Least squares standard uncertainties

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... 

It is notable that such kinds of error sources are fairly treated using the concept of measurement uncertainty which makes no difference between random and systematic . When simulated samples with known analyte content can be prepared, the effect of the matrix is a matter of direct investigation in respect of its chemical composition as well as physical properties that influence the result and may be at different levels for analytical samples and a calibration standard. It has long since been suggested in examination of matrix effects [26, 27] that the influence of matrix factors be varied (at least) at two levels corresponding to their upper and lower limits in accordance with an appropriate experimental design. The results from such an experiment enable the main effects of the factors and also interaction effects to be estimated as coefficients in a polynomial regression model, with the variance of matrix-induced error found by statistical analysis. This variance is simply the (squared) standard uncertainty we seek for the matrix effects. 

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... 

Nonetheless, the value for the standard error computed by the linear least square computations reveal an estimate of the uncertainty which is valid for comparison purposes among different experiments. 

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... 

If a calibration function is used with coefficients obtained by fitting the response of an instrument to the model in known concentrations of calibration standards, then the uncertainty of this procedure must be taken into account. A classical least squares linear regression, the default regression... 

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. 

Standard addition. To measure Ca in breakfast cereal, 0.521 6 g of crushed Cheerios was ashed in a crucible at 600°C in air for 2 h.22 The residue was dissolved in 6 M HC1, quantitatively transferred to a volumetric flask, and diluted to 100.0 mL. Then 5.00-mL aliquots were transferred to 50-mL volumetric flasks. Each was treated with standard Ca2+ (containing 20.0 pg/mL), diluted to volume with H20, and analyzed by flame atomic absorption. Construct a standard addition graph and use the method of least squares to find the x-intercept and its uncertainty. Find wt% Ca in Cheerios and its uncertainty. 

In order to estimate the uncertainty in the value obtained for the rate parameter K, one should also consider random errors. There are two sources of random error in this case the least-squares fitting which gives an error (dK)ls due to random noise in the experimental spectrum [equation (164)] and the measurement of w0 the line-width of the standard employed. The total random error in K is approximated by ... 

Abstract Since the uncertainty of each link in the traceability chain (measuring analytical instrument, reference material or other measurement standard) changes over the course of time, the chain lifetime is limited. The lifetime in chemical analysis is dependent on the calibration intervals of the measuring equipment and the shelf-life of the certified reference materials (CRMs) used for the calibration of the equipment. It is shown that the ordinary least squares technique, used for treatment of the calibration data, is correct only when uncertainties in the certified values of the measurement standards or CRMs are negligible. If these uncertainties increase (for example, close to the end of the calibration interval or shelf-life), they are able to influence significant-... 

When bond lengths with different chemical sorrounding are compared to each other, the experimental error limits for the individual values are of great importance. It has become common practice in ED studies to report estimated uncertainties, which usually are 2 to 3 times the standard deviation of the least-squares refinement. It is assumed that such uncertainties account for possible systematic errors. Most MW studies also report estimated uncertainties, which may be considerably larger than the standard deviations derived from fitting the rotational constants. In X-ray crystallography it is common use to cite standard deviations from the least-squares analysis of the structure factors. 

In the least squares analyses of eq. 3, the individual enthalpies were weighted inversely as the squares of the experimental uncertainty intervals. In all cases, r2 > 0.9999. The standard errors were generated from the unweighted enthalpies. nc is the number of carbon atoms in the compound. 

Evaluate a quantitative uncertainty in your value of A// /Z by the method of limiting slopes and compare this with the standard deviation obtained from the least-squares fit to your data. Comment on this uncertainty in relation to the variation with temperature calculated above. Discuss possible sources of systematic errors. 

M = Estimated uncertainties, in parentheses, represent one standard deviation in the least squares fit to the experimental data. Table IV summarizes several parameters of interest for... 

The standard uncertainties (or standard deviations) for each parameter determined according to the least squares method are calculated from... 

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. 

Spreadsheet Summary In Chapter 12 ot Applications of Microsoft Excel in Analytical Chemistry, we investigate the multiple standard additions method for determining solution concentration. A least-squares analysis of the data leads to the determination of the concentration of the analyte as well as the uncertainty of the measured concentration. 

Bakeeva, Pashinkin, Bakeev, and Buketov [73BAK/PAS] measured the selenium dioxide pressure over gold selenite in the interval 489 to 599 K by the dew point method. The pressure was calculated from the dew point temperature by the relationship for the saturated vapour pressure in [69SON/NOV]. The data in the deposited VlNITl document (No. 4959-72) have been recalculated with the relationship selected by the review. The enthalpy and entropy changes obtained from the temperature variation of the equilibrium constant are A //° ((V.123), 544 K) = (576.8 13.0) kJ-mol and A,S° ((V.123), 544 K) = (899.4 + 24.0) J-K -mor. The uncertainties are entered here as twice the standard deviations from the least-squares calculation. 

The uncertainties are entered here as twice the standard deviations from the least-squares calculation. The data yield Aj j//° (CoSe03,cr, 923 K) = (15.2 + 8.3) kJmol. ... 

Table A-119 Stability constants of cadmium thiocyanate complexes (mean values) in NaC104 medium with an extrapolation to / = 0 by the SIT method. The uncertainties correspond to twice the standard deviation from the least-squares calculation.

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