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STATISTICS

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Originally appearing in Volume V25, Page 811 of the 1911 Encyclopedia Britannica.
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STATISTICS . The word " statistic " is derived from the Latin status, which, in the See also:

middle ages, had come to mean a " See also:state " in the See also:political sense. " Statistic," therefore, originally denoted inquiries into the See also:condition of a state. Since the 18th See also:century the See also:denotation of the word has been extended, while at the same See also:time its See also:scope has become more definite, and may now be said, for all See also:practical purposes, to be fixed. See also:History.—The origin of what is now known as " statistics " (Ger. See also:die Statistik; Fr. la statistique; Ital. statistica) can only be'referred to briefly here. As human See also:societies became more and more highly organized, there can be no doubt that a very considerable See also:body of See also:official statistics must have come into existence, and been constantly used by statesmen, solely with a view to See also:administration. The See also:Romans were careful to obtain accurate See also:information, regarding the resources of the state, and they appear to have taken the See also:census with a regularity which has hardly been surpassed in See also:modern times. Statistics, or rather. the material for statistics, therefore existed at a very See also:early See also: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 See also:mere administration. A See also:volume compiled by See also:Francesco See also:Sansovino, entitled Del Govern et amministrazione di diversi regni et republicke, was printed in See also:Venice and bears the date 1583. Other See also:works of a similar See also:kind were published towards, the end of the 16th century in See also:Italy and See also:France. Works on state administration and See also:finance continued to be published during the first See also:half of the 17th century, and the tendency to employ figures, which were hardly used at all by Sansovino, became more marked, especially in See also:England, where the facts connected with " bills of mortality " had begun to attract See also:attention. G.

See also:

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 See also:work entitled Die gottliche Ordnung in den Veranderungen See also:des menschlichen Geschlechts aus der Geburt, dem Tode, and der Fortpfianzung desselben erwiesen. In this See also:book a systematic See also:attempt was made to make use of a class of facts which up to that time had been regarded as belonging to " political See also:arithmetic,'•' under which description some of the most important problems of what modern writers See also:term " vital statistics " had been studied, especially in England. Sussmilch had arrived at a See also:perception of the See also:advantage of studying what See also:Quetelet subsequently termed the " See also:laws of large See also: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 a statistical method. It is true that a few books were published attempt at generalizing from them. between 1830 and 185o in which the politico-See also:geographical Political arithmetic had come into existence in England description of a See also:country is spoken of as " statistics," which is in the middle of the 17th century. The earliest example thus distinguished from " political arithmetic." The See also:title of of this class of investigation is the work of See also:Captain See also:John Knies's See also:great work, Die Statistik als selbstandige Wissenschaft Graunt of See also:London, entitled Natural and Political Annota- (See also:Cassel, 1850), is especially noteworthy as showing that the lions made upon the Bills of Mortality, which was first published nature of the controversy was changing. Knies claimed in 1666. This remarkable work, which dealt with mortality that the really " scientific " portion of statistics consisted of in London only, ran through many See also:editions, and the See also:line- of the figures employed. As Haushofer says, " his starting point inquiry it suggested was followed up by various other writers, is political arithmetic." of whom the most distinguished was See also:Sir See also:William See also:Petty, who Some eminent statisticians of the latter half of the 19th published in 1683 his Five Essays in Political Arithmetick. century accepted the view of Knies, but the See also:majority of modern Other writers, of whom See also:Halley, the celebrated mathematician writers on the theory of statistics, especially in See also:Germany, have and astronomer, was one, entered on similar investigations, adopted a slightly different standpoint according to which and during the greater See also:part of the 18th century the number of statistics is at once a See also:science See also:relating to the social See also:life of See also:man persons who devoted themselves to " arithmetical " inquiries and a method of investigation applicable to all sciences. This into problems of the class now known as statistical was steadily view was ably maintained by von Mayr, Haushofer, Gabaglio increasing. Much attention was given to the construction and See also:Block, whose views, published fifteen to twenty years before of tables of mortality.

Attempts were also made to See also:

deal the See also:close of last century, still substantially represent the opinions with figures as the basis of political and fiscal discussion by held by the majority of statisticians in Germany, and probably See also:Arthur See also:Young, See also:Hume and other See also:historical writers, as well as by on the See also:European See also:continent. In France, however, several writers the two Mirabeaus. of importance have recently published works on the subject It is now necessary to return to Stissmilch, who, as already in which, in spite of the See also:influence of M. Block, the claim of mentioned, endeavoured to See also:form a See also:general theory of society, statistics to be considered as an See also:independent sociological science based on what were then termed " arithmetical " premises. has been rejected. There has been little systematic exposition In modern See also:language, he made use of quantitative aggregate- of the subject in the See also:United See also:Kingdom. Isolated dicta have been observation as an See also:instrument of social inquiry. It is true he furnished by authorities on the practice of statistics, such as did not enter on his investigation' with an " open mind." He the See also:late Dr W. A. See also:Guy, See also:Professor J. K. See also:Ingram, Sir Rawson W. desired to support a foregone conclusion, as the title of his work Rawson, Sir See also:Robert See also:Giffen and others, Professor Foxwell has shows. But nevertheless his work was a most valuable one, since lectured on statistics at University See also:College, London. The it pointed out a road which others who had no See also:desire to procure most important See also:English work dealing with the See also:matter is that See also:evidence in favour of a particular See also:system of thought were not of Mr A.

L. Bowley. His volume, Elements of Statistics (first slow to follow. Although for many years after the appear- published in 1901), is intended as a practical handbook for ance of Sussmilch's book there was a See also:

good deal of resistance teaching the principles on which statistics should be handled. to the introduction of " arithmetic " as the coadjutor of moral The nature of Mr Bowley's book is, indeed, an indication and political investigations, yet, practically there was a tacit of the fact that in the United Kingdom the study of statistics See also:admission of the usefulness of figures, even by the chiefs of the so- has been, in the See also:main, of a practical See also:character, the in-called " descriptive "school. On the other See also:hand, Sussmilch's See also:sue- vestigation of the theoretical basis of the statistical method See also:cess was the origin of a " mathematical " school of statisticians, attracting little See also:interest. On the other hand, numerous mono-some of whom carried their See also:enthusiasm for figures so far graphs have been published by English writers on particular that they refused to allow any See also:place for mere " descriptions " points connected with the technique of statistical investigation, at all. These two See also:schools have now coalesced, each admitting as was natural considering the excellence of the practical use the importance of the point of view urged by the other. They made of statistics in the United Kingdom. were, however, still perceptibly distinct even as late as 1850, and With regard to the few earlier invasions of the domain of the ignorant hostility with which many See also:people even among the theory attempted by English writers, it may be observed that cultivated classes still regard statistical inquiries into the nature the authorities above mentioned were not unanimous. Dr of human society may be regarded as a survival of the much Guy as well as Sir Rawson W. Rawson both claim that statistics stronger feeling which showed itself among "orthodox " See also:pro- is to be regarded as an independent science, apart from See also:sociology, fessors of See also:law and See also:economics on the publication of Sussmilch's while Professor Ingram maintained that statistics cannot See also:treatise. occupy a position co-See also:ordinate with that of sociology, and To the impulse given by the great Belgian, Quetelet, must be that they " constitute only one of the See also:aids or adminicula attributed the See also:foundation in 1834 of the Statistical Society of of science." Sir Robert Giffen has also expressed himself London, a body which, though it has contributed little to the adversely to the See also:continental See also:doctrine that there is an in-theory of statistics, has had a considerable influence on the dependent science of statistics, and this See also:opinion appears to practical work of carrying out statistical investigations in be the correct one, but, as Dr Guy and Sir Rawson W. Rawson the United Kingdom and elsewhere.

Quetelet was above had the support of the great body of systematic teaching emanaall things an exponent of the " laws of large numbers." He ting from distinguished continental statisticians in support of was especially fascinated with the tendency to relative See also:

con- their view, while their opponents have so far only the obiter stancy of magnitude displayed by the figures of moral statistics, dicta of a few eminent men to rely upon, it appears needful to especially those of See also:crime, which inspired him with a certain examine closely the views held by the continental authorities, degree of See also:pessimism. His conception of an See also:average man and the grounds on which they are based. (l'homme moyen) and his disquisition on the" See also:curve of possibility " The clearest and shortest See also:definition of the science of statistics were most important contributions to the technical development as thus conceived is that of M. Block, who describes it as " la of the statistical method. science de l'homme vivant en societe en tant qu'elle pent etre The influence exercised by Quetelet on the development exprimee See also:par See also:les chiffres." He proposes to give a new name of statistics is clearly seen from the fact that, though there is to the See also:branch of study thus defined, namely " See also:demography." still considerable controversy among statisticians, the old Von Mayr's definition is longer. He defines the statistical controversy between the " descriptive " and arithmetical science as " die systematische Darlegung and Erorterung der schools has disappeared, or perhaps we should say has been thatsachlichen Vorgange and der aus diesen sich ergebenden transformed into a discussion of another kind, the question now Gesetze des gesellschaftlichen menschlichen Lebens auf Grundlage at issue being whether there is a science of statistics as well as 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 See also:order by means of mathematical See also:induction." Most See also:German writers on the subject have endorsed the views of Block and von Mayr. Among them may be mentioned Professors J. See also:Conrad, Lexis and Westergaard, but Dr Angst Meitzen of See also: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 See also:Andre Liesse and Fernand See also: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 See also:

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 See also:reason either of convenience or See also:logic why the use of a certain scientific method should be held to have created a science in one See also: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 See also:meteorology, See also:medicine, and other See also:physical sciences statistical inquiries are plainly and obviously examples of the employment of a method, like microscopy, spectrum See also:analysis, or the use of the See also:telescope. Why should the fact of their employment in sociology be considered as authorizing the See also:classification of the phenomena thus dealt with to form a new science ? The most effective See also:argument put forward by the See also:advocates of this view is the assertion that statistics are merely a convenient aid to investigation in the majority of sciences, but are the See also:sole method of inquiry in the See also: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 scientific-ally organized branch—of sociology, and statistics are largely used in it, but no one, so far as we are aware, has proposed to See also:call economics a department of statistical science. Although, however, the above considerations forbid the See also: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 See also:

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 See also:population. But the nation, considered sociologically, remains the same in spite of this slow See also:change in the particles composing it, just as a human being is considered to be the same See also:person See also:year by year, although year by year the particles forming his er her body are constantly being destroyed and fresh particles substituted. Qf.course the See also: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 nearlyresemble some of the See also:lower forms of See also:animal life than the more highly organized forms of animal existence. There are organ-isms which are fissiparous, and when cut in two form two fresh independent organisms, so diffused is the vitality of the See also: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 See also:groups year by year observed, is the statistical method. Accordingly the correct view seems to be that it is the See also:function of this method to make perceptible facts regarding the constitution of society on which sociology is to See also:base its conclusions. It is not claimed, or ought not to be claimed, that statistical investigation can See also: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 See also:resolution of See also: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 See also:

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 See also:amateur handlers of statistics generally are in the See also:habit of See also: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 " See also: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 See also:aggregation not perceptible to the senses are rendered perceptible to the See also: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 See also:familiar example of this is the difficulty which is sometimes experienced in finding the large names, as of countries or provinces, on a See also: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 See also:standard of measurement implied by the terms in each particular See also:caste. A careful study of the first few pages of De See also:Morgan's See also:Differential and Integral Calculus will materially assist the student of statistics in attaining a grasp of the principles on which See also: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 See also:common standard.

It is also necessary that he should be acquainted with the theory of See also:

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. See also:Edgeworth, Professor Karl See also:Pearson, Dr John See also:Venn, Mr Udney See also:Yule and many other contributors to the Transactions of the Royal Society, the See also:Journal of the Royal Statistical Society, the Economic Journal, the Quarterly Journal of Economics and similar publications in different countries. See also:Sources whence Statistics are Derived.'—The term " statistics " in the See also:concrete sense means systematic arrangements of figures representing " See also: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 See also:accretion during a given period. As examples may be mentioned the number of deaths in a given See also:district during a given time, the number of pounds See also:sterling received by the London & See also:North Western railway during a given time, and the number of " inches of See also:rain " that See also:fell at See also:Greenwich during a given time. Other examples are the number of tons of See also:pig-See also:iron lying in a particular See also: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 See also: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 See also:Government Statistics.—1.

A vast See also:

mass of statistical material of more or less value comes into existence automatically in modern states in consequence of the See also:ordinary administrative routine of departments. To this class belong the highly important statistical information published in England by the registrar-general, the returns of See also:pauperism issued by the See also:local government See also: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 See also:annual statements of See also:trade and See also: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 See also: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 See also: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 See also:budget, and the statistics prepared and published by the commissioners of inland See also:revenue, by the See also:post See also:office, and by the See also:national See also: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 See also: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 See also: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 See also:

regular administrative duties of governments, though as the census is only taken once in a See also:series of years it must be mentioned under the See also: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 See also: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 See also:minister of the interior. The new regulations as to income-tax See also:assessment and the new See also:land taxes will furnish the government with much fresh information as to incomes; and the census of See also:production ordered in the session of 1907 and already carried out as regards a number of trades will also be useful. I3. 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 the Bank of England, by railway, See also:gas, See also:water, banking, See also: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 furnished809 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 association, some of them of great importance, such as See also:Lloyd's, the London Stock See also:Exchange, the See also:British Iron Trade Association, the London See also:Corn Exchange, the See also: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 See also:

Wagner in arriving at an approximate estimate of the population of the See also:earth does not belong to this See also:category, though its results are really primary statistical quantities. Many of these results have not been arrived at by a See also:direct See also: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 See also:Charles See also:Booth into labour and See also: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 See also: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 See also: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 See also:apt to underrate the difficulty of correctly practising it. See also:Simple Tables.—The first thing to be done in the construction of a table is to form a clear See also:idea of what the table is to show, and to See also:express that idea in accurate language. This is a matter which is often neglected, and it is a source of much See also:waste of time and occasionally of misapprehension to those who have to study the figures thus presented. No table ought to be considered See also:complete without a " heading " accurately describing its contents; and it is frequently necessary that such headings should be rather See also: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 See also: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 See also:

county of England on a given See also:day of a given year, and also, in another See also:column, the corresponding numbers for the same counties on the corresponding day of the tenth year subsequently, would be a simple See also:tabular statement of the general facts regarding the See also: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 See also: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 See also: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 See also: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 See also:

bad one, and its use should be discouraged as much as possible, the See also: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 arithmetical and weighted. An arithmetical mean is the sum of all the members forming the series 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 See also: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 See also:consideration. It was soon seen, however, that this procedure was fatally defective, inasmuch as it treated all prices as of equal See also:weight. See also:Cotton weighed no more than See also: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 See also:price each in the case of the See also: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 See also: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 See also: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 See also:

month by the Economist, and by Mr See also:Augustus Sauerbeck, which are weighted, are of great value; owing to the frequency of their See also:appearance they make it possible to See also: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 See also:harmony into the official statistics of the United Kingdom, and many years ago a See also:committee of the See also:treasury sat to inquire into the matter. The committee received a good deal of evidence, and presented a See also: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 See also:genesis of the official statistics of the country. The See also:International Institute of Statistics.—The See also:absence of uniformity in statistics which is See also: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 See also: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 See also:

object that statistical congresses have met from time to time since 1853. An endeavour was made at the See also:congress held in 1876 at See also:Budapest to arrange for the publication of a system of international statistics, each statistical See also: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 See also:left them little leisure for extra work. In 1885, at the See also:jubilee of the London Statistical Society, a number of eminent statistical officials from all parts of the See also:world except Germany were See also: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 securs See also:ing a proper See also: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 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 See also: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 See also:peculiar kind, viz. prices. The price of a given article is the approximate mathematical expression of the rates, in terms of See also:money, at which exchanges of the article for money were actually made at or about a given See also:hour on a given day. A See also:quotation of price such as appears in a daily price See also:list is, if there has been much fluctuation, only a very rough See also: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 See also:fair representative of the day's transactions, and let us further suppose that we desire to obtain the average price for See also: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 See also: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 See also:

chapter v. of Mr Bowley's volume, the subject is dealt with in a manner suitable for students. Before closing this See also: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 See also:i000 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. 11 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) 66o of them were within an See also:

inch of that height, either by excess or deficiency, while of the See also: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 See also:race whose average height was 5 ft. 8 in., and if they had been collected by See also: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. See also:

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 See also: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 See also: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 See also:Tooke, See also:Newmarch and others into the general 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 in-formation as to the constitution and See also:objects of the Institute reference may be made to a See also:paper by the late Dr F. X. von See also:Neumann-Spallart in vol. i. (1886) of the Bulletin de l'institut international de statistique (See also:Rome, 1886). Meetings of the Institute have been held annually ever since its formation in various cities of the world. Statistik," in Buntschli-Brater's Staatsworierbuch, vol. x.; H.

Westergaard, Die Grundzuge der Theorie der Statistik (See also:

Jena, 1890). (W. Ho.) .

End of Article: STATISTICS

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