The Most Probable Value

Given that the assumption of normally distributed data (see Section 1.2.1) is valid, several useful and uncomplicated methods are available for finding the most probable value and its confidence interval, and for comparing such results. 

When only a few measurements of a given property are available, and especially if an asymmetry is involved, the median is often more appropriate than the mean. The median, x , is defined as the value that bisects the set of n ordered observations, that is, 

The most useful characteristic of the median is the small influence exerted on it by extreme values, that is, its robust nature. The median can thus serve as a check on the calculated mean. 

The mean, Xmean, can be shown to be the best estimate of the true value /x it is calculated as the arithmetic mean of n observations  

Notice that by the inclusion of Tg, the mean is much more strongly influenced than the median. The value of such comparisons lies in the automatic processing of large numbers of small data sets, in order to pick out the suspicious ones for manual inspection. (See also the next Section.) 

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The microcanonical ensemble is a certain model for the repetition of experiments in every repetition, the system has exactly the same energy, Wand F but otherwise there is no experimental control over its microstate. Because the microcanonical ensemble distribution depends only on the total energy, which is a constant of motion, it is time independent and mean values calculated with it are also time independent. This is as it should be for an equilibrium system. Besides the ensemble average value (il), another coimnonly used average is the most probable value, which is the value of tS(p, q) that is possessed by the largest number of systems in the ensemble. The ensemble average and the most probable value are nearly equal if the mean square fluctuation is small, i.e. if... 

Millikan R A A new modification of the cloud method of determining the elementary electrical charge and the most probable value of that charge Phil. Mag. 19 209-28... 

The most probable value of the speed v p can be obtained by differentiation of the distribution function and setting dG(v)/dv = 0 (Kauzmann, 1966 Atkins 1990) to obtain... 

The standardized variable (the z statistic) requires only the probability level to be specified. It measures the deviation from the population mean in units of standard deviation. Y is 0.399 for the most probable value, /x. In the absence of any other information, the normal distribution is assumed to apply whenever repetitive measurements are made on a sample, or a similar measurement is made on different samples. 

Earlier we introduced the confidence interval as a way to report the most probable value for a population s mean, p, when the population s standard deviation, O, is known. Since is an unbiased estimator of O, it should be possible to construct confidence intervals for samples by replacing O in equations 4.10 and 4.11 with s. Two complications arise, however. The first is that we cannot define for a single member of a population. Consequently, equation 4.10 cannot be extended to situations in which is used as an estimator of O. In other words, when O is unknown, we cannot construct a confidence interval for p, by sampling only a single member of the population. 

About 1902, J. W. Gibbs (1839-1903) introduced statistical mechanics with which he demonstrated how average values of the properties of a system could be predicted from an analysis of the most probable values of these properties found from a large number of identical systems (called an ensemble). Again, in the statistical mechanical interpretation of thermodynamics, the key parameter is identified with a temperature, which can be directly linked to the thermodynamic temperature, with the temperature of Maxwell s distribution, and with the perfect gas law. 

At present, the most probable value of the reaction constant for the methoxy-dechlorination of 4-chloroquinolines is obtained from Eq. 

The relative error is the absolute error divided by the true value it is usually expressed in terms of percentage or in parts per thousand. The true or absolute value of a quantity cannot be established experimentally, so that the observed result must be compared with the most probable value. With pure substances the quantity will ultimately depend upon the relative atomic mass of the constituent elements. Determinations of the relative atomic mass have been made with the utmost care, and the accuracy obtained usually far exceeds that attained in ordinary quantitative analysis the analyst must accordingly accept their reliability. With natural or industrial products, we must accept provisionally the results obtained by analysts of repute using carefully tested methods. If several analysts determine the same constituent in the same sample by different methods, the most probable value, which is usually the average, can be deduced from their results. In both cases, the establishment of the most probable value involves the application of statistical methods and the concept of precision. 

Figure 14 shows the displacement of the distribution function towards high / , i.e. the uncoiling of molecules under the influence of stretching for polyethylene (A = 3 x 10-9 m, N = 100 and T = 420 K). This displacement will be characterized by the position of the maximum of the distribution curve, the most probable value of / , i.e. j3m, as a function of x (Fig. 15). Figure 15 also shows the values of stresses a that should be applied to the melt to attain the corresponding values of x (o = xkT/SL, where S is the transverse cross-section of the molecule). 

The most probable values of the parameters are those for which R is at a minimum, or... 

Figure 17. Plot of the potential of zero charge, EaJi, vs. the electron work function, . The point is the most probable value. Data for E0wq from Ref. 140 for Au (111) from Ref. 25 for Pt (111) from Ref. 867.

For standard deviations, an analogous confidence interval CI(.9jr) can be derived via the F-test. In contrast to Cl(Xmean), ClCij ) is not symmetrical around the most probable value because by definition can only be positive. The concept is as follows an upper limit, on is sought that has the quality of a very precise measurement, that is, its uncertainty must be very small and therefore its number of degrees of freedom / must be very large. The same logic applies to the lower limit. s/ ... 

Figure 6. Shown is the correlation between the liquid s fragility and the exponent p of the stretched exponential relaxations, as predicted by the RFOT theory, superimposed on the measured values in many liquids taken from the compilation of Bohmer et al. [50]. The dashed line assumed a simple gaussian distribution with the width mentioned in the text. The solid line takes into account the existence of the highest barrier by replacing the barrier distribution to the right of the most probable value by a narrow peak of the same area the peak is located at that most probable value. Taken from Ref. [45] with permission.

According to Eq. (5) the most probable value of x occurs at x = 0. As X increases in magnitude, W(x) decreases monotonically from its maximum at x = 0, the decrease being the more rapid the smaller n. Eq. (5) obviously becomes invalid at values of x approaching nl W x) remains finite, though very small, according to Eq. (5) even for x>nlj where it necessarily is zero. 

The most probable value r p of r for the Is state is found by setting the derivative of D Q(r) equal to zero... 

The ionic potentials can be experimentally determined either with the use of galvanic cells containing interfaces of the type in Scheme 7 or electroanalytically, using for instance, polarography, voltammetry, or chronopotentiometry. The values of and Aj f, obtained with the use of electrochemical methods for the water-1,2-dichloroethane, water-dichloromethane, water-acetophenone, water-methyl-isobutyl ketone, o-nitrotol-uene, and chloroform systems, and recently for 2-heptanone and 2-octanone [43] systems, have been published. These data are listed in many papers [1-10,14,37]. The most probable values for a few ions in water-nitrobenzene and water-1,2-dichloroethane systems are presented in Table 1. 

It is reasonable to assume that the most probable values of the parameters have normal distributions with means equal to the values that were obtained from well test and core data analyses. These are the prior estimates. Each one of these most probable parameter values (kBj, j=l,...,p) also has a corresponding standard deviation parameter estimate. As already discussed in Chapter 8 (Section 8.5) using maximum likelihood arguments the prior information is introduced by augmenting the LS objective function to include... 

Step 3. Provide estimates of the most probable values of the parameters and their standard deviation (k, okj, j=l,...,p). 

The distribution of distance h is characterized either by the most probable value /imax, defined by the condition dw(/i)/d/ = 0, or by the mean square of distance h, defined in the usual manner, h2 = J/i2w(/i) dh/ w(h) dh. The ratio (hmax)2/h2 gives the width of the distribution. The closer this value is to unity, the narrower the distribution. This ratio is equal to for the simplest model in statistics, the random walk . 

For a given value of a the probability is maximum when the sum in the exponent is minimum. Thus, the minimization of xf becomes the criterion for the most probable value obtainable from n equally reliable measurements. This result is the basis for the various curve-fitting procedures that are commonly used in the analysis of experimental data. 

In the first case, the minimum values for the economic yardsticks were evaluated assuming that a conservative 7300 bbl (= 8150 bbl -850 bbl 1160 m3 = 1295 m3 - 135 m3) of incremental oil had been produced by the end of the project. Maximum values for the economic data were calculated by assuming that 9000 bbl (= 8150 bbl + 850 bbl 1430 m3 = 1295 m3 + 135 m3) of incremental oil were produced by only 61.6 % of the TFSA which had been injected into Well TU-120 this assumption is based on the results of the tracer study which showed that as much as 38.4 % of the injected fluids flowed out of the pilot pattern. In the final case, the most probable values for the economic yardsticks were calculated assuming the 8150 bbl (1295 m3> of incremental oil were produced by 90 % of the TFSA. 

On the basis of Eq. (19), the plot of k0bS/[edta4 ] as a function of the hydrogen ion concentration on a logarithmic scale gave a straight line of slope — 2.0. The slope shows the most probable value for n. Thus, at pH 2 to 4 the substitution reaction proceeds as... 

The maximum occurs at the most probable value of the energy when dP/dH — 0, for the distribution function P, which has the form... 

We separate in a natural fashion the system in two parts, namely the hard-sphere fluid and the tube itself. Let F, Ft, and Fs denote the free energies of the system, the tube, and the spheres, respectively (all of which depend on n), such that we can write F = Ft+ Fs- The most probable value for n is obtained by... 

What is the most probable value for the polydispersity index for (a) a monodisperse polymer and (b) a polydisperse polymer synthesized by a condensation technique ... 

After completion of the MEM enhancement, it becomes possible to evaluate the reflections missing from the summation. In a Fourier summation, the amplitudes of the unobserved reflections are assumed to be equal to zero, while the MEM technique provides the most probable values. 

Figure 7 G(Si) value as a function of the carbon atom numbers in the molecules. When more than one measured value was published, we tried to select the most probable value. Alkanes (1) propane, (2) w-butane, (3) w-pentane, (4) cyclopentane, (5) w-hexane, (6) cyclohexane, (7) w-heptane, (8) cycloheptane, (9) methylcyclohexane, (10) w-octane, (11) cyclooctane, (12) isooctane, (13) w-decane, (14) cyclodecane, (15) cw-decalin, (16) trawx-decalin, (17) w-dodecane, (18) dicyclohexyl, (19) n-hexadecane. (From Refs. 18, 29, 65, 92, 148, and 155.)...

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