1902 Encyclopedia > Statistics

## Statistics

STATISTICS. The word "statistic" is derived from the Latin status, which, in the so-called Middle Ages, had come to mean a "state"in the political sense. "Statistic," therefore, originally denoted inquiries into the condition of a state. Since the beginning of the 18th century the denotation of the word has been extended so as to include subjects only indirectly connected with political organizations, while at the same time the scope of the investigations it implies has become more definite, and at the present day may be said, for practical purposes, to be fixed, though there are still controversies as to the position of statistical studies in relation to other departments of scientific procedure.

History.—The origin of what is now known as "statistic" (Ger. Die Statistik; Fr, La Statistique; Ital. Statistica) can only be referred to briefly here. As M. Maurice Block has observed in commencing his admirable treatise, "it is no exaggeration to say that statistic has existed ever since there were states." For the first administrative act of the first regular Government was probably to number its fighting men, and its next to ascertain with some degree of accuracy what amount of taxation could be levied on the remainder of the community. As human societies became more and more highly organized, there can be no doubt that a very considerable body of official statistics must have come into existence, and been con-stantly used by statesmen, solely with a view to administration. The Romans, who may be described as the most business-like people of antiquity, were careful to obtain accurate information regarding the resources of the state, and they appear to have carried on the practice of taking the census, a very comprehensive statistical operation, with a regularity which has hardly been surpassed in modern times. As to the efficiency of the work done we have unfortunately very little information, but those who are curious on the subject may be referred to an article by Dr Hildebrand, entitled "Die amtliche Bevölkerungs-statistik im alten Rom," printed in the Jahrbuch für Nationalökonomie und Statistik, 1866, p. 82.

Statistics, or rather the material for statistics, therefore existed at a very early period, but it was not until within the last three centuries that systematic use of the informa-tion available began to be made for purposes of investiga-tion and not of mere administration. According to M. Block, the earliest work in which facts previously known only to Government officials were published to the world was a volume compiled by Francesco Sansovino, entitled Del Governo et Amministrazione di Diversi Begni et Republiche, which was printed in Venice and bears the date 1583. Other works of a similar kind were published towards the end of the 16th century in Italy and France. Regarding these and other early books on the subject reference may be made to Fallati's Einleitung in die Wissenschaft der Statistik, Dr G. B. Salvioni's preface and notes to his translation into Italian of Dr Mayr's work on statistics, and other authors mentioned at the close of this article.

Works on state administration and finance continued to be published during the first half of the 17th century, and the tendency to employ figures, which were hardly used at all by Sansovino, became more marked, especially in England, where the facts connected with "bills of mortality " had begun to attract attention.

In the year 1660 Hermann Conring, "professor of medicine and politics," a rather odd combination, in the university of Helmstädt, was in the habit of giving lectures in which he analysed and discussed the circumstances existing in various countries, in so far as they affected the happiness of the inhabitants. Conring's example was followed by other writers, in Germany and elsewhere, to whom reference is made by Block (Traité, pp. 5, 6) and Haushofer (Lehr- und Handbuch, p. 10, note).

The best-known member of the "descriptive" school was Achenwall (1719-1772), who is sometimes spoken of as " the father of modern statistics," but, as his procedure was essentially the same as that of Conring, though it was carried out more fully, the title has not been unanimously granted. It is generally admitted, however, that Achenwall's work gave a great impulse to the pursuit of the studies which are now included under the title of statistics. He called his book Staatsverfassung der europäischen Reiche in the first two editions (1749, 1752), meaning "Constitution of the States of Europe." Subsequently he added " vornehmsten" and then "heutigen" before "europäischen," evidently with the desire of bringing his work, which may be regarded as the germ of such volumes as the Statesman's Year-Book, "up to date." Achenwall is usually credited with being the first writer who made use of the word "statistics," which he applied to his collection of "noteworthy matters regarding the state" (Staatsmerkwürdigkeiten), but the claim has been disputed by M. Block, who points out that the term collegium statisticum had been previously employed by Schmeitzel, a follower of Conring, whose lectures at Jena were no doubt attended by Achenwall.

In any case statistics, in the modern sense of the word, did not really come into existence until the publication by J. P. Süssmilch, a Prussian clergyman, of a work entitled Die göttliche Ordnung in den Veränderungen des Menschlichen Geschlechts aus dem Geburt, dem Tode, und der Fortpflanzung desselben erwiesen. In this book a systematic attempt was made to make use of a class of facts which up to that time had been regarded as belonging to "political arithmetic," under which description some of the most important problems of what modern writers term "vital statistics" had been studied, especially in England. Süssmilch had arrived at a perception of the advantage of studying what Quetelet subsequently termed the "laws of large numbers." He combined the method employed by the Conring-Achenwall school of "descriptive statistics," whose works were not unlike modern school-books of geography, with that of the "political arithmeticians," who had confined themselves to investigations into the facts regarding mortality and a few other similar subjects, without much attempt at generalizing from them.

Political arithmetic had come into existence in England in the middle of the 17th century, or about the time when Conring was instructing the students of Helmstädt. The earliest example of this class of investigation is the work of Captain John Graunt of London, entitled Natural and Political Annotations made upon the Bills of Mortality, which was first published in 1666. This remarkable work, which dealt with mortality in London only, ran through many editions, and the line of inquiry it suggested was followed up by other writers, of whom the most distinguished was Sir William Petty, whose active mind was naturally attracted by the prospect of making use of a new scientific method in the class of speculations which occupied him. Sir William was the first writer to make use of the phrase which for nearly a century afterwards was employed to describe the use of figures in the investigation of the phenomena of human society. He called his book on the subject, which was published in 1683, Five Essays in Political Arithmetick. Other writers, of whom Halley, the celebrated mathematician and astronomer, was one, entered on similar investigations, and during the greater part of the 18th century the number of persons who devoted themselves to "arithmetical" inquiries into problems of the class now known as statistical was steadily increasing. Much attention was given to the construction of tables of mortality, a subject which had a great attraction for mathematicians, who were eager to employ the newly-discovered calculus of probabilities on concrete problems. Besides Halley, De Moivre, Laplace, and Euler busied themselves with this branch of study. Attempts were also made to deal with figures as the basis of political and fiscal discussion by Arthur Young, Hume, and other historical writers, as well as by the two Mirabeaus.

M. Block is of opinion that the descriptive school, by whom figures are regarded merely as accessories to and illustrations of the text, would have maintained its position even now but for the establishment of official statistical offices and the influence of the great Belgian Quetelet. Quetelet's work was certainly "epoch-making" in a far higher degree than that of any of his predecessors. To the impulse created by him must be attributed the foundation in 1835 of the Statistical Society of London, a body which, though it has contributed little to the discussion of the theory of statistics, has had a considerable and very useful influence on the practical work of carrying out statistical investigations in the United Kingdom and elsewhere. Quetelet's works were numerous and multifarious, but his most important contribution to the growth of statistical inquiry was his investigation of the theory of pro-babilities as applied to the "physical and social" sciences, contained in a series of letters to the duke of Saxe-Coburg and Gotha, and published in 1846. Quetelet was above all things an exponent of the "laws of large numbers." He was especially fascinated with the tendency to relative constancy of magnitude displayed by the figures of moral statistics, especially those of crime, which inspired him with a certain degree of pessimism. His conception of an average man (I'homme moyen) and his disquisition on the "curve of possibility" were most important contributions to the technical development of the statistical method, though, as M. Block observes, their value may have been somewhat exaggerated by subsequent writers (Block, ch. i. p. 16, and ch. v. p. 112 sq.). It is not possible to enter at length into Quetelet's work in connexion with statistical science. At the close of this article will be found a list including those of his works which are likely to be of use to students of statistics.

The influence exercised by Quetelet on the development of statistics is clearly seen from the fact that, though there is still considerable controversy among statisticians, the old controversy between the "descriptive" and arithmetical schools has disappeared, or perhaps we should say has been transformed into a discussion of another kind, the question now at issue being whether there is a science of statistics as well as a statistical method. It is true that a few books were published between 1830 and 1850 in which the politico-geographical description of a country is spoken of as "statistics," which is thus distinguished from "political arithmetic." The title of Knies's great work, Die Statistik als selbständige Wissenschaft (Cassel, 1850), is especially noteworthy as showing that the nature of the controversy was changing. The opponents of Süssmilch maintained that "political arithmetic" ought not to be spoken of as statistics at all. They clung to the conceptions of Conring and Achenwall, to whom "statistics" represented "Staatenkunde" or " Staatszustandskunde," or, as Herzberg, one of Achenwall's followers, called it, "die Kenntniss von der politischen Verfassung der Staaten." Knies claimed that the really "scientific" portion of statistics consisted of the figures employed. As Haushofer says, "his starting point is political arithmetic."

Some eminent statisticians of the latter half of the present century agree with Knies, but the majority of the modern writers on the theory of statistics have adopted a slightly different view, according to which statistics is at once a science relating to the social life of man and a method of investigation applicable to all sciences. This view is ably maintained by Mayr, Haushofer, Gabaglio, and Block, who may be taken to represent the opinions held by the majority of statisticians on the Continent.

Having dealt as far as was possible, within the limits of this article, with the history of statistics, we may here enter a little more minutely into the views of the existing Continental school. This is all the more necessary because, singular to say, there has been no systematic exposition of the subject in England. Isolated dicta have been furnished by high authorities, such as the late Dr W. A. Guy, Prof. Ingram, Sir Rawson W. Bawson, Mr Robert Giffen, and to some extent also by John Stuart Mill, Buckle, Sir George Cornewall Lewis, and other historical and economic writers. There are also monographs on particular points connected with the technique of statistical investigation, such as the contribution made by Mr F. Y. Edgeworth to the discussions at the jubilee of the Statistical Society in 1885, and some of the observations contained in a paper by Mr Patrick Geddes, entitled An Analysis of the Principles of Economics, read before the Royal Society of Edinburgh in 1884. Prof. Foxwell has also lectured on the subject of statistics in his capacity of Newmarch lecturer at University College, London. But there has been no attempt to deal with the subject in a systematic way. The practice of statistical inquiry, on the other hand, has been carried on in England with a high degree of success.

With regard to the few invasions of the domain of theory attempted by English writers, it may be observed that the authorities above mentioned are not unanimous. Dr Guy as well as Sir Rawson Rawson, who handled the subject with great ability at the jubilee meeting of the London Statistical Society in June 1885, both claim that statistics is to be regarded as an independent science, apart from sociology, while Prof. Ingram, who presided over Section F at the Dublin meeting of the British Association in 1878, maintained that statistics cannot occupy a position co-ordinate with that of sociology, and went on to say that they "constitute only one of the aids or adminicula of science." Mr Giffen has also expressed himself adversely to the Continental doctrine that there is an independent science of statistics, and this opinion appears to be the correct one, but, as Dr Guy and Sir Rawson Rawson have the support of the great body of systematic teaching emanating from distinguished Continental statisticians in support of their view, while their opponents have so far only the obiter dicta of a few eminent men to rely upon, it appears needful to examine closely the views held by the Continental authorities, and the grounds on which they are based.

The clearest and shortest definition of the science of statistics as thus conceived is that of M. Block, who describes it as "la science de l'homme vivant en société en tant qu'elle peut être exprimée par les chiffres." He proposes to give a new name to the branch of study thus defined, namely, "Demography." Mayr's definition is longer. He defines the statistical science as " die systematische Darlegung und Erörterung der thatsächlichen Vorgänge und der aus diesen sich ergebenden Gesetze des gesellschaftlichen menschlichen Lebens auf Grundlage quantitativer Massenbeobachtungen" (the systematic statement and explanation of actual events, and of the laws of man's social life that may be deduced from these, on the basis of the quantitative observation of aggregates). Gabaglio's view is practically identical with those adopted by Mayr and Block, though it is differently expressed. He says "statistics may be interpreted in an extended and in a restricted sense. In the former sense it is a method, in the latter a science. As a science it studies the actual social-political order by means of mathematical induction."

This discussion regarding the nature of statistics is to a large extent a discussion about names. There is really no difference of opinion among statistical experts as to the subject-matter of statistics, the only question being— Shall statistics be termed a science as well as a method? That there are some investigations in which statistical procedure is employed which certainly do not belong to the domain of the supposed statistical science is generally admitted. But, as already shown, an attempt has been made to claim that the phenomena of human society, or some part of those phenomena, constitute the subject-matter of an independent statistical science. It is not easy to see why this claim should be admitted. There is no reason either of convenience or logic why the use of a certain scientific method should be held to have created a science in one department of inquiry, while in others the said method is regarded merely as an aid in investigation carried on under the superintendence of a science already in existence. It is impossible to get over the fact that in meteorology, medicine, and other physical sciences statistical inquiries are plainly and obviously examples of the employment of a method, like microscopy, spectrum analysis, or the use of the telescope. Why should the fact of their employment in sociology be considered as authorizing the classification of the phenomena thus dealt with to form a new science?

The most effective argument put forward by the advocates of this view is 'the assertion that statistics are merely a convenient aid to investigation in the majority of sciences, but are the sole method of inquiry in the case of sociology. Dr Mayr especially (Gesetzmässigkeit, &c, p. 14 sq.) makes use of this argument, and illustrates it with his usual ability; but his reasoning is very far from being conclusive. When, indeed, it is tested by reference to the important class of social facts which are named economic, it becomes obvious that the argument breaks down. Economics is a branch—the only scientifically organized branch—of sociology, and statistics are largely used in it, but no one, so far as we are aware, has proposed to call economics a department of statistical science. Sir Rawson W. Rawson, it is true, has boldly proposed to throw over the term " sociology"altogether, and to describe the study of man in the social state as "statistics," but common usage is too firmly fixed to make this alteration of nomenclature practicable even if it were desirable. The existence of the works of Mr Herbert Spencer and Dr Schäffle alone would render the attempted alteration abortive.

Although, however, the above considerations forbid the acceptance of the Continental opinion that the study of man in the social state is identical with statistics, it must be admitted that without statistics the nature of human society could never become known. For society is an aggregate, or rather a congeries of aggregates. Not only that, but the individuals composing these aggregates are not in juxtaposition, and what is, from the sociological point of view, the same aggregate or organ of the "body politic" is not always composed of the same individuals. Constancy of social form is maintained concurrently with the most extensive changes in the collocation and identity of the particles composing the form. A "nation" is really changed, so far as the individuals composing it are concerned, every moment of time by the operation of the laws of population. But the nation, considered sociologically, remains the same in spite of this slow change in the particles composing it, just as a human being is considered to be the same person year by year, although year by year the particles forming his or her body are constantly being destroyed and fresh particles substituted. Of course the analogy between the life of a human being and the life of a human community must not be pressed too far. Indeed, in several respects human communities more nearly resemble some of the lower forms of animal life than the more highly organized forms of animal existence. There are organisms which are fissiparous, and when cut in two form two fresh independent organisms, so diffused is the vitality of the original organism ; and the same phenomenon may be observed in regard to human communities.

Now the only means whereby the grouping of the individuals forming a social organism can be ascertained, and the changes in the groups year by year observed, is the statistical method. Accordingly the correct view seems to be that it is the function of this method to make perceptible facts regarding the constitution of society on which sociology is to base its conclusions. It is not claimed, or ought not to be claimed, that statistical investigation can supply the whole of the facts a knowledge of which will enable sociologists to form a correct theory of the social life of man. The statistical method is essentially a mathematical procedure, attempting to give a quantitative expression to certain facts; and the resolution of differences of quality into differences of quantity has not yet been effected even in chemical science. In sociological science the importance of differences of quality is enormous, and the effect of these differences on the conclusions to be drawn from figures is sometimes neglected, or insufficiently recognized, even by men of unquestionable ability and good faith. The majority of politicians, social "reformers," and amateur handlers of statistics generally are in the habit of drawing the conclusions that seem good to them from such figures as they may obtain, merely by treating as homogeneous quantities which are heterogeneous, and as comparable quantities which are not comparable. Even to the conscientious and intelligent inquirer the difficulty of avoiding mistakes in using statistics prepared by other persons is very great. There are usually "pit-falls " even in the simplest statistical statement, the position and nature of which are known only to the persons who have actually handled what may be called the "raw material" of the statistics in question; and in regard to complex statistical statements the "outsider" cannot be too careful to ascertain from those who compiled them as far as possible what are the points requiring elucidation.

The Statistical Method.—This method is a scientific procedure (1) whereby certain phenomena of aggregation not perceptible to the senses are rendered perceptible to the intellect, and (2) furnishing rules for the correct performance of the quantitative observation of these phenomena. The class of phenomena of aggregation referred to includes only such phenomena as are too large to be perceptible to the senses. It does not, e.g., include such phenomena as are the subject-matter of microscopy. Things which are very large are often quite as difficult to perceive as those which are very small. A familiar example of .this is the difficulty which is sometimes experienced in finding the large names, as of countries or provinces, on a map. Of course the terms "large," "too large," "small," and "too small" must be used with great caution, and with a clear comprehension on the part of the person using them of the standard of measurement implied by the terms in each particular case. A careful study of the first few pages of De Morgan's Differential and Integral Calculus will materially assist the student of statistics in attaining a grasp of the principles on which standards of measurement should be formed. It is not necessary that he should become acquainted with the calculus itself, or even possess anything more than an elementary knowledge of mathematical science, but it is essential that he should be fully conscious of the fact that "large" and "small" quantities can only be so designated with propriety by reference to a common standard.

Sources whence Statistics are Derived.—The term "statistics" in the concrete sense means systematic arrangements of figures representing "primary statistical quantities." A primary statistical quantity is a number obtained from numbers representing phenomena, with a view to enable an observer to perceive a certain other phenomenon related to the former as whole to parts. They represent either a phenomenon of existence at a given point of time or a phenomenon of accretion during a given period. As examples may be mentioned the number of deaths in a given district during a given time, the number of pounds sterling received by the London and North Western Railway during a given time, and the number of "inches of rain" that fell at Greenwich during a given time. Other examples are the number of tons of pig-iron lying in a particular store at a given date, the number of persons residing (the term "residing" to be specially defined) in a given territory at a given date, and the number of pounds sterling representing the ''private deposits " of the Bank of England at a given date.

Primary Statistical Quantities are the result of labours carried on either (A) by Governments or (B) by individuals or public or private corporations.

A. Government Statistics.—(1) A vast mass of statistical material of more or less value comes into existence automatically in modern states in consequence of the ordinary administrative routine of departments. To this class belong the highly important statistical information published in England by the registrar-general, the returns of pauperism issued by the Local Government Board, the reports of inspectors of prisons, factories, schools, and those of sanitary inspectors, as well as the reports of the commissioners of the customs, and the annual statements of trade and navigation prepared by the same officials. There are also the various returns compiled and issued by the Board of Trade, which is the body most nearly resembling the statistical bureaus with which most foreign Governments are furnished. Most of the Government departments publish some statistics for which they are solely responsible as regards both matter and form, and they are very jealous of their right to do so, a fact which is to some extent detrimental to that uniformity as to dates and periods which should bo the ideal of a well-organized system of statistics. Finally may be mentioned the very important set of statistical quantities known as the budget, and the statistics prepared and published by the commissioners of inland revenue, by the post office, and by the national debt commissioners. All these sets of primary statistical quantities arise out of the ordinary work of departments of the public service. Many of them have been in existence, in some form or other, ever since a settled Government existed in the country. There are records of customs receipts at London and other ports of the time of Edward III., covering a period of many years, which leave nothing to be desired in point of precision and uniformity. It may be added that many of these sets of figures are obtained in much the same form by all civilized Governments, and that it is often possible to compare the figures relating to different countries, and thus obtain evidence as to the sociological phenomena of each, but in regard to others there are differences which make comparison difficult.

(2) Besides being responsible for the issue of what may be called administration statistics, all Governments are in the habit of ordering from time to time special inquiries into special subjects of interest, either to obtain additional information needful for administrative purposes, or, in. countries possessed of representative institutions, to supply statistics asked for by parliaments or congresses. It is not necessary to refer particularly to this class of statistical information, except in the case of the census. This is an inquiry of such great importance that it may be regarded as one of the regular administrative duties of Governments, though as the census is only taken once in a series of years it must be mentioned under the head of occasional or special inquiries undertaken by Governments. In the United Kingdom the work is done by the registrars-general who are in office when the period for taking the census comes round. On the Continent the work is carried out by the statistical bureaus of each country,—except France, where it is under the supervision of the minister of the interior. For further information on this subject reference may be made to the excellent chapter in M. Maurice Block's Traité entitled ''Recensement." See also "Instructions to the Superintendent Registrar of Births and Deaths as to his duties in taking the Census," 1871; also CENSUS, vol. v. p. 334 sq.

B. The primary statistical quantities for which individuals or corporations are responsible may be divided into three categories.

(1) Among those which are compiled in obedience to the law of the land are the accounts furnished by municipal corporations, by railway, gas, water, banking, insurance, and other public companies making returns to the Board of Trade, by trades unions, and by other bodies which are obliged to make returns to the registrar of friendly societies. The information thus obtained is published in full by the departments receiving it, and is also furnished by the companies themselves to their proprietors or members.

(2) An enormous mass of statistical information is furnished voluntarily by public companies in the reports and accounts which, in accordance with their articles of association, are presented to their proprietors at stated intervals. With these statistics may be classed the figures furnished by the various trade associations, some of them of great importance, such as Lloyd's, the London Stock Exchange, the British Iron Trade Association, the London Corn Exchange, the Institute of Bankers, the Institute of Actuaries, and other such bodies too numerous to mention.

(3) There are cases in which individuals have devoted themselves with more or less success to obtaining original statistics on special points. The great work done by Messrs Behm and Wagner in arriving at an approximate estimate of the population of the earth does not belong to this category, though its results are really primary statistical quantities. Many of these results have not been arrived at by a direct process of enumeration at all, but by ingenious processes of inference. It need hardly be said that it is not easy for individuals to obtain the materials for any primary statistical quantity of importance, but it has been done in some cases with success.

Operations Performed on Primary Statistical Quantities.—Only a brief description of matters connected with the technique of the statistical method can be given in this article. In order to form statistics properly so called the primary statistical quantities must be formed into tables, and in the formation of these tables lies the art of the statistician. It is not a very difficult art when the principles relating to it have been properly grasped, but those who are unfamiliar with the subject are apt to underrate the difficulty of correctly practising it.

Simple Tables.—The first thing to be done in the construction a table is to form a clear idea of what the table is to show, and to express that idea in accurate language. This is a matter which is often neglected, and it is a source of much waste of time and occasionally of misapprehension to those who have to study the figures thus presented. No table ought to be considered complete without a ''heading" accurately describing its contents, and it is frequently necessary that such headings should be rather long. It has been said that "you can prove anything by statistics." This statement is of course absurd, taken absolutely, but, like most assertions which are widely believed, it has a grain of truth in it. If this popular saying ran "you can prove anything by tables with slovenly and ambiguous headings," it might be assented to without hesitation. The false "statistical" facts which obtain a hold of the public mind may often be traced to some widely circulated table, to which either from stupidity or carelessness an erroneous or inaccurate "heading" has been affixed.

A statistical table in its simplest form consists of "primaries " representing phenomena of the same class, but existing at different points of time, or coming into existence during different portions of time. This is all that is essential to a table, though other things are usually added to it as an aid to its comprehension. A table stating the number of persons residing in each county of England on a given day of a given year, and also, in another column, the corresponding numbers for the same counties on the corresponding day of the tenth year subsequently, would be a simple tabular statement of the general facts regarding the total population of those counties supplied by two successive censuses. Various figures might, however, be added to it which would greatly add to its clearness. There might be columns showing the increase oi decrease for each county and for the whole kingdom during the ten years, and another column showing what proportion, expressed in percentages, these increases or decreases bore to the figures for the earlier of the two years. Then there might be two columns showing what proportions, also expressed as percentages, the figures for each county bore in each year to the figures for the whole kingdom. The nine-column table thus resulting would still be simple, all the figures being merely explicit assertions of facts which are contained implicitly in the original '' primaries."

Complex Tables.—Suppose now we have another table precisely similar in form to the first, and also relating to the counties of England, but giving the number of houses existing in each of them at the same two dates. A combination of the two would form a complex table, and an application of the processes of arithmetic would make evident a number of fresh facts, all of which would be implied in the table, but would not be obviour to most people until explicitly stated.

The technical work of the statistician consists largely in opera-tions of which the processes just referred to are types.

Proportions.—The most usual and the best mode of expressing the proportion borne by one statistical quantity to another is to state it as a percentage. In some cases another method is adopted —namely, that of stating the proportion in the form "one in so many." This method is generally a bad one, and its use should be discouraged as much as possible, the chief reason being that the changing portion of this kind of proportional figure becomes greater or less inversely, and not directly, as the phenomenon it represents increases or diminishes.

Averages.—Averages or means are for statistical purposes divided into two classes, the geometrical and arithmetical. An arithmetical mean is the sum of all the members forming the series of figures under consideration divided by their number, without reference to their weight or relative importance among themselves. A geometrical mean is the sum of such figures divided by their number, with due allowance made for their weight. An example will make this clear, and the simplest example is taken from a class of statistical quantities of a peculiar kind — namely, prices. The price of a given article is the approximate mathematical expression of the rates, in terms of money, at which exchanges of the article for money were actually made at or about a given hour on a given day. A quotation of price such as appears in a daily price list is, if there has been much fluctuation, only a very rough guide to the actual rates of exchange that have been the basis of the successive bargains making up the day's business. But let us suppose that the closing price each day may be accepted as a fair representative of the day's transactions, and let us further suppose that we desire to obtain the average price for thirty days. Now the sum of the prices in question divided by thirty would be the arithmetical mean, and its weak point would be that it made no allowance for the fact that the business done on some days is much larger than that done on others; in other words, it treats them as being all of equal weight. Now if, as is actually the case in some markets, we have a daily account of the total quantities sold we can weight the members accurately, and can then obtain their geometrical mean. There are cases in which the careless use of arithmetical means misleads the student of the social organism seriously. It is often comparatively easy to obtain arithmetical means, but difficult to obtain geometrical means. Inferences based on the former class of average should be subjected to the most rigid investigation.

Before closing this short survey of the very important subject of averages or means, it is needful to discuss briefly the nature of the phenomena which they may safely be regarded as indicating, when they have been properly obtained. Given a geometric mean of a series of numbers referring to no matter what phenomenon, it is obvious that the value of the mean as a type of the whole series will depend entirely on the extent of divergence from it of the members of the series as a body. If we are told that there are in a certain district 1000 men, and that their average height is 5 feet 8 inches, and are told nothing further about them, we can make various hypotheses as to the structure of this body from the point of view of height. It is possible that they may consist of a rather large number of men about 6 feet high, and a great many about 5 feet 5 inches. Or the proportions of relatively tall and short men may be reversed, that is, there may be a rather large number of men about 5 feet 4 inches, and a moderate number of men about 5 feet 11 inches. It is also possible that there may be very few men whose height is exactly 5 feet 8 inches, and that the bulk of the whole body consists of two large groups—one of giants and the other of dwarfs. Lastly, it is possible that 5 feet 8 inches may really give a fair idea of the height of the majority of the men, which it would do if (say) 660 of them were within an inch of that height, either by excess or deficiency, while of the remainder one half were all above 5 feet 9 inches and the other half all below 5 feet 7 inches. This latter supposition would most likely be found to be approximately correct if the men belonged to a race whose average height was 5 feet 8 inches, and if they had been collected by chance. The extent of the divergence of the items composing an average from the average itself may be accurately measured and expressed in percentages of the average, the algebraic signs + and - being employed to indicate the direction of the variation from the mean. An average may, therefore, advantageously be supplemented—(1) by a figure showing what proportion of the members from which it is derived differ from the average by a relatively small quantity, and (2) by figures showing the maximum and minimum deviations from the average. The meaning of the term "relatively small" must be considered inde-pendently in each investigation. Further remarks on averages will be found in the works mentioned at the conclusion of this article.

Prices.—Reference has already been made to the peculiar class of statistical quantities known as prices. Prices in their widest sense include all figures expressing ratios of exchange. In modern society the terms of exchange are always expressed in money, and the things for which money is exchanged are—(1) concrete entities with physical attributes, such as iron or wheat; (2) immediate rights, such as those given by interest-bearing securities of all kinds, by bills of exchange, by railway or steamship contracts to carry either passengers or goods, and by bargains relative to the foreign exchanges; (3) contingent rights, such as those implied in policies of insurance. All these rates of exchange belong to the same category, whether they are fixed within certain limits by law, as in the ease of railway charges, or are left to be determined by the "higgling of the market." All these cases of price may conceivably come within the operation of the statistical method, but the only matter connected with price which it is necessary to refer to here is the theory of the index number.

Index Numbers.—The need for these became conspicuous during the investigations of Tooke, Newmarch, and others into the general cyclical movements of the prices of commodities; and to construct a good system of these may be said to be one of the highest technical aims of the statistical method. In comparing the prices of different years it was soon observed that, though whole groups of articles moved upwards or downwards simultaneously, they did not all move in the same proportion, and that there were nearly always cases in which isolated articles or groups of articles moved in the opposite direction to the majority of articles. The problem presented to statisticians therefore was and is to devise a statistical expression of the general movement of prices, in which all prices should be adequately represented. The first rough approximation to the desired result was attained by setting down the percentages representing the movements, with their proper algebraic signs before them, and adding them together algebraically. The total with its proper sign was then divided by the number of articles, and the quotient represented the movement in the prices of the whole body of articles during the period under consideration. It was soon seen, however, that this procedure was fatally defective, inasmuch as it treated all prices as of equal weight. Cotton weighed no more than pimento, and iron no more than umbrellas. Accordingly an improvement was made in the procedure, first by giving the prices of several different articles into which cotton, iron, and other important commodities entered, and only one price each in the case of the minor articles, and secondly by fixing on the price of some one article representing iron or cotton, and multiplying it by some number selected with the view of assigning to these articles their proper weights relatively to each other and to the rest. The objection to both these plans is the same,—that the numbers attached to the various articles or groups of articles are purely arbitrary; and of late years attempts have been made to obtain what may be called natural index numbers, the most successful so far being that of Mr Robert Giffen, whose index numbers are obtained from the declared values of the imports or exports into or from the United Kingdom of the articles whose prices are dealt with. In the case of both imports and exports Mr Giffen worked out the proportion borne by the value of each article to the total value for a series of years. Deducting the "unenumerated " articles, a series of numbers was thus obtained which could be used as the means of weighting the prices of the articles in an investigation of a movement of prices. This procedure is no doubt susceptible of further improvement, like its predecessors, but it is a great advance on the arbitrary systems of index numbers employed in them.

The Desirability of Increased Uniformity in Statistics. — One of the most serious difficulties in connexion with statistical investigations is the variety of the modes in which primaries of the same order are obtained, as regards dates and periods. This is a matter of which all persons who have occasion to use statistics are made painfully aware from time to time. Some attempts have lately been made to introduce more harmony into the official statistics of the United Kingdom, and some years ago a committee of the Treasury sat to inquire into the matter. The committee received a good deal of evidence, and presented a report, from which, how-ever, certain members of the committee dissented, preferring to express their views separately. The evidence will be found very interesting by all who wish to obtain an insight into the genesis of the official statistics of the country. The report and evidence were published in the June number of the Journal of the Statistical Society for 1881, as well as in the usual official form.

The International Institute of Statistics. —The absence of uniformity in statistics which is felt in England is not so marked in foreign countries, where the principle of centralization in arrangements of a political character is more powerful than it is here. In several Continental countries and in the United States there are statistical bureaus with definite duties to perform. In the United Kingdom, as already remarked, the nearest approach to a central statistical office is the Commercial and Statistical Department of the Board of Trade, on which the work of furnishing such statistics as are not definitely recognized as within the province of some other state department usually falls. Various attempts have been made to introduce more uniformity into the statistics of all countries. It was with this object that statistical congresses have met from time to time since 1853. An endeavour was made at the congress held in 1876 at Budapest to arrange for the publication of a system of international statistics, each statistical bureau undertaking a special branch of the subject. The experiment was, however, foredoomed to be only a very partial success, first because all countries were not then and are not yet furnished with central statistical offices, and secondly because the work which fell on the offices in existence could only be performed slowly, as the ordinary business of the offices necessarily left them little leisure for extra work. In 1885, at the jubilee of the London Statistical Society, a number of eminent statistical officials from all parts of the world except Germany were present, and the opportunity was taken to organize an International Institute of Statistics with a view to remedying the defects already ascertained to exist in the arrangements made by the congresses. The only obstacle to securing a proper representation of all countries was the absence of any German delegates, none of the official heads of the German statistical office being allowed to attend,—apparently on political grounds. Since then assurances of a satisfactory kind have been given to the German Government that their servants would be in no way committed to any course disapproved by that Government if they gave their assistance to the institute, from the formation of which it is hoped that much advantage may result. For information as to the constitution and objects of the institute reference may be made to a paper by Dr F. X. von Neumann-Spallart in vol. i. (1886) of the Bulletin de l'Institut International de Statistique (Rome, 1886).

Literature. — Maurice Block, Traité Théorique et Pratique de Statistique, Paris, 1878; Luigi Bodio, Della Statistica nei suoi Rapporti coll' Economía Politica, &c, Milan, 1869; Antonio Gabaglio, Storia e Teovia Generale della Statistica, Milan, 1880; Max Hausbofer, Lehr- u. Handbuch der Statistik, 2d ed., Vienna, 1882; K. Knies, Die Statistik als selbständige Wissenschaft, Cassel, 1850; Georg Mayr, Die Gesetzmässigkeit im Gesellschaftsleben, Munich, 1877 (abridged translation in Journ. Stat. Soc., Sept. 1883; the work has also been translated into Italian with valuable notes by G. B. Salvioni, Turin, 1886); Adolphe Quetelet, various works, but especially that entitled Sur l'Homme et le Développement de ses Facultés, ou Essai de Physique Sociale, 2 vols., Paris, 1835, and Letters on the Theory of Probabilities, already referred to; Albert. C. F. Schäme, Bau und Leben des socialen Körpers, Tübingen, 1881; Herbert Spencer, Principles of Sociology, especially part ii. pp. 465 sq.; Adolf Wagner, article "Statistik" in Buntschli-Brater's Staatswörterbuch, vol. x. (W. HO.)

The above article was written by: Wynyard Hooper, M.A.