Appendix to Chapter II, § I (to Chapter II, § 3) The Concept of an Average

The subject of averages or means is so important—both theoretically and practically—and so little upon it is readily available for economic readers that a short statement of its fundamental principles may be fitly inserted here.

There are numerous kinds of averages or means. Among them are the arithmetical, geometrical, and harmonical; and of each of these there are many different varieties. The simple arithmetical mean of a specific series of terms is found by adding the terms together and dividing by their number. Thus, suppose it is desired to find a mean of 2 and 8. It is evidently

This is, as a matter of fact, the mean most commonly employed.

The simple geometrical mean is obtained by multiplying all the terms together and extracting that root of the product which corresponds to the number of terms. Thus, the geometrical mean of 2 and 8 is or 4.

The simple harmonical mean of any number of terms is the reciprocal of the arithmetical average of their reciprocals. For 2 and 8 it is

or 3 1/5.

The weighted arithmetical mean is a modification of the simple arithmetical mean. Suppose it is desired to find the mean height of two groups of trees, one tall, the other short. The tall group is 8 yards high, the short, 2. The simple arithmetical mean as we have seen is 5. But this mean treats both groups as of equal importance. Let us suppose that there are twenty of the two-yard trees and ten of the eight-yard trees, and let us seek a mean of the two heights, such as will give equal importance to each tree. This will give the short group of twenty trees twice the importance of the tall group of ten trees. We shall be giving equal importance to each tree if we take the simple arithmetical mean of the thirty trees. But this simple mean of thirty trees will be a weighted mean of the two groups of trees. It is to be found by adding their heights together (twenty heights of two yards plus ten of eight) and dividing by the number of trees (20 + 10). That is, the mean height is

and this (considered as an average of the two groups instead of that of the thirty trees) is said to be the weighted arithmetical mean of 2 and 8, the 2 being weighted twenty times, and the 8, ten times. The weighted mean of the two groups means the simple mean of the thirty trees. In other words when we "weight" the various terms averaged, we no longer count these terms once each, but we count one term as though it were (say) twenty, and another as though it were (say) ten and the number of times we count a term is its "weight." In the same way we may define the weighted geometrical and weighted harmonical means. Taking the same example of the trees, we find the results to be respectively or 3.175 and or 2 2/3.

The same results would have been obtained in each case if, instead of the weights 20 and 10, we had taken, as weights, 2 and 1.

Since there are so many different kinds of means, the question arises, What is the meaning of an average or mean in general? We answer: Any mean of a series of terms must be obtainable from them by a mathematical rule such that, when applied to a series of identical terms, it will make their mean identical with each of them. Any rule of averaging is admissible which is consistent with this condition (that the average of identical terms must be identical with each). We know that the simple arithmetical mean A, of a, b, and c is

It is easy to see that this formula meets the required test. Substituting A for each of the three magnitudes a, b, and c in (a + b + c)/3, we obtain

which is evidently equal to A; thus the test is satisfied.

Again, let G be the geometrical mean of a, b, and c; so that G = This formula also conforms to the definition of a mean because G =.

Similarly, the harmonical average (which we may call H) of a, b, and c is

This also conforms, because

For a weighted arithmetical average Aw of a, b, c, the weights being a, b, g, we have the formula

which conforms to our test, since evidently

By applying this general rule, we can make at will innumerable kinds of averages. It is only necessary to write any formula twice, once using the terms to be averaged and once using, instead, the required average, and then equate the two. Thus, let us take the complicated formula

This may be employed to obtain a new species of average (x) of a, b, and c, simply by equating it with the similar form

That x as determined by this equation will conform to our definition of an average is evident, since substituting x for a, b, and c, the equation becomes a truism, showing that the proposed new average of the identical terms x is x.

A special case of the definition, requiring particular mention, is that in which two or more means (not necessarily of the same kind) are related to one another. In order that A should be a mean of a1, a2, a3,...when we know that B is a mean of b1, b2, b3,...it is only necessary to have a determining formula such that if a1 = a2 = a3...and at the same time b1 = b2 = b3...(each of which by hypothesis must be equal to B), then A shall also be equal to each of the magnitudes a1, a2, a3, etc. Many examples of pairs of means like A and B will be given in Chapter X (on the construction of index numbers). The following is a simple example:—

Let nAB = a1b1 + a2b2 + a3b3 +...and let B be the arithmetical mean =

(n being the number of terms). Then A is a (new) sort of mean of a1, a2, a3; for, substituting A for a1, a2, a3,... and B for b1, b2, b3, in the equation nAB = a1b1 +..., the equation is satisfied.