The probable error

Some observations deviate so little from the mean that we may consider that value to be a very close approximation to the truth, in other cases the arithmetical mean is worth very little. The question, therefore, to be settled is, What degree of confidence may we have in selecting this mean as the best representative value of a series of observations In other words, How good or how bad are the results  

We could employ Gauss absolute measure of precision to answer this question. It is easy to show that the measure of precision of two series of observations is inversely as their accuracy. If the probabilities of an error xv lying between 0 and lv and of an error xa, between 0 and lr are respectively 

One way of showing how nearly the arithmetical mean represents all the observations, is to suppose all the errors arranged in their order of magnitude, irrespective of sign, and to select a quantity whioh will occupy a place midway between the extreme limits, so that the number of errors less than the assumed error is the same as those which exceed it. This is called the probable 

The probable error determines the degree of confidence we may have in using the mean as the best representative value of a series of observations. For instance, the atomic weight of oxygen (H = 1) is said to be 15-879 with a probable error + 0 0003. This means that the arithmetical mean of a series of observations is 15 879, and the probability is —that is, the odds are even, or you may bet 1 against 1—that the true atomic weight of oxygen lies between 15 8793 and 15 8787. 

Referring to Fig. 171, let MP and M P be drawn at equal distances from Oy in such a way that the area bounded by these lines, the curve, and the a-axis (shaded part in the figure), is equal to half the whole area, bounded by the whole curve and the rc-axis, then it will be obvious that half the total observations will have errors numerically less than OM, and half, numerically greater 

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Gani" derived a group contribution method that shows promise. Initial evaluations show average errors of 5 to 10 percent (10-30 K) on a wide variety of compounds, but larger errors can occur. It is recommended that severaf compounds of Known melting point in the same or a similar family be predic ted in order to estimate the probable error. 

Finally, force field methods are zero-dimensional . It is not possible to asses the probable error of a given result within the method. The quality of the result can only be judged by comparison with other calculations on similar types of molecules, for which relevant experimental data exist. 

A. Standard series method (Section 17.4). The test solution contained in a Nessler tube is diluted to a definite volume, thoroughly mixed, and its colour compared with a series of standards similarly prepared. The concentration of the unknown is then, of course, equal to that of the known solution whose colour it matches exactly. The accuracy of the method will depend upon the concentrations of the standard series the probable error is of the order of + 3 per cent, but may be as high as + 8 per cent. 

The reliability factor B was 0276 after the first refinement and 0-211 after the fourth refinement. The parameters from the third and fourth refinements differed very little from one another. The final values are given in Table 1. As large systematic errors were introduced in the refinement process by the unavoidable use of very poor atomic form factors, the probable errors in the parameters as obtained in the refinement were considered to be of questionable significance. For this reason they are not given in the table. The average error was, however, estimated to be 0-001 for the positional parameters and 5% for the compositional parameters. The scattering power of the two atoms of type A was given by the least-squares refinement as only 0-8 times that of aluminum (the fraction... 

The quantitative comparison of measured. To values and 5 values calculated for models A for mesitylene and hexamethylbenzene is given in Tables XII and XIII. With omission of the innermost maximum and minimum, the unreliable third minimum, and the very weak fifth maximum for each substance, the average values sa/s0 = 1.000 and 1.002, respectively, are found. These correspond to the interatomic distances C — = 1.54 0.01 A. and Car-Car = 1.39 A. in both substances, the 0 —Qa distance being equal to the single-bond distance in aliphatic compounds and the Car—distance to that in benzene to within the probable error of the determination. 

Student5 (1908) The probable error of a mean. Biometrika 6 1 Thompson M (1995) Uncertainty in an uncertain world. Analyst 120 117N... 

One then proceeds to calculate a value of the rate constant for each pair of points separated by a time A [i.e., a value is calculated from the points corresponding to (0 and A), and (A + tj, 2ti and (A + 2tfj, etc.]. The arithmetic mean of these values is a good representative value of the rate constant. In this technique each data point is used once and only once, and the probable errors of the quantities that are averaged are all of the same order of magnitude. For the first-order case it is apparent from equation 3.1.8 that the average value of the rate constant is given by... 

We keep learning more about the history of noise calculations. It seems that the topic of the noise of a spectrum in the constant-detector-noise case was addressed more than 50 years ago [1], Not only that, but it was done while taking into account the noise of the reference readings. The calculation of the optimum absorbance value was performed using several different criteria for optimum . One of these criteria, which Cole called the Probable Error Method, gives the same results that we obtained for the optimum transmittance value of 32.99%T [2], Cole s approach, however, had several limitations. The main one, from our point of view, is the fact that he directed his equations to represent the absorbance noise as soon as possible in his derivation. Thus his derivation, as well as virtually all the ones since then, bypassed consideration of the behavior of noise of transmittance spectra. This, coupled with the fact that the only place we have found that presented an expression for transmittance noise had a typographical error as we reported in our previous column [3], means that as far as we know, the correct expression for the behavior of transmittance noise has still never been previously reported in the literature. On the other hand, we do have to draw back a bit and admit that the correct expression for the optimum transmittance has been reported. 

The volume fraction of each phase was taken from the fractional area in the transmission electron micrographs. Combined with the values shown in Table 1, the compositions within each phase were calculated and are shown in Table 2. Overall, the results suggest variably 0-20% actual molecular mixing. Noting the probable errors in estimating the experimental Tg s, the and W2 values are probably correct to within +0.05. Thus mixing plays an important role in interpenetration and influences the reinforcement within each phase. 

Probable errors are 0.2 Hz or less for all values reported here except the 13C-F couplings for which the probable errors are 0.4 Hz. b) Values reported by Frankiss, Ref. 63, for 95 % solutions in cyclohexane. 

In the following we review the on-line mono-exponential evaluation procedure we have chosen for on-line used on Stelar instruments. We believe that the qualitative features of this algorithm, such as the method used to estimate the probable error of the relaxation rate, represent a good example of how data should be handled in more complex cases also. 

It is possible not only to identify the components of a mixture, but also to estimate the proportions of the different components from the relative intensities of the patterns. No simple mathematical relationship between thq proportions of the components and the relative intensities of particular diffraction arcs can be given, and therefore the method of analysis must be empirical when the constituents have been identified, powder photographs of mixtures containing known proportions of the constituents must be taken, and that of the unknown mixture compared with them. Estimates of proportions to within 5 per cent, can be made by visual comparison, but the probable error can be reduced to the order of 1 per cent, by measuring the intensities of selected diffraction arcs by means of a micro-photometer the relation between known composition and the relative photographic densities of particular arcs is found empirically, and the composition of the unknown mixture... 

An estimate of the probable errors in the correction factors and cutoff values follows. From Equation 3 one sees that the fractional errors in both are of the same order of magnitude as the fractional error in the velocity, Av/v, averaged over the region of motion. There are three main contributions to this error. One comes from the approximation to the Davies equations (8 and 10). The average fractional error is of the order of Av/v —5%, the minus sign occurring since Equations 9 and 10 underestimate the true values of Re and v. The other error contributions come from the approximations for air density and viscosity. One sees from Equations 7-9 that the first-order term in v is independent of p and has a 1/rj dependence. The second-order term is directly proportional to P. Since this term contributes a maximum of 30% to the velocity and the maximum error in p is 8%, this contribution to Av/v should be... 

Moreover, covalent bond distances are often related to one another in an additive manner the bond distance A- -B is equal to the arithmetic mean of the distances A—A and B —B. For example, the C—C distance in diamond is 1.542 A and the Cl—Cl distance in Clj is 1.988 A. The arithmetic mean of these, 1.765 A, is identical with the Cl—Cl distance 1.766 0.003 A found in carbon tetrachloride to within the the probable error of the experimental value.5 In consequence, it becomes possible to assign to the elements covalent radii such that the sum of two radii is approximately equal to the equilibrium inter-nuclear distance for the two corresponding atoms connected by a single covalent bond. 

It is a useful indicator of the probable error of a measurement. Standard deviation is often transformed to standard deviation of the mean or standard error. This is defined by Equation 1.4, where n is the number of measurements. 

The cntical properties of methyl acetate and ethyl acetate have been reported in the literature.Estimation methods were used to calculate the critical properties or butyl and vinyl acetate.1 The probable error is d C on the critical temperature. 15-25 psi on the critical pressure, and U.002 grams-milliliter on (he critical density. 

Although the crystalline structures of many polymers have been determined, it is unusual for these to include any estimate of the probable error in the various parameters. But unless these are known, deductions which are sometimes made from the values of the parameters (e.g. the assignment of infra-red absorption band frequencies) can be misleading. 

Statistical estimation uses sample data to obtain the best possible estimate of population parameters. The p value of the Binomial distribution, the p value in Poison s distribution, or the p and a values in the normal distribution are called parameters. Accordingly, to stress it once again, the part of mathematical statistics dealing with parameter distribution estimate of the probabilities of population, based on sample statistics, is called estimation theory. In addition, estimation furnishes a quantitative measure of the probable error involved in the estimate. As a result, the engineer not only has made the best use of this data, but he has a numerical estimate of the accuracy of these results. 

DallaValle and Goldman (1939) that for representative samples having the same size-distribution the probable error of a9 may exceed 10 percent. 

The probable error of a single determination was found to be = =3.1. Eq (14-1) shows that the finest particles and the organic matter on a weight-for-weight basis overbalance all the other constituents combined. 

The results of different workers in early measurements of the radiative lifetime of N02 showed considerable disparity ranging from 40 to 90 pis [82, 85—89]. From considerations of the geometries used in the different studies, Sackett and Yardley [83] were able to estimate the probable error in these measurements caused by using too small a fluorescence cell. The results of their calculations showed that the range of lifetimes measured in various studies were consistent with a true lifetime of 65 pis or longer when geometrical effects were taken into account. 

Details of the way in which the calculations are performed may be found in a textbook on statistics. The method resembles the well-known method of least squares, but it has the characteristic feature that a system of rationally determined weight factors is introduced. If the calculations are carried through to completion, the method gives not only the most probable values of the constants themselves but also the probable errors of the different constants. 

Error in Calculated Rate Constants. The slope of the straight-line section in the reaction rate curves should be equal to the calculated values of k2. To obtain an idea of the probable error in the calculated values for k2 (Table I), they can be compared with the slopes of these lines. The slopes (and standard errors) were determined by least-squares regression analysis. Table II lists the values for k> from Table I, the least-squares slope, and the standard error of the least-squares value. In most cases, k2 agrees with the least-squares slope to within the standard error (4% or less). 

Its square root is the estimated standard deviation, or sample standard deviation, a, which is universally quoted as a good estimate of the probable error e in the measurement ... 

The probable errors of the refined chemical-shifts and coupling-constants are printed in the output of the LAOCOON programs, but these errors are now generally believed246 to be unreasonably small, and they have sometimes been multiplied by a factor of two248 or five247 when reported. 

Fig. 7.27. Dependence of the isotopic exchange rate constants (k) and the stability constants (K) of the complexes R(EDTA) on the atomic number of the rare earths. The vertical bars shown in the figure represent the probable error in the k values. The dotted line describing the behavior of the elements lanthanum, neodymium and europium is from the comparison of exchange reactivity attempted by Fomin.

Table 3 and is the basis for the estimated fixed-capital cost tabulation given in Table 4. The probable error in this method of estimating the fixed-capital investment is as much as + 30 percent.

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