Best fit straight line

A variety of experimental data has been found to fit the Langmuir equation reasonably well. Data are generally plotted according to the linear form, Eq. XVn-9, to obtain the constants b and n from the best fitting straight line. The specific surface area, E, can then be obtained from Eq. XVII-10. A widely used practice is to take to be the molecular area of the adsorbate, estimated from liquid or solid adsorbate densities. On the other hand, the Langmuir model is cast around the concept of adsorption sites, whose spacing one would suppose to be characteristic of the adsorbent. See Section XVII-5B for an additional discussion of the problem. 

Using Numbers Using the three points, draw the best-fit straight line through the points. Find the slope of the line. What are the units for the slope The value of the slope should look like another value you have previously calculated. Which one is it ... 

The mean values of the duplicate measurements are plotted with respect to time and the best-fit straight line drawn through the points. The point at which the line reaches a value of 3.v less than the initial value gives the maximum holding time. Figure 3.2 shows a graph of this type. 

The probit relationship of Equation 2-4 transforms the sigmoid shape of the normal response versus dose curve into a straight line when plotted using a linear probit scale, as shown in Figure 2-10. Standard curve-fitting techniques are used to determine the best-fitting straight line. 

In the vicinity of the minimum, the H should be positive-definite. This may not be the case everywhere in which case there is a small but real danger of iterating towards a saddle instead of the minimum. It is therefore highly advisable, especially when the data scatter about the best-fit straight line, plane, or hyper-plane, to use the best possible initial estimate. Most commonly, one of the linear estimates (Section 5.1) will be good enough. 

A numerical description of how closely the points adhere to the best fit straight line on a correlation plot (r). 

Calibration graphs must be drawn as best-fit straight lines. 

Plotting these data as in Figure 16-1, and using the method of least squares (see p 72), we obtain a best-fit straight line whose slope is -1809.5 K and whose y... 

Select the most rapid portion of oxygen uptake following the short lag to draw a best-fit straight line for determination of activity (Figure C4.2.2). If desired, determine protein concentration (unitbu) of enzyme solution to express activity per mg protein. 

Relationship between strength and toughness for glass/alumina composites with best fit straight line to eq. (4.2). Data from [29],... 

The slope and intercept of the best-fit straight line are then given by... 

Note that the coefficients are different from those of Section 2.1.2. One reason is that there are still a number of interferents, from the other PAHs, in the spectrum at 336 nm, and these are modelled partly by the intercept term. The models of Sections 2.1.1 and 2.1.2 force the best fit straight line to pass through the... 

We then want to find the best fitting straight line for the data. A possible line is shown in Figure 14.9. To determine how good a fit this line achieves, we determine the vertical distance (deviation) between each data point and the line. For example, the point corresponding to the lowest rainfall (0.51 cm/week) deviates very slightly... 

In regression analysis, we identify the best fitting straight line through the observed data points. This is selected on the basis that it minimizes the sum of the squared vertical deviations of the points from the line - the least squares fit . The goodness of fit is reported as an -squared value which can vary between 0 (no fit) and 100 per cent (perfect fit of line to points). 

Best fit straight lines for classical and inverse calibration data for pyrene at 335 nm, no intercept... 

One interesting and important consideration is that the apparent root mean square error in Sections 5.2.2 and 5.2.3 is only reduced by a small amount, yet the best fit straight line appears much worse if we neglect the intercept. The reason for this is that there is still a considerable replicate error, and this cannot readily be modelled using a... 

Best fit straight line using inverse calibration data of Figure 5.5 and an intercept term... 

Fig. 7 Graph showing the linear correlation between 31P chemical shift (in ppm) and the number of hydrogen bonds N-H 0=P per Ar3PO molecule [142]. Experimental data points for cocrystals of known structure are denoted +. The best fit straight line through these points is shown...

Experimentally, E is found by plotting measured values of In k against /T and computing the slope of the best-fit straight line through the data points (slope = E/R ). After E has been obtained, A may be calculated from equation (58) and the measured values of /c. Often E can be determined within a few percent however, difficulties in measuring the absolute value of the reaction-rate constant often produce uncertainties in A that exceed a factor of 2. If the reaction were not an elementary step then E obtained in this way would be called an overall activation energy. 

A calibration curve for 1 -naphthyl glucuronide is shown in Figure 4. For this data, the best fit straight line has a correlation coefficient of 0.989. When fitted to a quadratic equation the correlation coefficient was 1.00. Quadratic calibration curves were also seen for the other compounds in this study, and this is consistent with another report in this symposium (Brown, M. A., et al. this volume). 

Fitting the best fit straight line to a set of points involves calculating the values for intercept a and slope b in the following line equation ... 

The equation for the best fit straight line from these calculated values is y = —0.028 + 1.0114x. This equation suggests that a constant error of —0.028 is evident regardless of the true concentration and this is a fixed bias of —0.028 and a relative bias of 1.14%. Using these figures it is possible to calculate the bias at any particular concentration. 

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