Mean harmonic

Thus the average cost per share for John is the arithmetic mean of pi, po,. . . , pn, whereas that for Mary is the harmonic mean of these n numbers. Since the harmonic mean is less than or equal to the arithmetic mean for any set of positive numbers and the two means are equal only i pi=po = =pn, we conclude that the average cost per share for Mary is less than that for John if two of the prices Pi are distinct. One can also give a proof based on the Gaiichy-Schwarz inequality, To this end, define the vectors... 

The use of the harmonic mean often leads to better predictions of interfacial tensions between polymers and better contact angles between liquids and polymer solids, but the criterion for maximization of the work of adhesion is the same as... 

Harmonic mean approximation Dispersive and polar components of solid surface energy are found by solving yiv(l +COS0) = y"y, 1 ys"vH ys" + yiv Ys + yiv Similar to geometric mean approximation. [85]... 

Geometric mean method Harmonic mean method ... 

Krug et al. have shown that a plot of AG against AH [see Eq. (7-75)] is not subject to the error correlation problem of the AH vs. AS plot provided AG is evaluated at the harmonic mean temperature of the experimental range. (The harmonic mean is the reciprocal of the mean of the reciprocals, i.e., = <1/... 

Thus, y is the slope of the plot of A// against AG at the harmonic mean temperature, and from y the isokinetic temperature P is calculated. Tomlinson has shown many examples of this type of analysis. 

While many data are suggestive of chain length dependence, the data are not usually suitable for or have not been tested with respect to model discrimination. Values of ,u have been determined for a variety of small monomeric radicals to be ca I09 M s 1.4 Taking kt0 as Jk,lj and a as 1.0 in the geometric expression yields values of ,iJ as shown in Figure 5.4a.49 Use of the Smoluchowski mean or the harmonic mean approximation prediets a shallower dependence of k 1 on the chain length (Figure 5.4b). All expressions yield the same dependence for j=i. 

Figure 5.4 Chain length dependence of A,I J predicted by (a) the geometric mean (eq, 25) or (b) the harmonic mean approximation (eq. 22) or the Smoluchowski mean (eq. 23) with a=1.0 and to=109 i and j are the lengths of the reacting chains.

Exactly, the mean temperature should be defined by 2/T = 1/Tj + 1/T in the case of two temperatures, but the difference from the harmonic mean is small. 

It may be mentioned here that the mode which represents the most commonly occurring size in a given distribution is not of much use in mineral processing since it does not describe fully the characteristics of a group of particles. The arithmetic mean diameter suffers from the same limitation except when the distribution is a normal one. The harmonic mean diameter is related to the specific surface area. It is, therefore, useful in such mineral processing operations where surface area is an important parameter. 

The unbound PCP concentration was calculated by multiplying the total unchanged PCP concentration In serum by the unbound fraction in serum at each time point. The harmonic means for the terminal elimination half-life for unchanged total PCP and unbound PCP were virtually the same (3.5 and 3.3 hours, respectively). The arrows indicate the time of Fab administration. 

Polyethylene beads of relatively narrow size distribution with a harmonic mean diameter of 2800 mm and a particle density of 910 kg/m3 were used as the bed material. A static bed height of 1.4 m was employed. 

This is a harmonic mean because it is really the mean free path that is relevant. At high temperatures (> 105 K), the main sources of opacity and approximate formulae for them (the first two originally due to H. Kramers) are ... 

The goal of harmonization is to bring the policies, standards, monograph specifications, analytical methods, and acceptance criteria of pharmacopoeias into agreement. Such imity may, however, not always be achievable. Where imity cannot be achieved, harmonization means agreement based upon objective comparability and a clear statement of any differences. The goal, therefore, is harmony, not imison. 

Besides the above-mentioned measures for the location there are several other possible parameters. The geometric mean is the arithmetic mean of the logarithms of the data (or the n root of the product of all data). The harmonic mean is the reciprocal of the mean of the reciprocals of all single values. 

A final measure of location is the harmonic mean. This is rarely used explicitly although again it may be implicitly used. For example, when considering the analysis of heart rate data, many statisticians would recommend that the reciprocal of the heart rate be analysed rather than the heart rate itself. Again, this has to do with an attempt to make the distribution of the transformed variable be more s)unmetric. The... 

Mean any one of a number of estimates of the centre of a set of observations (e.g. arithmetic mean, geometric mean, harmonic mean). 

Harmonic Mean - the reciprocal of the average of the reciprocals of the observed values. 

Results of this type have proved of value in experimental investigations involving surface-volume relations. Of particular interest is the fact that specific surface is inversely proportional to the first moment of the surface-weighted size distribution, and this moment, in turn, is equal to the harmonic mean of the volume-weighted size distribution. 

Table I lists some of the basic mathematical expressions of importance in droplet statistics. The expressions are given in terms of an arbitrary ptb-weighted size distribution. The specific forms are obtained for various integral values of p. For example, the substitution of p = 2 into the equations of Table I yields the cumulative distribution, arithmetic mean, variance, geometric mean, and harmonic mean of the surface-weighted size distribution. Analogous expressions valid for frequencies or mass distributions are obtained by setting p equal to 0 or 3, respectively.

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