I do not know what attempts may have been made to answer this question in its generalized form, but I believe that no method has hitherto been suggested, or at any rate used, for measuring the sub-homogeneity of mixed populations, or of the social mind. I venture therefore to suggest that the problem admits of solution and to offer a formula for finding a numerical measure or coefficient of heterogeneity, or, as I think we may more accurately say, of sub-homogeneity.

When we plot the surface of magnitude (the polygon of distribution as defined by Galton), we measure magnitudes, or we determine successive positions, by vertical distances from a baseline. Since equal magnitudes must be measured, and positions that indicate equal degrees of relationship must be determined, by equal vertical distances, the plotted points indicating such equal measures must lie in a horizontal straight line. Therefore, the horizontal straight line is the graphic expression of equality or of homogeneity. Departures from equality, or degrees of sub-homogeneity, are best plotted as minus or negative magnitudes, graphically expressed by points placed at proper distances below a horizontal line, which itself expresses the ideal homogeneity from which departures are supposed to be made.

Let us then suppose that we have to describe a group of human beings, twelve in number, of whom three are unobjectionable in conduct; three are vicious persons, or minor misdemeanants; two are petty criminals or major misdemeanants; two are felons, not capital; and two are felons, capital. The group as a whole is sub-homogeneous. Plotting the data, we get the arrangement shown in Fig. IX.

Imagine now, that by expending one unit of some kind of effort, we could lift any one vicious person up to the level of the men whose conduct is unobjectionable; that by expending two units of the same kind of effort we could lift any one major misdemeanant to the same standard level; that by expending three units of the same kind of effort, we could lift any one of the minor felons, and by expending four units of the same kind of effort we could lift any one of the major

felons, to our standard level. Then, by expending 1 unit 3+2 units x2+3 units x2+4 units × 2, or 21 units in all, or 1.75 units per capita, for the whole group of twelve persons, we should convert the entire sub-homogeneous group of twelve persons into a group perfectly homogeneous in respect of a standardized conduct.

2L

3

་

4

Fig. 9. Measurement of Sub-Homogeneity.

Students of the physical sciences who are accustomed to measure physical phenomena of every description by the number of units of effort, or of energy, necessary to transform them from one state into another, will assent to the proposition that if we could thus actually transform any heterogeneous group of human beings into a homogeneous group, the number of units of effort necessarily expended in the process could be taken as an accurate measure of the total sub-homogeneity of the original group, and that the total number of units of effort so expended, divided by the total number of individuals in the transformed group, could be taken as a measure of the per-capita degree of sub-homogeneity of the original group.

Is it legitimate to conceive of any heterogeneous group as ideally transformable by such a procedure, and then to assume that we may measure its sub-homogeneity by (1) multiplying each successive numerical mark on a marking-scale of resemblance positions by the "frequency" or number of individuals

assigned to that position, (2) obtaining the sum of the produets, and (3) dividing it by the whole number of individuals in the group or population?

It is, I think, an adequate and satisfactory answer to this question to observe that the conception and the assumption. are legitimate, if mankind is warranted in believing that by an expenditure of educational and reformatory effort, it can standardize knowledge and conduct, and can assimilate alien habits and ideals to prevailing or national types. If the validity of this pragmatic belief be conceded, there can be no objection to conceiving of an average and abstract unit of standardizing effort, practically unchanging throughout the same group or population, living under practically constant conditions.

If so much be granted, we may write the formula for measuring sub-homogeneity as follows:

Designate positions on the marking-scale by the numerals, 0, 1, 2, 3, 4,

n.

Designate frequencies by K., K1, K2, K3, K4,
Designate total individuals, or Population by P.
Designate per capita degree of sub-homogeneity by S.

Then :

S= K1 + 2 K2+3K, +

P

+NKn

Kn.

Applying this formula to the data presented in the tables of ethnic distribution by resemblance position, we obtain for 1890 the coefficient .01579 as our expression for the per-capita degree of sub-homogeneity, and for 1900 the coefficient .01552. Applying the formula to the data presented in the tables of religious distribution by resemblance positions, we obtain for 1890 the coefficient 1.833, and for 1906 the coefficient 1.953. These coefficients indicate that in ethnic composition the people of the United States are now becoming, contrary to the prevailing impression, slightly more homogeneous, but that in religious persuasion they are still becoming slightly more heterogeneous.

THE NECESSITY OF THE SUPERVISION OF WEIGHTS AND MEASURES.

BY FRITZ REICHMANN, State Superintendent of Weights and Measures, Albany, N. Y.

Statistical results and deductions there from are based upon the assumption of the uniformity of the units, the uniformity of the standards of measurement. When these deductions or results are of a very general character and no close degree of accuracy is intended this assumption may be safe. There are many phases of commercial activity in which there is little or no uniformity of standards and there exists a greater degree of heterogeneity than was formerly true in physical or chemical work where it was often difficult to interpret or repeat certain investigations because of the lack of knowledge of what the standards were.

In the ultimate analysis, practically every transaction of barter or trade is based upon some standard of quantity. How many cubic feet of gas? How many kilowatt hours of electricity? How many miner's inches of water? How many tons of coal? How many acres of land? How many bushels of certain farm produce? All of these questions must be answered in terms of quantity. It would seem, therefore, highly important to have definite, uniform, reliable, consistent standards by which transactions, wholesale or retail, are gauged. This is particularly true in a democratic form of government where there is still in the minds of many a confusion of liberty and license. There are many, even some in exalted position, who decry any form of necessary restraint. The framers of the constitution of the United States when defining the powers of Congress very properly provided for the establishment and enforcement of uniform standards by placing in the same sub-section and in the same sentence with its power to coin money the power to establish the use of standards of

weight and measure. This covers the subject completely, as every standard for whatever use is some form of measure. Congress has never availed itself of this power, except in a single instance, namely, the establishment of the Troy pound for the purpose of coinage, although it has legalized many suggestions that have been made.

Commercial electrical standards in the main have been unified and in a number of instances legalized. The various electrical associations of America and Europe have by national and international conferences brought about a more or less definite uniformity of standards, although there are still a great many very faulty instruments used for the measurement of these quantities. Unfortunately such instruments are often placed in the hands of those who have no knowledge of their manipulation and thus give unreliable results; yet, on the whole, in electrical fields the instrumental, or measurement side has approximately kept pace with production. In most of the cities there is a regulation of the electrical meters in use. Furthermore the production of electrical energy is in the hands of a limited number who are more or less closely checked unofficially by those who have a technical knowledge or interest in the subject of electricity.

Gas has been the subject of control by the states for a great many years and there is a large degree of uniformity in the standards by which gas is sold. There is, however, no systematic or rigorous inspection of gas meters by competent officials, although this would be highly desirable. Furthermore, the inspection of the gas meters is in many cases of secondary importance to the inspection of the heating and lighting qualities of the gas, which are the two factors which should be primarily considered.

The measurement of water has been and is today an example of the greatest lack of uniformity and lack of accuracy in methods of measurement. Statistics of water supply in cities are largely a matter of guess work except in those instances where reliable meters are used. The majority of cities still sell water by the so-called "flat rate," and the con