Slopes errors

Slope re-plot (Figure 5.22) Calculated from (5.138) Error (%) Slope re-plot (Figure 5.22) Calculated from (5.139) Error (%)... 

In the graph of AH versus AS, large deviations in the direction of T are thus admissible, while much smaller ones in the perpendicular direction are not. Hence, sequences of points with the slope T can easily result from experimental errors only this is why the value of T is called error slope (1-3,115, 116, 118, 119). Isokinetic relationships with slopes close to T should be viewed with suspicion, but they have been reported frequently. However, we shall see later that even correlations with other slopes are only apparent, or at least the isokinetic temperature is determined erroneously from the plot of AH versus AS. 

The method outlined is quick and useful for testing isokinetic relationships described in the literature and for finding approximate values of j3 (149). It should replace the incorrect plotting of E versus log A, which gives fallacious results for the value of (3 and which usually simulates better correlations than in fact apply. Particularly, the values of correlation coefficients (1) in the E versus log A plane are meaningless. As shown objectively in Figures 9-12, the failure of this plotting is not caused by experimental errors only (3, 143, 153), nor is it confined to values of j5 near the error slope or within the interval of experimental temperatures (151). 

The two beams, which form the difference pattern (one beam - -, and one beam —), cross each other at less than 3 dB below their peaks for maximum error slope-sum pattern product and to give increased angle tracking accuracy. This can be seen in Fig. 17.61 in Skolnik (1962) and in Fig. 17.62 in Rhodes (1980), where a comparison is shown between squint angle and the two difference beam magnitude crossover. 

Threshold detection is easy to set up but is affected by changes in base line level, caused by long term drift or zero calibration errors. Slope detection is less affected by base line changes but may be inadvertently triggered by a small peak, shoulder, or other signal noise, not present when the set up parameters were entered. 

In Figure 4 the measured attenuation values (TT) and the corresponding estimates are plotted against each other. Ideally (with error free estimates) all sample points should lie on the straight line through the origin with unit slope. Clearly there is a strong correlation between the estimates and the true values. 

Thus D(r) is given by the slope of the V versus P plot. The same distribution function can be calculated from an analysis of vapor adsorption data showing hysteresis due to capillary condensation (see Section XVII-16). Joyner and co-woikers [38] found that the two methods gave very similar results in the case of charcoal, as illustrated in Fig. XVI-2. See Refs. 36 and 39 for more recent such comparisons. There can be some question as to what the local contact angle is [31,40] an error here would shift the distribution curve. 

The mathematical requirements for unique determination of the two slopes mi and ni2 are satisfied by these two measurements, provided that the second equation is not a linear combination of the first. In practice, however, because of experimental error, this is a minimum requirement and may be expected to yield the least reliable solution set for the system, just as establishing the slope of a straight line through the origin by one experimental point may be expected to yield the least reliable slope, inferior in this respect to the slope obtained from 2, 3, or p experimental points. In univariate problems, accepted practice dictates that we... 

Subtracting the slope matrix obtained by the multivariate least squares tieatment from that obtained by univariate least squares slope matiix yields the error mahix... 

The scatter of the points around the calibration line or random errors are of importance since the best-fit line will be used to estimate the concentration of test samples by interpolation. The method used to calculate the random errors in the values for the slope and intercept is now considered. We must first calculate the standard deviation Sy/x, which is given by ... 

The method of standard additions can be used to check the validity of an external standardization when matrix matching is not feasible. To do this, a normal calibration curve of Sjtand versus Cs is constructed, and the value of k is determined from its slope. A standard additions calibration curve is then constructed using equation 5.6, plotting the data as shown in Figure 5.7(b). The slope of this standard additions calibration curve gives an independent determination of k. If the two values of k are identical, then any difference between the sample s matrix and that of the external standards can be ignored. When the values of k are different, a proportional determinate error is introduced if the normal calibration curve is used. 

The most commonly used form of linear regression is based on three assumptions (1) that any difference between the experimental data and the calculated regression line is due to indeterminate errors affecting the values of y, (2) that these indeterminate errors are normally distributed, and (3) that the indeterminate errors in y do not depend on the value of x. Because we assume that indeterminate errors are the same for all standards, each standard contributes equally in estimating the slope and y-intercept. For this reason the result is considered an unweighted linear regression. 

A more useful representation of uncertainty is to consider the effect of indeterminate errors on the predicted slope and intercept. The standard deviation of the slope and intercept are given as... 

Equations 5.13 for the slope, h, and 5.14 for the y-intercept, ho, assume that indeterminate errors equally affect each value of y. When this assumption is false, as shown in Figure 5.11b, the variance associated with each value of y must be included when estimating [3o and [3i. In this case the predicted slope and intercept are... 

It has been shown that for most acid-base titrations the inflection point, which corresponds to the greatest slope in the titration curve, very nearly coincides with the equivalence point. The inflection point actually precedes the equivalence point, with the error approaching 0.1% for weak acids or weak bases with dissociation constants smaller than 10 , or for very dilute solutions. Equivalence points determined in this fashion are indicated on the titration curves in figure 9.8. 

As predicted by the Arrhenius equation (Sec. 4), a plot of microbial death rate versus the reciprocal or the temperature is usually linear with a slope that is a measure of the susceptibility of microorganisms to heat. Correlations other than the Arrhenius equation are used, particularly in the food processing industry. A common temperature relationship of the thermal resistance is decimal reduction time (DRT), defined as the time required to reduce the microbial population by one-tenth. Over short temperature internals (e.g., 5.5°C) DRT is useful, but extrapolation over a wide temperature internal gives serious errors. 

To obtain the corrosion current from Rp, values for the anodic and cathodic slopes must be known or estimated. ASTM G59 provides an experimental procedure for measuring Rp. A discussion or the factors which may lead to errors in the values for Rp, and cases where Rp technique cannot be used, are covered by Mansfeld in Polarization Resistance Measurements—Today s Status, Electrochemical Techniques for Corrosion Engineers (NACE International, 1992). 

In evaluating the effect of these errors, it must be remembered that these values are errors in retention time. The error in molecular weight will depend on the slope of the calibration curve, but will usually be considerably larger. 

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