Least squares standard deviations

Using a relative rate method, rate constants for the gas-phase reactions of O3 with 1- and 3-methylcyclopentene, 1-, 3- and 4-methylcyclohexene, 1-methylcycloheptene, cw-cyclooctene, 1- and 3-methylcyclooctene, cycloocta-1,3- and 1,5-diene, and cyclo-octa-l,3,5,7-tetraene have been measured at 296 2 K and atmospheric pressure. The rate constants obtained (in units of 10-18 cm3 molecule-1 s-1) are as follows 1-methylcyclopentene, 832 24 3-methylcyclopentene, 334 12 1-methylcyclohex-ene, 146 10 3-methylcyclohexene, 55.3 2.6 4-methylcyclohexene, 73.1 3.6 1-methylcycloheptene, 930 24 d.s-cyclooclcnc, 386 23 1-methylcyclooctene, 1420 100 3-methylcyclooctene, 139 9 d.v.d.v-cycloocta-1,3-diene, 20.0 1.4 cycloocta- 1,5-diene, 152 10 and cycloocta-l,3,5,7-tetraene, 2.60 0.19 the indicated errors are two least-squares standard deviations and do not include the uncertainties in the rate constants for the reference alkenes (propene, but-l-ene, d.s-but-2-ene, trans-but-2-ene, 2-methylbut-2-ene, and terpinolene). These rate data were compared with the few available literature data, and the effects of methyl substitution have been discussed.50... 

Throughout this paper, the experimental errors are quoted in the following way Least-squares standard deviations in parentheses as units of the last digit, e.g., 1.107(1) A, and estimated total errors are quoted as error limits, e.g., 1.184 0.003 A. 

The second example is shown in Figure 5.10 and Table 5.13. The observed Bragg peak positions were determined using a profile fitting procedure and the least squares standard deviations in the observed Bragg angles did not exceed 0.003° of 29. Considering the results shown in Table 5.13, the column labeled contains nearly whole numbers, but unlike... 

It is worth noting that parameters identical to those listed in our examples can be expected only when the same computer codes are used to perform full profile refinement due to small but detectable differences in the implementation of the Rietveld method by various software developers. Furthermore, even when the same version of an identical computer program is employed to treat the same set of experimental data, small deviations may occur due to subjective decisions, such as when to terminate the refinement. In the latter case, however, the differences should be within a few least squares standard deviations. 

Distances quoted are rg unless indicated otherwise distances in the literature were converted to i>. Three times least squares standard deviations as reported in the literature. ... 

In this table, the numbers in parentheses give statistical errors (least squares standard deviations) expressed in the last significant digit. Neutron powder diffraction data. 

The least sum of squared standard deviations between measured and calculated rate constants... 

For a specified mean and standard deviation the number of degrees of freedom for a one-dimensional distribution (see sections on the least squares method and least squares minimization) of n data is (n — 1). This is because, given p and a, for n > 1 (say a half-dozen or more points), the first datum can have any value, the second datum can have any value, and so on, up to n — 1. When we come to find the... 

When experimental data is to be fit with a mathematical model, it is necessary to allow for the facd that the data has errors. The engineer is interested in finding the parameters in the model as well as the uncertainty in their determination. In the simplest case, the model is a hn-ear equation with only two parameters, and they are found by a least-squares minimization of the errors in fitting the data. Multiple regression is just hnear least squares applied with more terms. Nonlinear regression allows the parameters of the model to enter in a nonlinear fashion. The following description of maximum likehhood apphes to both linear and nonlinear least squares (Ref. 231). If each measurement point Uj has a measurement error Ayi that is independently random and distributed with a normal distribution about the true model y x) with standard deviation <7, then the probability of a data set is... 

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.)... 

We wish to apply weighted linear least-squares regression to Eq. (6-2), the linearized form of the Arrhenius equation. Let us suppose that our kinetic studies have provided us with data consisting of Tj, and for at least three temperatures, where o, is the experimental standard deviation of fc,. We will assume that the error in T is negligible relative to that in k. For convenience we write Eq. (6-2) as... 

Cliebysliev s theorem provides an interpretation of the sample standard deviation, tlie positive square root of the sample variance, as a measure of the spread (dispersion) of sample observations about tlieir mean. Chebyshev s tlieorem states tliat at least (1 - 1/k ), k > 1, of tlie sample observations lie in tlie... 

Coefficients and standard deviation,s, of the least-squares second-degree polynomial representing the titration curve of PGA with different strong bases at 298 K. 

The parameter values found by the two methods differ slightly owing to the different criteria used which were the least squares method for ESL and the maximum-likelihood method for SIMUSOLV and because the T=10 data point was included with the ESL run. The output curve is very similar and the parameters agree within the expected standard deviation. The quality of parameter estimation can also be judged from a contour plot as given in Fig. 2.41. 

In Equation 14.27, cT, oP and ax are the standard deviations of the measurements of T, P and x respectively. All the derivatives are evaluated at the point where the stability function cp has its lowest value. We call the minimization of Equation 14.24 subject to the above constraint simplified Constrained Least Squares (simplified CLS) estimation. 

Copp and Everet (1953) have presented 33 experimental VLE data points at three temperatures. The diethylamine-water system demonstrates the problem that may arise when using the simplified constrained least squares estimation due to inadequate number of data. In such case there is a need to interpolate the data points and to perform the minimization subject to constraint of Equation 14.28 instead of Equation 14.26 (Englezos and Kalogerakis, 1993). First, unconstrained LS estimation was performed by using the objective function defined by Equation 14.23. The parameter values together with their standard deviations that were obtained are shown in Table 14.5. The covariances are also given in the table. The other parameter values are zero. 

A central concept of statistical analysis is variance,105 which is simply the average squared difference of deviations from the mean, or the square of the standard deviation. Since the analyst can only take a limited number n of samples, the variance is estimated as the squared difference of deviations from the mean, divided by n - 1. Analysis of variance asks the question whether groups of samples are drawn from the same overall population or from different populations.105 The simplest example of analysis of variance is the F-test (and the closely related t-test) in which one takes the ratio of two variances and compares the result with tabular values to decide whether it is probable that the two samples came from the same population. Linear regression is also a form of analysis of variance, since one is asking the question whether the variance around the mean is equivalent to the variance around the least squares fit. 

One can apply a similar approach to samples drawn from a process over time to determine whether a process is in control (stable) or out of control (unstable). For both kinds of control chart, it may be desirable to obtain estimates of the mean and standard deviation over a range of concentrations. The precision of an HPLC method is frequently lower at concentrations much higher or lower than the midrange of measurement. The act of drawing the control chart often helps to identify variability in the method and, given that variability in the method is less than that of the process, the control chart can help to identify variability in the process. Trends can be observed as sequences of points above or below the mean, as a non-zero slope of the least squares fit of the mean vs. batch number, or by means of autocorrelation.106... 

To construct the Hill plot (Figure 5.10E), it was assumed that fimax was 0.654 fmol/mg dry wt., the Scatchard value. The slope of the plot is 1.138 with a standard deviation of 0.12, so it would not be unreasonable to suppose % was indeed 1 and so consistent with a simple bimolecular interaction. Figure 5.10B shows a nonlinear least-squares fit of Eq. (5.3) to the specific binding data (giving all points equal weight). The least-squares estimates are 0.676 fmol/mg dry wt. for fimax and... 

If separate D-RESP charge sets are fitted for every single one of the 36 frames, the standard deviation of the electrostatic field generated varies between 3.5 and 5% with respect to the full quantum reference. This accuracy is the best (in the least-squares sense) that can be obtained if the system is modeled with time-dependent atomic point-charges and represents the accuracy limit for a fluctuating point charge model of the dipeptide. 

Preparation of a calibration curve has been described. From the fit of the least-squares line we can estimate the uncertainty of the results. Using similar equations we can determine the standard deviation of the calibration line (similar to the standard deviation of a group of replicate analyses) as... 

Also determine the slope and intercept of the least-squares line for this set of data. Determine the concentration and standard deviation of an analyte that has an absorbance of 0.335. 

White noise signifies that the experimental standard deviation, ay, is normally distributed and the same for all individual measurements, yij. The traditional least-squares fit delivers the most likely parameters only under the condition of white noise. 

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