal magnitude (e.g., weighing the captain of the ship by weighing the ship with the captain on board and subtracting the mass of the ship when the captain is not on board), is notoriously beset by large uncertainty. 

For addition and subtraction, the uncertainty in the answer is obtained from the absolute uncertainties of the individual terms as follows ... 

For addition and subtraction, use absolute uncertainty. Relative uncertainty can be found at the end of the calculation. 

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