TABLE IV.-Modes and Means, with their Probable Errors.

TABLE V.-Probable Errors of m, and m.; Details of Range; Skewness.

DISCUSSION on MR. YULE'S PAPER.

MR. FRANCIS GALTON' said that statisticians owed a large debt of gratitude to University College for the variety of the statistical investigations carried on there, both mathematical and experimental. Those pursued by Professor Karl Pearson, assisted by the author of the present paper, and those by Professor Weldon, had greatly raised the level of statistical science; they had enlarged its powers and widely extended its applications. Ordinary practical statisticians need not be alarmed at the technical terms, which would become familiar in time, nor at the high mathematics of the original investigations, with whose results they would be chiefly concerned.

The arithmetical operations were laborious; on the other hand, shorter methods sufficiently close for most purposes were almost sure to be discovered, such as that curious but as yet wholly empirical method of deducing the position of the mode from those of the median and the mean, which was found to hold good in a large class of cases. Mr. Yule's memoir was very welcome as an illustration of the process by which the new method was applied, and it enabled statisticians to form a useful idea of its value as compared to those of other methods. He (Mr. Galton) would venture to criticise one of the conclusions, namely, as to the trustworthiness of the calculated position of the mode, which was reckoned in the present cases to be nearly equal to that of either the mean or median. The values assigned to the mode for the years 1850 and 1860 differed however so largely from the judgment of his eye in respect to Plates II and III, that he could not accept them, and was compelled to suspect some insecurity in the foundation upon which the calculations were based. He should in both cases locate the mode much more to the left.

His next criticism was of a different kind, namely, an expression of doubt whether the new method when applied to the particular cases before them gave results that were better or even as good as the method of percentiles or, on this occasion perhaps, of deciles. Professor K. Pearson's formulæ were admirably applicable whenever large series of observations of the same variable always yielded the same smoothly running curve; but the present case was very different. The data ran roughly, and were unchecked by others of a similar kind, so that they did not seem to justify elaborate work. It might reasonably be maintained that such data as these admitted of being interpreted almost equally well in a variety of ways. Quite possibly the interpretation here given might be the most probable of all, and yet other interpretations might differ little from it in their respective probabilities. After

1 In response to a request made at the meeting, Mr. Galton has written the memorandum on p. 392, giving fuller information as to the method of percentiles.

all, the conclusions derived by this method seemed to be little if at all more quantitative or more complete for ordinary statistical purposes than those derived from the method of deciles, while the diagrams which expressed them were certainly less graphic and less suitable for inter-comparison.

Mr. A. H. BAILEY said that without meaning to deny the great value of the paper, he thought that it was impossible to follow it sufficiently upon the reading for its discussion. He certainly had not had time to study it thoroughly, but his own work had familiarised him with the method suggested in the paper. The law of mortality had been represented by a curve in which the abscissæ represented the years of life, and the ordinates the corresponding mortality at each age. That was of course a simpler matter, showing clearly the law of mortality from the observations. Now there was no law of pauperism, and therefore the cases were not parallel.

Mr. A. W. FLUX felt very strongly how great an advantage it was to be able to describe a complicated set of figures by a curve, or by two or three standard measurements which should define that curve, as had been done in the paper before them. But the facts represented in the curves were not absolutely the fiets recorded in the statistical returns, although the smooth-running curve did keep very close to the polygon formed by the actual figures. He felt very strongly, too, how far that simple diagram, which Mr. Galton had given them, went towards expressing nearly, if not quite, all that there was in this valuable paper, and yet the paper undoubtedly pointed out out a way to develop

statistical records in a more effective fashion than had hitherto been done. It seemed to him that the three measurements which practically characterised the curve, viz., the mode, the standard deviation, and the range, left out a great deal which was very valuable. With regard to pauperism in particular, they wanted to know what was the effect of a given policy; but it was absolutely impossible to represent that in these curves. Those parts of the country where various causes had reduced pauperism most rapidly were not separated out distinctly as places, and any connection between the cause and the result in particular districts was necessarily hidden from the representation here given. The figures varied, but the distribution of the figures only was shown. If there had been a definite change in the policy of the whole kingdom, then they might be able to trace the results on the whole kingdom to that definite change, but in some places there had been very little change, and in others great change of policy, and they were not able by this method to connect these changes with the results. It would have been interesting if they could havė seen how the mode and the standard deviation varied in the course of a year, but there would be a difficulty very similar to that which Mr. Yule found in dealing with the figures of 1886. In the course of a year the population in many unions changed a great deal and, as they did not know what these changes were,

VOL. LIX. PART II.

2 A

it was

not possible to get the percentages of pauperism to population. The great difficulty in dealing with the figures for 1886 was due to the fact that, although they knew the number of paupers in each union for that year, they did not know the population sufficiently accurately to be able to get the percentages. Was it not possible that the method which had been adopted of using the figures of population for 1851, combined with the figures of pauperism for 1850, and similarly for 1860 and 1870, might give results for those three periods not quite comparable with the results for 1881 and 1891, and that the changes of population in the course of that one year might have caused these results to be misleading? Following the suggestion of Mr. Yule in the paper, he had very hastily compared one hundred unions in Yorkshire, Lancashire, and Cheshire with those for the whole country, and the results he obtained were very curious. He found that the standard deviation for this area for 1891 was less than any of the standard deviations that Mr. Yule had worked out. Another very curious thing was that, in the case of these one hundred unions in 1881, the median was not on the same side of the mean as in all the five cases which Mr. Yule had worked out, but fell on the other side. He should hesitate in reference to the conclusion that the side of the mean on which the median lay indicated the tendency of the development, because that would mean that the policy adopted in Yorkshire, Lancashire, and Cheshire tended to increase the pauperism of those districts. Taking Mr. Charles Booth's figures, from which he understood that Mr. Yule had worked, and comparing 1881 with 1891, he found that the criterion rose very considerably. But it was quite possible that one hundred unions was too small a number to yield trustworthy results."

Mr. C. S. LOCH said that be was not familiar with the methods adopted by Mr. Yule. He was greatly interested in the subject; but, so far as he had dealt with it statistically, he had worked only on the rules of simple arithmetic. The effect of increase or decrease of population upon statistics of pauperism had often puzzled him, and he felt sure that many deductions had been made respecting the latter which were fallacious. Results sometimes seemed satisfactory which were due primarily to the diminution or increase of population, and not to the causes assigned to account for them. The same remark applied also to local statistics of ratings and other matters. It was to be hoped that expert statisticians might enable them to deal more satisfactorily with a point of this kind. He was much interested in what the last speaker had said about one hundred unions in the north of England. The results were just what he would have expected. The deviation would be less, and ought to be less. Where there was least pauperism there would actually be least deviation. He believed that Mr. Yule's figures expressed total numerical results irrespective of causes;

2 A further examination of the figures shows that this discrepancy really does arise from the smallness of the number of unions, and a calculation of the median by a method better adapted to this small number, gives a result in accordance with those obtained by Mr. Yule.-A.W.F.

but it was interesting to attempt, if possible, to isolate causes. With the aid of local knowledge it might be possible to work carefully over the whole area of England, and group unions on a logical and scientific method from this point of view; and either Mr. Yule's method or Mr. Galton's, which seemed to him extraordinarily clear, might be applied so as to find some numerical expression of the cause. No one who had tried to work out these figures by arithmetical means only, could help feeling that instruments of much greater precision were wanted. As a very important contribution towards this result he welcomed Mr. Yule's paper very heartily. He hoped that it might be followed by others that would in an equal degree force them to strenuous thinking and a reconsideration of statistical methods.

Mr. W. F. SHEPPARD said that he had two criticisms to make on Mr. Yule's method. In the first place, he was not satisfied as to the à priori grounds on which the kind of curve used by Mr. Yule was taken to be the appropriate curve for the particular data. It seemed to him that the predominance of the normal symmetrical curve, both in theory and in actual statistics, made it desirable to test the data first for such a curve, and then to consider whether the discrepancies might not be due to paucity of observations. The total number of unions was not very great, and this would account for a good deal of irregularity. Secondly, in the absence of these à priori grounds for supposing the proper curve to be a curve of the particular kind adopted, the median seemed to be a better criterion of the degree of pauperism than the mode. The median could be obtained, within small limits of error, from the actual observations, whereas in determining the maximum ordinate from the observations an appreciable error might be caused by inaccuracy in the amount of pauperism in a few unions. Thus, for the year 1860, an alteration of the relative position of some ten unions, by perhaps a very slight alteration in their actual pauperism, would make a great change in the position of the maximum ordinate.

Mr. J. A. BAINES differed from Mr. Bailey with regard to this paper, for he thought that it made more widely known to a class that did not habitually deal with higher mathematics, the value of such methods when adapted to actual statistics. As the last speaker but one (Mr. Loch) had truly observed, the results might not hold good for a special area, but the avowed object of the paper was to reduce figures for local areas to one general total, and to show the progress or retrocession of pauperism for the whole of England. No doubt it was easy to test general results by a group of unions, but the methods set forth in this paper seemed almost the only means of combining the whole body of the statistics into one concrete form. He had often experienced the impossibility of getting accurate results from interpolated figures, and that was an additional argument, if indeed one was needed, for the contention so often and strongly put forward by the Society for a quinquennial census, which he hoped they would never cease striving