this is very doubtful. It seems on the whole more likely, though the question requires further study, that, in order to bring it into general accord with other measures, the Pareto measure should be inverted, so that, the greater a, the smaller inequality. But such an inversion will explode Professor Pareto's alleged economic harmonies and it will follow, according to his law, that increased production per head will always mean increased inequality! According to Professor Gini,1 many actual distributions of income approximate to the formula

where s is the total income of the n richest incomereceivers and & and c are constants for any given distribution. He proposes as a measure of inequality, or "index of concentration," as he prefers to call it, such that, the greater &, the greater inequality. This formula is a more convenient variant of Professor Pareto's, such

a that &= a- I'

and, as a diminishes from any quantity

greater than one down to one, & increases up to infinity. The equation log n = log s - log c is easily transformed into that of a Lorenz curve. For, if N is the total number of income-receivers and S the total income, we have

Thus, the greater 8, the larger is the above area, and the larger the relative mean difference. There is thus some ground for believing, though I do not here definitely commit myself to the belief, that the reciprocal of Professor Pareto's measure is a mere variant of the relative mean difference, in the particular case when distribution is approximately according to Pareto's law. In this particular case, then, Professor Pareto's measure would have no independent significance, and, in the more general case, when distribution may depart widely from Pareto's law, the measure has, of course, no general significance at all. It will, therefore, be provisionally set aside in this discussion.

12. Returning to the four measures set out in order of merit at the end of Section 10, this order is based on theoretical advantages. But account must also be taken of practical applicability to statistics. Both the relative mean deviation and the quartile measure are more easily applicable than either of their two rivals to perfect statistics, and applicable, with less risk of serious error, to imperfect statistics. As regards perfect, or nearly perfect, statistics, the advantage of the former pair over the latter relates only to laboriousness and not to accuracy, and is not, therefore, a matter of great importance. But, as regards markedly imperfect statistics, such as are actually

This index has been used by several Italian writers in enquiries into distributions of income. See, e.g., Savorgnan, La Distribuzione dei Redditi nelle Provincie e nelle Grandi Citta dell' Austria, and Porru, La Concentrazione della Ricchezza nelle Diverse Regioni d'Italia.

available, the advantage relates to accuracy as well as to laboriousness and is, therefore, vital.

The provisional conclusion which suggests itself, is as follows. When statistics are so imperfect that neither the relative standard deviation nor the relative mean difference can be applied with any expectation of reasonable accuracy, we must make shift with the relative mean deviation and the quartile measure. It is some palliation of the comparative insensitiveness to transfers which is a defect of both the latter measures, that each is sensitive to many possible transfers, to which the other is insensitive. If, therefore, both give the same result in any particular comparison, their evidence is to some extent corroborative.

If statistics are so far improved that the relative standard deviation and the relative mean difference are applicable, these are to be preferred to the two measures just mentioned. If a single measure is to be used, the relative mean difference is, perhaps, slightly preferable, owing to the graphical convenience of the Lorenz curve. Probably, however, it will be desirable, at any rate for some time to come, not to rely upon the evidence of a single measure, but upon the corroboration of several. Given perfect, or nearly perfect, statistics, it is worth while considering whether corroboration may not also be sought from the measure

log *, applied, for the sake of

log xx simplicity, to total incomes, and not to surplus incomes in excess of the requirements of "bare subsistence." For this measure passes our test of proportionate additions to incomes, which none of the other four survivors do. In most practical cases, no doubt, these five measures will give results pointing in the same direction, but in some cases they may not do so.

Meanwhile, the chief practical necessity is the improvement of existing statistical information, especially as regards the smaller incomes. This paper may be compared to an essay in a few of the principles of brickmaking. But, until a greater abundance of straw is forthcoming, these principles cannot be put to the test of practice.

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