Finding the Best Straight Line

The method of least squares finds the best straight line through experimental data points. We will apply this procedure to analytical chemistry calibration curves in the next section. 

Some of the deviations are positive and some are negative. To minimize the magnitude of the deviations irrespective of their signs, we square the deviations to create positive numbers  

Because we minimize the squares of the deviations, this procedure is called the method of least squares. 

When we use such a procedure to minimize the sum of squares of the vertical deviations, the slope and the intercept of the best straight line fitted to n points are 

Quantities required for propagation of uncertainty with Equation 4-19  

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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. 

Figure 4-11). Use the least-squares procedure to find the best straight line through the linear portion of the data, up to and including 20.0 p.g of protein (14 points, including the 3 corrected blanks, in the shaded portion of Table 4-7). Find the slope and intercept and uncertainties with Equations 4-16, 4-17, 4-20, 4-21, and 4-22. 

Find the best straight line through the following data points. 

A common application of LINEST is to find the best straight line through a set of data points, i.e., to find the regression parameters m and h of the best straight line y = mx + b through the data points. 

The data from these experiments were analyzed using the statistical methods described in Chapters 5, 6, and 7. For each of the standard arsenic solutions and the deer samples, the average of the three absorbance measurements was calculated. The average absorbance for the replicates is a more reliable measure of the concentration of arsenic than a single measurement. Least-squares analysis of the standard data (see Section 8C) was used to find the best straight line among the points and to calculate the concentrations of the unknown samples along with their statistical uncertainties and confidence limits. 

Use LSQUARES to find the best straight line fit through the experimental X, y data points. 

To see how least-squares linear regression works, consider that we are given a set of N experimental data values (x , y ), i = We wish to find the best straight line (Equation... 

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. 

Let s use these equations to find the slope and intercept of the best straight line through the four points in Figure 4-9. The work is set out in Table 4-6. Noting that n = 4 and putting the various sums into the determinants in Equations 4-16, 4-17. and 4-18 gives... 

If we were dealing with real experimental data, there would be some error in each of the data points on the plot. For this reason, we should wot use experimental data points to determine the slope. (Random experimental errors of only 10% can introduce errors of more than 100% in slopes based on only two points.) Rather we should draw the best straight line and then use points on that Une to find its slope. Errors are further minimized by choosing points that are widely separated. 

We draw the best straight line with IIA = 403 erg when /7 = 0. From the slope of this line we find... 

Trace the results on to a piece of paper, and plot the best straight line through the points for each vapour. Find the slope of each line. 

The aim of LDA is to find the optimal linear surface in the multidimensional space (the best straight line in two dimensions, when considering the example described in Figure 3) to separate the region corresponding to class... 

How do we find the best estimate for the relationship between the measured signal and the concentration of analyte in a multiple-point standardization Figure 5.8 shows the data in Table 5.1 plotted as a normal calibration curve. Although the data appear to fall along a straight line, the actual calibration curve is not intuitively obvious. The process of mathematically determining the best equation for the calibration curve is called regression. 

Thus finding the best values for the slope, b, and intercept, a, of a straight line simply involves developing the sums indicated and combining them in equations 83 and 84. 

Drawing straight lines through data points is a slightly arbitrary procedure. The slope of the straight line does not depend very much on this arbitrariness but the value of the intercept usually depends very much on it. Consequently, the value of the kinetic parameter related to the intercept will be estimated with the accuracy of the eyes capability of finding the best fit between experimental points and those lying on the line drawn. An objective method of parameter estimation consist in evaluation of the minimum of the function ... 

Find the best least-squares straight-line fit to this data ... 

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... 

Besides finding the best parameter values according to a Hill-De Boer plot, one should of course also try the localized version, i.e. the FFG plot, A1.5al. For two values of monolayer capacity (39 and 40 gmol g" ) the data are replotted as ln[0/(l- 0] p /p)l as a function of 6, which should yield a straight line. As with the localized case, the isotherms appeared more or less linear if considered over short ranges of 6, although linearity over the entire range is difficult to achieve. 

A similar method has been proposed in the PF (10). The technique just described differs from the PF proposal in that it does not rely on subjectivity to draw the best-fitting straight line through the data points instead, it uses simple linear regression to find the best fit. Also, the scale for the cumulative percent finer is linear and not logarithmic, which the subsequent discussion will show leads to a more thorough analysis and interpretation. 

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