This November 2017, I found a third average, the arithmetic mean, for
error-reducing purposes, in my system of Binomial STV (BTV). (This could be
noted as BTV3.)

A previous discussion proposed an arithmetic mean of redistribution counts,
with respect to all the candidates having their votes redistributed This
averaging of all possible redistribution results, would be more stable than
redistribution only of the most prefered candidates votes, in an election
count, or the most un-prefered candidates votes in an exclusion count.

The geometric mean was used to average candidates keep values from both election and exclusion counts. The harmonic mean neutralised democratic objections to the maximum and minimum proportional counts by the Hare and Droop quotas.

When young, I wrote an article that the single transferable vote uniquely follows the four main scales of measurement, widely accepted in the sciences. (It has a copyright UNESCO, 1981. The French translation is still extant, on my Democracy Science website, and appended in my e-book: Peace-making Power-sharing.)

It occurred to me that the three averages corresponded to these recognised scales. The harmonic series, which the harmonic mean averages, is a ratio scale, which is the fourth scale of measurement.

The third scale, the interval scale, is also a proportional scale, but lacks a true zero, unlike the ratio scale. It so happens that the geometric mean cannot work with a zero, in a given series, because that would reduce the geometric mean to zero, a non-result.

The first, and simplest of the four scales, the nominal or classificatory scale is the most obvious candidate to correspond to the simplest average, the arithmetic mean, of the distribution of a class of objects.

The second scale of measurement is the ordinal scale or ranking. If this also has a corresponding average, there should be 4 averages for 4 scales. But what is the missing average? (To date, 21 November 2017, Wikipedia article, Averages, groups together just three basic averages: arithmetic, geometric, and harmonic means.)

So, I played blind mans buff, fumbling for a fourth basic average in the group, and wondering whether there really was one. Anyway, after a bit of thinking on the character of the geometric mean, I stumbled upon this apparently unknown average, making-up the quartet. (It seemed a long time, over an uncertain effort, but the discovery of a fourth average came only a few days after I realised that the arithmetic mean was needed as a third average for BTV.)

I already had a clue, because the members of the harmonic series are the inverse of the members of the arithmetic series. Therefore, it made sense that the members of the unknown series, to be averaged, should be the inverse of the members of the geometric series.

This indeed turned out to be the case.

The geometric mean is a power arithmetic mean. So, the unknown mean is a power harmonic mean.

This is explained with an example.

An arithmetic series increases by a constant amount, for instance, two, in the following distribution: 2, 4, 6, 8 ,10, etc.

The next distribution is a geometric series, because it does not merely add a constant number, to every successive member. The next series does not add by two but multiplies by two. The series doubles, in size with every successive member:

2, 4, 8, 16, 32.

The average member of this geometric series can be found by the geometric mean. This example is a completely regular series, so it can be calculated most simply by multiplying the two end members of the range, 2 by 32 equals 64, and taking the square root, for a geometric mean of 8.

This calculation can also be done, by using indices, where 2 = 2^1 (which means 2 to the power of 1) and 32 = 2^5 (which means 2 to the power of 5).

Then, the geometric mean equals: {(2^1)(2^5)}^1/2 = 2^(1+5)/2 = 2^3 = 8, which is the result, for the geometric mean, which we had before.

Notice that the calculation, in the power index, is the same as the calculation for the arithmetic mean.

The calculation for the unknown average, or unknown mean, is therefore the calculation, in the power index, for the harmonic mean.

Before that, let us do the calculation for the arithmetic mean and the harmonic mean. Taking the simple case again, of a perfectly regular series, let the minimum be N = 2, and the maximum be X = 32.

The arithmetic mean is: (N+X)/2.

The division is by two, because N & X are two items. Three items would be divided by three, and so on.

In the above example, AM = (2+32)/2 = 17.

This would be the average of a uniform increase from two to thirty-two.

The harmonic mean is: 1/{(1/N)+(1/X)}/2.

In the above example: HM = 1/{(1/2)+(1/32)}/2 =

1/{(17/32)}/2 = 64/17 ~ 3.76.

Now for the unknown average, which is the power harmonic mean:

2^[1/{(1/n)+(1/x)}/2].

For the above example, N = 2^n. Here, 2 = 2^1. And X = 2^x. Here, 32 = 2^5.

The power harmonic mean is:

2^[1/{(1/1)+(1/5)}/2] = 2^[1/(6/10)] = 2^(5/3) ~ 3.17.

(Note that two is not the only base, to the power, that could be used. It applies to the above series, which gives the first few sums of the binomial theorem, for ascending powers, assuming the two terms, in its factor, are both equal to 1.)

From the above example, the four averages are:

AM = 17; GM [power arithmetic mean] = 8; HM ~ 3.76; new average [power harmonic mean] ~ 3.17.

The three recognised averages AM, GM, HM are generally of descending magnitude. As to be expected, my innovation of the power harmonic mean has the smallest magnitude of the four averages.

What to call the power harmonic mean? The inverted terms of the geometric series as exponential series (whose sum is the exponent, e) is a factorial series: 0! 1! 2! 3! 4! 5! …

So, prime candidate, for naming the power harmonic mean, would be "the factorial mean."

The ordinary number system, 1,2,3,4,5,..., can be expressed in ratios of factorial numbers: 1!/0!, 2!/1!, 3!/2!, 4!/3!, 5!/4!... The troublesome-looking factorial zero, 0!, by convention is valued at one.

The term, factorial mean, might not be entirely satisfactory, given that the series for the exponent to the power of one, generalises to the exponent to the power of x: e^x. Taking into account negative and inverse modifications, this series gives the successive orders of differentiation or anti-differentiation in calculus: a differentiations series (or "derivatives" series).

I once translated traditional differentiation into ultimate averagings either of the arithmetic mean or the harmonic mean. Differential calculus could be given a statistical basis. It also followed that there should be a geometric mean differentiation, and I tried to secure a landing on that unknown continent of knowledge.

Now that I have also discovered a factorial mean, this opens-up the prospect of a factorial mean differentiation. Presumably, this would be to find, or to “differentiate,” the ultimate factorial mean. Perhaps this is the ultimate power harmonic mean of the series of ultimate harmonic means.

At any rate, the successive differentiations series is an ordered power series, which is to say that it is on an ordinal scale of powers. To some extent, this relates the power harmonic mean to the ordinal scale, so that all 4 basic averages correspond to the 4 main scales of measurement.

This is all very tentative, I know. But I have found that putting mathematics on a statistical basis works well.

Finally, I might say a few words about how the factorial mean might be
applied, as the fourth average to be used on Binomial STV (BTV). There are
orders of BTV, which are bound to get slightly different keep values for the
candidates, tho not much alter the results. Based on successive expansions of
the binomial theorem, there is first, second, third etc order BTV, whose
respective keep value results might be averaged by the factorial mean.

This practical possibility strengthens the supposition that this new average is
related to the ordinal scale, so that all four scales of measurement have a
related basic average.

I thought that higher orders of BTV would be ever more refined. However, possible application of a factorial mean suggests rather that first approximations remain important.

Just take the first and second order BTV counts, as all that’s needed to give an idea of the principle. Let K be a first order keep value, where K = 2^a, and let k be a second order keep value, where k = 2^b.

The factorial mean keep value might be something like this:

FM = 2^[1/{(1/a)+(1/b)}/2].

This concludes an explanation of my new average, a so-called (by me) factorial mean and its possible first application, by way of the averaged keep values of successive orders of Binomial STV.

Application of all four averages to Binomial STV might be given the notation: BTV4.

This study illustrates that a particular practical problem can discover a result of general application. The factorial mean represents a fourth distinct data distribution pattern from the three traditionally recognised in statistics.

*Richard Lung*

*21/11/2017*