inebriates is 12.1 years, while among those who are at least 55 years of age it is 18.5 years, and this gives no support to such a view.

The importance of these partial correlation coefficients in this as in every branch of social investigation is so great that it seems desirable that some illustrations of their use should be given directly from the data. Table II in the Appendix gives the relationship between age and duration of alcoholism. If we deal with successive columns, we get the distribution of duration of alcoholism among groups of inebriates who are, within narrow limits, of the same ages, while if we consider successive rows, we get the distribution of age among inebriates whose alcoholism has lasted the same length of time. We can now go a step further and break up the numbers in each cell into the fit and the unfit. We can thus find from the columns the average duration of alcoholism of fit and unfit among inebriates who are of constant age*, while from the rows we get the average ages of inebriates with the same duration of alcoholism. Using all the rows and columns from which averages may safely be calculated, we get:

There is of course some variation within the three-year age groups, but this is small and for present purposes may be neglected.

We see then that when we take age constant, the differences between the average duration of alcoholism among fit and unfit are all small. It is essential to notice that every difference is small; if we sum up the results by means of an average of these differences, when we take constant ages, the average duration of alcoholism among the fit is 113 years, or about six weeks, longer than among the unfit. This corresponds to a partial correlation coefficient of +04. On the other hand, when we make duration of alcoholism constant, we find that the average age of the unfit is 6.042 years greater than that of the fit, corresponding to a partial correlation coefficient of -'44. We see then that the increase in the percentage of the unfit with every additional decade of drunkenness is merely an indirect effect of age.

The point is so important that it has been further illustrated graphically. In Fig. 5 we have the relationship between age and physical fitness. At 16 At 16 years of age less than 10% of the inebriates are unfit for hard work and the percentage rises steadily until at 70 over 95% are unfit for hard work. are unfit for hard work. In Fig. 6 the continuous line gives the relationship between duration of alcoholism and physical unfitness. Figs. 5 and 6 are drawn so that they may be directly compared, and it is at once seen that the relationship between physique and age is much closer than that between physique and duration of alcoholism. The dotted line in Fig. 6 shows how much of the latter relationship is due to the indirect effect of age. It was reached in the following way. I calculated in turn the average ages of the women in each duration of alcoholism group. From Fig. 5 I then obtained the percentages of women who are unfit at these ages. This gives the percentage of women who would be unfit independent of duration of alcoholism and these figures were then plotted on the diagram and joined by a dotted line. It is then clear that age is sufficient to account for all the observed relationship. It should be clearly understood that although averages and percentages are given in these illustrations of the use of partial correlation coefficients, as measures of degrees of relationship they are of no value apart from the correlation coefficients and can only be used after the latter have been determined.

V. The Relationship between Alcoholism and Mental Defect in the General Population.

Before we discuss the relationships between age, number of convictions, mental and physical condition and so forth, among the inebriates, it is desirable that we should consider, just as was done in the "Preliminary Study," the extent to which the inebriates are differentiated from the general population in respect of mental defect.

We must take in the first place a sample of the women of the general population of such a size that it will include, in the proper proportion, 865 convicted inebriates. This sample will then be made up of four classes. We shall have convicted inebriates who are of normal intelligence, convicted inebriates who are mentally defective, and women who are not convicted inebriates, normal and mentally defective. We have already seen that out of 865 inebriates, 311 are normal and 554 are mentally defective.

We do not know directly the numbers of normal and mentally defective women among those who are not inebriates. Let these be x and y, and we can form a fourfold table as follows:

It remains to determine the values of x and y. They are not known directly but can be obtained if we know (a) the proportion of mentally defective women in the general population and (b) the proportion of convicted inebriates in the general population. The amount of mental defect in the general population was considered very fully in our "Preliminary Study" on the basis of the statistics of mental defect among school children. It was considered safe to take 618% and 1.09%, the percentages of mentally defective children in Liverpool and London respectively, as limiting values. In further justification of the use of these figures it may be said that in the Report of the Royal Commission on the FeebleMinded, it is stated that, in England and Wales, "the total number of mentally defective persons, including certified lunatics, may be estimated to be 271,607 or 83% of the population." In the present investigation I have taken 5% and 1.5° as the extreme values, so that there can be no doubt that the actual percentage of mental defect in the general population really lies between these limits.

I

P

The determination of the ratio of convicted inebriates to the general population is more troublesome. Let this ratio =R. Then it may fairly be assumed that this ratio will be nearly stable and hence that the ratio of the increase in the number of convicted inebriates, less the deaths among the total convicted inebriates, in or out of Reformatories, to the increase in the population from which the inebriates are drawn, will also be equal to R. According to this sample of inebriates, their ages run from 17 to 75. The statistics refer to the period 1st Jan. 1907 to 31st Dec. 1909, so that we can use the year 1908 in estimating the increase of population. The increase in the population between the ages of 17 and 75 is equal to the number of woman aged 16 in 1907 less the number of women aged 75 in 1908, less the number of women aged 16-74 who died in 1908. We thus find that the increase in the population is approximately 202,847.

The increase in the number of convicted inebriates equals the number of new convictions (A,) less the number of deaths of convicted inebriates in or out of Reformatories. It was found that the average number of new convictions was 274.875. Let AP be the number of deaths in the general population between 17 and 75, λ,I the number of deaths among convicted inebriates. Then I=RP and λI:

=

λ

R. XP.

Let =K, the ratio of the death-rate among the convicted inebriates to that among

the general population at the same ages. Now AP, the number of deaths in the general population aged 17-75, =129,193.

Now if the proportion of inebriates in the general population is fairly constant, we shall have that, very nearly,

What value are we to assign to K, the ratio of the death-rate among inebriates to the death-rate in the general population? This cannot be determined directly, but we shall be safe in assuming that the true value of K lies between 1 and 2, i.e., that the death-rate among inebriates is not less than that of the general population and not greater than twice that of the general population.

On this assumption we get

R=000,827,837 when K=1,

R='000,595,957 when K=2.

But the total population between 17 and 75 in 1908 was estimated at 11,439,103; thus

or

I=9470 if K=1

= 6817 if K=2.

It is now possible to complete the fourfold table. We have estimated that the proportion of mentally defective women in the general population is

and that the ratio of convicted inebriates to the whole is

865 =(a) '000,827,837 or (b) 000,595,957.

865+x+y

Combining these two pairs of equations we obtain four values for x and y and hence four possible fourfold tables.

But the maximum correlation will be got by taking the minimum percentage of mental defect with the minimum death-rate among inebriates, while the minimum correlation will be obtained when the mental defect and the death-rate among inebriates are at a maximum, and the limits of the correlation will be obtained from these two cases.

We thus obtain the following tables: