Straight standard deviation

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.)... 

Although etch rate data for a particular polymer film yielded straight-line kinetic plots, the data from one film to another for any given polymer exhibited considerable scatter this is indicated by the large standard deviations in Tables I and II. Because of the scatter, the cause of which is under investigation, the etch rate data reported here have only semiquantitative significance. 

Plotting droplet size data on an arithmetic-probability graph paper will generate a straight line if the data follow normal distribution. Thus, the mean droplet diameter and standard deviation can be determined from such a plot. 

Figure 12.1 shows how well the predicted results agree with the experimental values listed in Table 12.2. The predictions consistently overestimate experimental dielectric constant values for this series of samples. The difference varies from 12 to 27% of the experimental values. Qualitatively, predictions follow the same trend as experiment. The calculated slope for the best straight-line fit through the data points is equal to 0.822 with a standard deviation of 0.088. The correlation coefficient is equal to 0.881. For several of the polyimides tested here the method... 

Figure 4-11 Calibration curve for protein analysis in Table 4-7. The equation of the solid straight line fitting the 14 data points (open circles) from 0 to 20 y.g, derived by the method of least squares, is y = 0.016 3 ( 0.000 22)x t-0.004, (+0.0024). The standard deviation of y is sy = 0.0059. The equation of the dashed quadratic curve that fits all 17 data points from...

The method of least squares is used to determine the equation of the best straight line through experimental data points. Equations 4-16 to 4-18 and 4-20 to 4-22 provide the least-squares slope and intercept and their standard deviations. Equation 4-27 estimates the uncertainty in x from a measured value of y with a calibration curve. A spreadsheet greatly simplifies least-squares calculations. 

Here is a least-squares problem that you can do by hand with a calculator. Find the slope and intercept and their standard deviations for the straight line drawn through the points (x.y) = (0,1), (2,2), and (3,3). Make a graph showing the three points and the line. Place error bars ( sv) on the points. 

SD standard deviation SDE simultaneous distillation extraction SDS sodium dodecyl sulfate SFC solid fat content SFI solid fat index SHAM salicylhydroxamic acid SIM selected ion monitoring SNIF-NMR site-specific natural isotope fractionation measured by nuclear magnetic resonance spectroscopy SP-HPLC straight-phase high-performance liquid chromatography... 

Figure 5. Variation of Mott-Schottky intercept with pCl for (100) orientation n-GaAs, 40°C. Circles denote intercept values from automated admittance measurements. Bars signify standard deviation of least-squares straight line.

Very often it is not possible a priori to separate contaminated and uncontaminated soils at the time of sampling. The best that can be done in this situation is to assume the data comprise several overlapping log-normal populations. A plot of percent cumulative frequency versus concentration (either arithmetic or log-transformed values) on probability paper produces a straight line for a normal or log-normal population. Overlapping populations plot as intersecting lines. These are called broken line plots and Tennant and White (1959) and Sinclair (1974) have explained how these composite curves may be partitioned so as to separate out the background population and then estimate its mean and standard deviation. Davies (1983) applied the technique to soils in England and Wales and thereby estimated the upper limits for lead content in uncontaminated soils. 

Figure 7.7. Total, elastic, and viscous stress-strain curves for uncrosslinked self-assembled type I collagen fibers.Total (open squares), elastic (filled diamonds), and viscous (filled squares) stress-strain curves for self-assembled uncrosslinked collagen fibers obtained from incremental stress-strain measurements at a strain rate of 10%/min. The fibers were tested immediately after manufacture and were not aged at room temperature. Error bars represent one standard deviation of the mean value for total and viscous stress components. Standard deviations for the elastic stress components are similar to those shown for the total stress but are omitted to present a clearer plot. The straight line for the elastic stress-strain curve closely overlaps the line for the viscous stress-strain curve. Note that the viscous stress-strain curve is above the elastic curve suggesting that viscous sliding is the predominant energy absorbing mechanism for uncrosslinked collagen fibers.

Standard Curve Chromatograph aliquots of each Standard Preparation as directed under Procedure. Measure the peak areas for each Standard Preparation. Plot a standard curve using the concentration, in milligrams per kilogram, of each Standard Preparation versus its corresponding peak area, and draw the best straight line. To ensure that the relative standard deviation does not exceed 2.0%, chromatograph a sufficient number of replicates of each Standard Preparation, and record the areas as directed under Procedure (below). 

If a straight line can be fitted to the plot, then the median particle diameter can be determined as being the 50 percent value on the plot (remember that when you are plotting number distribution, geometric mean and median for the number distribution are the same if there is a lognormal distribution). The geometric standard deviation is determined by the ratio... 

If unit weights are employed, aU Wj are deleted and S Wj is replaced by m.) These equations may be useful when a simple straight-line fit ( linear regression ) is being done with a spreadsheet program or a pocket calculator. Many calculators accumulate most or aU of the sums required in Eqs. (20) to (22) some complete the calculation and offer both the refined parameters and their estimated standard deviations. 

Eor the two-parameter case of the linear relationship we may apply Eqs. (28) and (40) to the estimation of the standard deviations in the intercept uq and the slope of the corresponding straight line. Erom Eq. (22) we have... 

A value of 5.2A for the ion-size parameter a yielded straight-line plots of E0 vs. m at each temperature and solvent composition. The intercepts were obtained by standard linear regression techniques. A graphical representation of the data for each of the H20/NMA solvent mixtures at 25°C is shown in Figure 1. The calculations were performed with the aid of a PDP-11 computer with a teletype output. The intercepts (E°) and the standard deviations of the intercepts are summarized in Table III. 

This equation of a straight line takes into account a slight difference of the base lines by the constant a. By applying least-squares regression the constants a and b are determined. The variance of A(R) indicates the degree of conformity between the two spectra. If sample and reference spectrum differ only slightly, the spectra should be compared blockwise, and the standard deviations of the different blocks should be used. This method is useful for spectra which show a sufficient number of sharp bands. It may fail if there are broad bands, in which case it is necessary to compare the second derivatives of the spectra. The algorithm is shown in Fig. 5.1-15. 

---