Best straight line

Expand the three detemiinants D, Dt, and for the least squares fit to a linear function not passing through the origin so as to obtain explicit algebraic expressions for b and m, the y-intercept and the slope of the best straight line representing the experimental data. 

The variables that are combined hnearly are In / 17T, and In C, Multilinear regression software can be used to find the constants, or only three sets of the data smtably spaced can be used and the constants found by simultaneous solution of three linear equations. For a linearized Eq. (7-26) the variables are logarithms of / C, and Ci,. The logarithmic form of Eq. (7-24) has only two constants, so the data can be plotted and the constants read off the slope and intercept of the best straight line. 

An alternative method is to fit the best straight line through the linearized set of data assoeiated with distributional models, for example the Normal and 3-parameter Weibull distributions, and then ealeulate the correlation coejficient, r, for eaeh (Lipson and Sheth, 1973). The eorrelation eoeffieient is a measure of the degree of (linear) assoeiation between two variables, x and y, as given by equation 4.4. 

The simplest procedure is merely to assume reasonable values for A and to make plots according to Eq. (2-52). That value of A yielding the best straight line is taken as the correct value. (Notice how essential it is that the reaction be accurately first-order for this method to be reliable.) Williams and Taylor have shown that the standard deviation about the line shows a sharp minimum at the correct A . Holt and Norris describe an efficient search strategy in this procedure, using as their criterion minimization of the weighted sum of squares of residuals. (Least-squares regression is treated later in this section.)... 

Once a linear relationship has been shown to have a high probability by the value of the correlation coefficient (r), then the best straight line through the data points has to be estimated. This can often be done by visual inspection of the calibration graph but in many cases it is far better practice to evaluate the best straight line by linear regression (the method of least squares). 

Example 10. Calculate by the least squares method the equation of the best straight line for the calibration curve given in the previous example. 

From plots of In g(a) against In T with n = 0, 1 and 2, the plot which gives the best straight line identifies the value of n and from the slope Um and hence E is calculated. (This approach can also be regarded as a reference temperature method , see also Sect. 6.2.4). 

Fig. 3.—A. Initial Slope Approximation to Determine the Initial, Nonselective, Spin-Lattice Relaxation Rate of H-S of 2,3 S,6-Di-0-isopropylidene-a-D-mannofuranose (2) in Me2SO-d Solution. (Points between 0.01 and l.SS s were selected for tracing the best straight line.) B. The Same as in A for H-1 of a Partially Deuterated Sample of 1,6-Anhydro- -cellobiose Hexaacetate (3). [Note that the relaxation of H-1 is strongly dependent on the choice of I value. An R (ns) value of 0.24s was obtained from the data points 0 t 5s, where a value of 0.18 s was obtained from the terminal decay 5 lOs (see text).]...

Fit the best straight line to these data T) represents —log C, and Yi represents E. We will perform the calculation manually, using the following tabular lay-out. 

A value of n is assumed and values of the left hand side are plotted against (t-t0). The correct value of n has been found when the data are colllnear. The correct value of k is found from the slope of the best straight line. 

Another problem with real data is that due to random indeterminate errors (Chapter 1), the analyst cannot expect the measured points to fit a straight-line graph exactly. Thus it is often true that we draw the best straight line that can be drawn through a set of data points and the unknown is determined from this line. A linear regression, or least squares, procedure is then done to obtain the correct position of the line and therefore the correct slope, etc. 

When plotting the results of the measurement of a series of standard solutions, why do we draw the best straight line possible through the points rather than just connect the points ... 

The method of least squares is a procedure by which the best straight line through a series of data points is mathematically determined. More details are given in Section 6.4.4. It is useful because it eliminates guesswork as to the exact placement of the line and provides the slope and y-intercept of the line. 

The best straight line fit is obtained when the sum of the squares of the individual y-axis value deviations (deviations between the plotted y values and the values on the proposed line) is at a minimum. 

Example The results obtained from the determination of concentration of the standard solutions and measurements of corresponding peak areas with a GC are recorded in Table 3.1 and plotted in Figure 3.3 where the former is represented along the x-axis and the latter along the y-axis. How to draw the best straight line through all these points ... 

It makes sense to start with the well known task of finding the best straight line through a set of (x,y)-data pairs. We can refer back to Figure 4-3 which displays the sum of squares, ssq, as a function of the two parameters defining a straight line, the slope and the intercept. The task is to find the position of the minimum, the values for slope and intercept that result in the least sum of squares. 

With this method, the best straight line is fitted to a set of points that are linearly related as y = mx + b , where/ is the ordinate and x is the abscissa datum point, respectively. The slope (m) and the intercept (b) can be calculated by least squares analysis using Eqs. (50) and (51), respectively [23] ... 

Draw a Heyrovsky-Ilkovic plot of In [(/j - /)//] against E. /d = 25.01 pA, because the current becomes limiting at higher E. The best straight line should have an intercept of 1/2 = -1.00 V and, from the gradient, the number of electrons (n) passed is 2. 

The method of least squares permits us to calculate the best function of a given form for the set of data at hand, but it does not help us decide which form of analytic function to choose. Inspection of a graph of the data is helpful in such a choice. Figure A.l shows the data of Table A.l as well as the best straight line and the best quadratic curve, with the latter represented by Equation (A. 12), both are fitted to the data by the method of least squares. 

Because the determinant is equal to zero, the (X X) matrix cannot be inverted, and a unique solution does not exist. An interpretation of the zero determinant is that the slope P, and the response intercept Po are both undefined (see Equations 5.14 and 5.15). This interpretation is consistent with the experimental design used and the model attempted the best straight line through the two points would have infinite slope (a vertical line) and the response intercept would not exist (see Figure 5.8). 

As shown in Fig. 17, the number of protons from a series of silica-alumina samples with differing BET surface areas is a nearly linear function of surface area, indicating that essentially all of the protons are situated in the surface phase of the solid. The samples corresponding to the data of Fig. 17 were prepared by steam sintering of the above-mentioned 425 meterVgram sample at various temperatures above 500° so as to reduce the surface area to the values given and then soaked in water for 4 hours, oven-dried at 110° and evacuated at 500° for 16 hours. A plot somewhat similar to Fig. 17 has been obtained for silica gel. The slope of the best straight line repre-... 

Fiq. 17. Number of protons in silica-alumina (12.5 wt. % AljOj) dehydrated at 500° versus BET surface area. Dashed line is the best straight line representing the data (173). 

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