The PRESIDENT said he believed that, compared with very old days, the Society was rather stronger in the official world, but rather less strong in the House of Commons and House of Lords.

The motion was then put, and carried unanimously.

Mr. REW having read out the list of names of defaulters, the PRESIDENT declared that they had ceased to be Fellows of the Society.

Mr. REW announced that the subject selected for essays in competition for the Howard Medal in 1908-09 was "A Statistical Study of Infantile Mortality in Great Britain and Ireland, and of its Causes."

The ballot was then taken, and the Scrutineers (Messrs. SYDNEY YOUNG and DAVID HERON) reported that the proposed President, Council, and Honorary Officers for the ensuing session had been unanimously elected.

The PRESIDENT, in returning thanks to the Society for electing him President for a second term, said that he had last year explained his Presidency as connected with a movement to promote the better organisation of Government statistics. The preference had been expressed by him for a permanent statistical office, undertaking among other duties those connected with the Census. From the creation of such an office it was known that the Treasury had always shrunk, and little co-ordination of the statistics of the various offices had been obtained, except through the Board of Trade. Mr. Bowley now took up the theme, to which that distinguished statistician had in past years devoted much attention. He (Sir Charles Dilke) had been asked to continue in office for a year, although unable to give time to prepare a fresh Address on the subject-covered indeed by him last year as fully as his knowledge would allow. On the same points Mr. Bowley was able to express similar views with a better scientific backing. The labours of the Census Committee of the Royal Statistical Society were also directed to the same end, and were now drawing to a close. They were still without public information as to definite steps yet taken by the Government to meet the admitted need for change. The Treasury had considered the matter, and the Board of Inland Revenue, as they knew, was revising statistical methods of finance. The Prime Minister had announced in the Budget speech a change between the Inland Revenue and the Customs, which would in the long run have statistical importance. The Presidential Address of last November had been circulated by the Home Office, and had also been officially made known by some of the departments of the Board of Trade.

A correspondence had taken place between himself and others, when the Cabinet enquiry into the allocation of work among

Government departments was announced by Mr. Asquith, and an arranged question had been answered by a promise that the co-ordination of Government statistics would be considered in the course of the enquiry. Personal changes in office had prevented that enquiry yet taking formal shape, although preliminary work had been done by some departments in advance.

A fresh Select Committee, the third or fourth, on Government Publications was now sitting, and the Presidential Address of November last had been formally brought before it. One of the principal official witnesses had prepared for the Committee a memorandum giving an outline of such duplications as those mentioned last year between, for example, the publications of the Board of Trade and of the Local Government Board. As the Home Office and the Board of Trade, as well as the Prime Minister and Treasury, had to some extent taken up the co-ordination of statistics, the ground was now prepared for reforms, which the preparation of the next Census Bill ought to, and he hoped would, inaugurate. Constant pressure was, however, needed, and, above all, a Treasury decision. The Treasury appeared to contemplate a special enquiry, but that would mean delay. The discussion on Mr. Bowley's paper might advance the question, in which that gentleman had been a pioneer, and which he had much at heart. The distinguished statistician who was Agent-General of New South Wales, and a Vice-President of the Society, Mr. Coghlan, might also possibly find time to give the Society a contribution upon the subject from the point of view of his wide knowledge of that which had actually been done elsewhere.

Mr. REGINALD HOOKER, in proposing a hearty vote of thanks to the President and the Council for their services during the past year, said that, having been once himself in close contact with the Council and Secretariat, he was in a better position than most Fellows to testify to the large amount of work which fell on the officers and the Council generally.

Mr. E. L. WALFORD, in seconding the motion, said he would remind the Council that the Meetings of the Society formerly took place in the evening, and the result was that a large number of City men attended their Meetings. He thought it might be worth considering whether it would not be advisable to revert to the old system of evening Meetings.

The motion was carried unanimously.

The PRESIDENT thanked the Members on behalf of the Officers, and said that the suggestion just made by Mr. Walford would be brought before the Council at the next Meeting.

I.-On the Probable Errors of Frequency-Constants.

By PROFESSOR F. Y. EDGEWORTH.

Conspectus of Contents.

The "probable error" is a well-chosen index of the belief-or, rather, the credibility-that a value which has been obtained for a frequency-constant characterising a group of statistics has a certain degree of accuracy, will not differ by more than a certain extent from the result of continued observations in pari materiâ. The apparatus for testing this credibility-the received Law of Error, with a certain other law of great numbers -is exemplified by the following problems:-(1) Given a set of observations ranging under a normal error-curve, and given the coefficient of deviation for that curve, to determine the average to which the observations if indefinitely continued would tend; (2) Given a normal set of observations as before, and given the average to which they tend, to determine the coefficient of deviation; (3) Given a normal set of observations, but given neither the average nor the coefficient of deviation, to determine both those frequency constants; (4) Given two sets of observations ranging under a normal surface, and given the average and the coefficient of deviation pertaining to each set, to determine the coefficient of correlation; (5) Given, as before, two normal sets of observations, but not any of the frequency-constants, to determine them all; (6) Given one or more sets of observations ranging under any given (not in general normal) laws of frequency, to determine all the frequency-constants; (7) Given one or more sets of observations, but not given their laws of frequency, to determine the averages to which they tend (the Method of Least Squares); (8) To determine a coefficient of correlation between two given sets of observations for which the laws of frequency are not given (Yule's method). The mathematical treatment of credibility in Statistics is comparable to the mathematical treatment of utility in Economics.

THE "probable error" is not in favour with some high authorities. Mr. Galton denounces the term as a "cumbrous, slipshod, and misleading phrase."* He refers to Dr. Venn, who also regards the "probable error" as a "highly misleading term."†

* Natural Inheritance, p. 58.

+ Logic of Chance, ed. 3, pp. 446-47.

"Such an error," he observes, "is not in any strict sense 'probable.' It is, indeed, highly improbable that in any particular instance we should happen to get this error. Nor can it be said to be probable that we shall be within this limit of the truth, for by definition we are just as likely to exceed as to fall short." These and other eminent writers on Statistics and Probabilities who have protested against the use of the "probable error" are doubtless right with respect to the subject which they have in view, the normal law of error considered as a statistical fact. They are also well-advised in not altogether discarding the term against which they protest. As Dr. Venn says in the context of the passage above quoted, "it is best to stand by accepted nomenclature." The case, I think, is one in which there is a danger of incurring what Mill calls the "evil consequences of casting off any portion of the customary connotation of words."* In the weighty section of his Logic devoted to that topic, he points out that in the case of words which "in their original acceptation connoted a complication. of outward facts and inward feelings," the latter portion of the meaning is apt to be obscured "by the incautious proceedings of mere logicians." The "inward feeling," the subjective phenomenon of belief or credibility, is not so well suggested by the term "quartile "much less by the term "standard deviation," or some multiple thereof-as by a phrase of which the word "probable" is a part. It is not merely that the name recalls that species of psychical measurement which characterises the Calculus of Probabilities, but also the conception defined is peculiarly appropriate to that use of the Calculus. The "probable error corresponds to the one definite notch in the scale of credibility, the point of complete uncertainty whether an event such as the occurrence of an error in excess of the assigned limit will or will not occur. In the scale of credibility this point has much the same significance as the point of "indifference" in the scale of economic utility. The "standard deviation," of which the "probable error" is sometimes described as a mere appendage, has not this sort of advantage. The probability of an error exceeding (in one direction or the other) the standard deviation is about 0.317, a degree of no particular significance in the scale of credibility. This justification of the adjective in the phrase under consideration carries with it an apology for the substantive. The "error" which is defined as probable, or rather not improbable, is not the particular error which is just equal to the assigned deviation from the average, say, nor yet an error in the immediate neighbourhood of q, say between q and q + Ar, where A is very small, the minimum sensibile on the abscissa along which is measured; but an error of the class which is defined by excess above (or defect below) q. When Laplace says it is a million to one that the mass of Jupiter as deduced by him from certain observations "is not in error by

*Logic, Book iv, ch. iv, sec. 6 (Contents).

† The fundamental character of this conception in economic theory is particularly well shown by Professor Pareto.

more

I per cent. of its value,' "* he does not mean the error 0.02 to the exclusion of the next degree of error, say 0-021, but the whole class of errors which exceed 0:02. Dr. Venn, indeed, has anticipated this use of the term "error," and has directed against it an objection, above quoted, which is perhaps formally valid. That a future error shall be within the defined limit is not indeed probable, only not improbable. But the former statement differs by a negligible quantity from the latter, where, as usual, the extent of deviation varies, if not continuously, at least by very small degrees. The slight inaccuracy is fully excused by usage and the need of brevity. Associated with the "probable error -more useful in practice, if less appropriate for definition-is the conception of improbable error, that extent of deviation which is hardly credible. Such a measure of incredibility is afforded wherever the normal law of error prevails. Its perhaps most important use is to afford assurance that measurements effected by several observations, or generally statistical determinations of frequency-constants, are trustworthy within assigned limits, may be relied on not to exceed those limits in future experience (in pari materia). The earliest and most familiar instance, the leading case, is the measurement of an objective quantity, such as the angular distance between two stars. Physicists require to have the sort of assurance which Laplace expresses in the passage above quoted that the measures which they have obtained will not differ from the thing measured by more than a certain extent. The methods which they successfully employ for this purpose are extended to the analogous case of types, such as the mean dimensions of men or crabs, which do not correspond to any one real objective thing. Though the thing measured has in this case, unlike the first case, no separate existence apart from the measurements, yet if these observations have the sort of stability, the sort of unity in the midst of plurality, which characterises fallible observations relating to one and the same real thing, we may still regard the mean value as a substantive thing about whose dimensions assurance is desiderated and obtained. This remark may be extended from the simpler kinds of averages to what may be called the secondary frequency-constants,† such as the coefficient of dispersion, or the coefficient of correlation, for the normal law of frequency, and other constants for other definite species of groupings. But a doubt may arise how far probability in the proper sense of the term as distinguished from objective statistical frequency, probability as understood by the older writers, is applicable to these newer results. In fact, with regard to the secondary frequency-constants at least, it is often not obvious where the normal curve occurs in virtue of which we are entitled to predicate probability, or improbability, of certain deviations. Even with regard to primary frequency-constants, and even with regard to the measurements of real objects, when the errors of

* Théorie Analytique des Probabilités, Supplement I.

+ Compare Prof. Karl Pearson's distinction between "organs' and "constants" in a passage to be quoted below.