Statistics

Statistics. The word " statistic " is derived from the Latin status, which, in the 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 18th century the denotation of the word has been extended, while at the same time its scope has become more definite, and may now be said, for all practical purposes, to be fixed.

History

The origin of what is now known as " statistics " (Ger. die Statistik; Fr. la statistique; Ital. statistica) can only be referred to briefly here. 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 constantly used by statesmen, solely with a view to administration. The Romans were careful to obtain accurate information regarding the resources of the state, and they appear to have taken the census with a regularity which has hardly been surpassed in modern times.

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 information available began to be made for purposes of investigation and not of mere administration. A volume compiled by Francesco Sansovino, entitled Del Govern() et amministrazione di diversi regni et republiche, 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. 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.

G. Achenwall is usually credited with being the first to use the word " statistics," but statistics, in the modern sense of the word, did not really come into existence until the publication (1761) by J. P. Sussmilch, a Prussian clergyman, of a work entitled Die glittliche Ordnung in den Veranderungen des menschlichen Geschlechts aus der Geburt, dem Tode, and 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. Sussmilch had arrived at a perception of the advantage of studying what Quetelet subsequently termed the " laws of large numbers." He combined the method of " descriptive statistics " 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. 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 various other writers, of whom the most distinguished was Sir William Petty, who published in 1683 his 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. 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.

It is now necessary to return to Siissmilch, who, as already mentioned, endeavoured to form a general theory of society, based on what were then termed " arithmetical " premises. In modern language, he made use of quantitative aggregateobservation as an instrument of social inquiry. It is true he did not enter on hit, investigation' with an " open mind." He desired to support a foregone conclusion, as the title of his work shows. But nevertheless his work was a most valuable one, since it pointed out a road which others who had no desire to procure evidence in favour of a particular system of thought were not slow to follow. Although for many years after the appearance of Siissmilch's book there was a good deal of resistance to the introduction of " arithmetic " as the coadjutor of moral and political investigations, yet, practically there was a tacit admission of the usefulness of figures, even by the chiefs of the socalled " descriptive " school. On the other hand, Siissmilch's success was the origin of a "mathematical" school of statisticians, some of whom carried their enthusiasm for figures so far that they refused to allow any place for mere " descriptions " at all. These two schools have now coalesced, each admitting the importance of the point of view urged by the other. They were, however, still perceptibly distinct even as late as 1850, and the ignorant hostility with which many people even among the cultivated classes still regard statistical inquiries into the nature of human society may be regarded as a survival of the much stronger feeling which showed itself among " orthodox " professors of law and economics on the publication of Siissmilch's treatise.

To the impulse given by the great Belgian, Quetelet, must he attributed the foundation in 1834 of the Statistical Society of London, a body which, though it has contributed little to the theory of statistics, has had a considerable influence on the practical work of carrying out statistical investigations in the United Kingdom and elsewhere. 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 (l'homme moyen) and his disquisition on the " curve of possibility " were most important contributions to the technical development of the statistical method.

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 selbstandige Wissenschaft (Cassel, 1850), is especially noteworthy as showing that the nature of the controversy was changing. 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 igth century accepted the view of Knies, but the majority of modern writers on the theory of statistics, especially in Germany, have adopted a slightly different standpoint 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 was ably maintained by von Mayr, Haushofer, Gabaglio and Block, whose views, published fifteen to twenty years before the close of last century, still substantially represent the opinions held by the majority of statisticians in Germany, and probably on the European continent. In France, however, several writers of importance have recently published works on the subject in which, in spite of the influence of M. Block, the claim of statistics to be considered as an independent sociological science has been rejected. There has been little systematic exposition of the subject in the United Kingdom. Isolated dicta have been furnished by authorities on the practice of statistics, such as the late Dr W. A. Guy, Professor J. K. Ingram, Sir Rawson W. Rawson, Sir Robert Giffen and others, Professor Foxwell has lectured on statistics at University College, London. The most important English work dealing with the matter is that of Mr A. L. Bowley. His volume, Elements of Statistics (first published in 1901), is intended as a practical handbook for teaching the principles on which statistics should be handled. The nature of Mr Bowley's book is, indeed, an indication of the fact that in the United Kingdom the study of statistics has been, in the main, of a practical character, the investigation of the theoretical basis of the statistical method attracting little interest. On the other hand, numerous monographs have been published by English writers on particular points connected with the technique of statistical investigation, as was natural considering the excellence of the practical use made of statistics in the United Kingdom.

With regard to the few earlier invasions of the domain of theory attempted by English writers, it may be observed that the authorities above mentioned were not unanimous. Dr Guy as well as Sir Rawson W. Rawson both claim that statistics is to be regarded as an independent science, apart from sociology, while Professor Ingram maintained that statistics cannot occupy a position co-ordinate with that of sociology, and that they " constitute only one of the aids or adminicula of science." Sir Robert 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 W. Rawson had 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 societ y en tant qu'elle peut titre exprimee par les chiffres." He proposes to give a new name to the branch of study thus defined, namely " demography." Von Mayr's definition is longer. He defines the statistical science as " die systematische Darlegung and Erdrterung der thatsachlichen Vorgange and der aus diesen sich ergebendcn 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 von 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." Most German writers on the subject have endorsed the views of Block and von Mayr. Among them may be mentioned Professors J. Conrad, Lexis and Westergaard, but Dr Augst Meitzen of Berlin, a second edition of whose Geschichte, Theorie and Technik der Statistik was published in 1903, makes a much less wide claim. In France opinions are divided, Professors Andre Liesse and Fernand Faure and others accepting the view that statistics is essentially a method.

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. 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.

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 er 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 caste. 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. It is also necessary that he should be acquainted with the theory of probability as applied to statistical investigations, the need of which is well set forth by Mr A. L. Bowley in Part II. of his work, already referred to, and by other writers. Valuable instruction on this technical subject can be obtained from monographs by Professor F. Y. Edgeworth, Professor Karl Pearson, Dr John Venn, Mr Udney Yule and many other contributors to the Transactions of the Royal Society, the Journal of the Royal Statistical Society, the Economic Journal, the Quarterly Journal of Economics and similar publications in different countries.

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 & 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. - i. 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 be 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. The new regulations as to income-tax assessment and the new land taxes will furnish the government with much fresh information as to incomes; and the census of production ordered in the session of 1907 and already carried out as regards a number of trades will also be useful.

B. The primary statistical quantities for which individuals or corporations are responsible may be divided into three categories: t. Among those which are compiled in obedience to the law of the land are the accounts furnished by municipal corporations, by the Bank of England, 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 thecompanies themselves to their proprietors or members.

2. An enormous mass of statistical information is furnished xxv. 26 a 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. The investigations of Mr Charles Booth into labour and wages questions, carried out with care over many years, are a remarkable example of this.

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 of 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 or 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 obvious to most people until explicitly stated.

The technical work of the statistician consists largely in operations 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, viz. 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.

A verages. - Averages or means are for statistical purposes divided into two classes, the arithmetical and weighted. 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 weighted 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, viz. 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 weighted 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 weighted means. Inferences based on the former class of average should be subjected to the most rigid investigation.

There are many methods of weighting averages; for descriptions of these statistical processes the reader must be referred to the works on the technique of statistics. In chapter v. of Mr Bowley's volume, the subject is dealt with in a manner suitable for students.

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 weighted 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 moo men, and that their average height is 5 ft. 8 in., 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 ft. high, and a great many about 5 ft. 5 in. 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 ft. 4 in., and a moderate number of men about 5 ft. II in. It is also possible that there may be very few men whose height is exactly 5 ft. 8 in., 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 ft. 8 in. 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 ft. 9 in. and the other half all below 5 ft. 7 in. This latter supposition would most likely be found to be approximately correct if the men belonged to a race whose average height was 5 ft. 8 in., 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 differs 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 independently in each investigation. Fuller 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 case 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 attempts have been made to obtain what may be called natural index numbers, the most successful so far being that of Sir Robert Giffen, whose index numbers were 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 Sir Robert 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. The index numbers prepared and published every month by the Economist, and by Mr Augustus Sauerbeck, which are weighted, are of great value; owing to the frequency of their appearance they make it possible to watch the tendency of prices closely.

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 many 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, however, 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 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. 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 the late Dr F. X. von NeumannSpallart in vol. i. (1886) of the Bulletin de l'institut international de statistique (Rome, 1886). Meetings of the Institute have been held annually ever since its formation in various cities of the world.

Literature. - E. Blaschke, Vorlesungen fiber mathematische Statistik (die Lehre von den statistischen Masszahlen) (Leipzig and Berlin, 1906); Maurice Block, Traite theorique et pratique de statistique (Paris, 1878); Luigi Bodio, Della Statistica nei suoi rapporti toll' economia politica, &c. (Milan, 1869); Arthur L. Bowley, Elements of Statistics (London, 1901); J. Conrad, " Statistik," Grundriss zum Studieren der politischen Oekonomie, vierter Teil; (2nd ed., Jena, 1902) (vierter Teil); Elderton (W. Palin and Ethel M.), Primer of Statistics (London, 1910); F. Faure, Elements de statistique (Paris, 1906); A. Gabaglio, Storia e teoria della statistics (Milan, 1880); Max Haushofer, Lehrand Handbuch der Statistik (2nd ed., Vienna, 1882); K. Knies, Die Statistik als selbstandige Wissenschaft (Cassel, 1850); A. Liesse, La Statistique (Paris, 1905); R. Mayo-Smith, Science of Statistics (1895); G. von Mayr, Die Gesetzmassigkeit im Gesellschaftsleben (Munich, 1877), abridged translation in Journ. Roy. Stat. Soc. (Sept. 1883); idem, Statistik and Gesellschaftslehre, pts. i. and ii. (Freiburg, 1897); A. Meitzen, Geschichte, Theorie and Technik der Statistik (2nd ed., Stuttgart and Berlin, 1903); A. Quetelet, various works, but especially that entitled Sur l'homme et le developpement de ses facultes, ou Essai de physique sociale (Paris, 1837); and Letters on the Theory of Probability; idem, Lettres d s.a.r. le duc regnant de Saxe-Coburg et Gotha sur la theorie des probabilites (Brussels, 1846); A. C. F. Schaffle, Ban und Leben des socialen Kiirpers (Tubingen, 1881), especially pt. ii. pp. 463 seq., and pt. iv. pp. 493 sqq.; Herbert Spencer, Principles of Sociology (London, 1877), especially pt. ii. p. 465 seq.; A. Wagner, article " Statistik," in Buntschli-Brater's StaatswOrterbuch, vol. x.; H. Westergaard, Die Grundziige der Theorie der Statistik (Jena, 1890).

(W. Ho.)