STATISTICS .465 of interest, either to obtain additional information needful for its clearness. There might be columns showing the increase or administrative purposes, or, in countries possessed of representative decrease for each county and for the whole kingdom during the institutions, to supply statistics asked for by parliaments or con- ten years, and another column showing what proportion, expressed gresses. It is not necessary to refer particularly to this class of in percentages, these increases or decreases bore to the figures for statistical information, except in the case of the census. This is the earlier of the two years. Then there might be two columns an inquiry of such great importance that it may be regarded as one showing what proportions, also expressed as percentages, the of the regular administrative duties of Governments, though as the figures for each county bore in each year to the figures for the census is only taken once in a series of years it must be mentioned whole kingdom. The nine-column table thus resulting would under the head of occasional or special inquiries undertaken by still be simple, all the figures being merely explicit assertions of Governments. In the United Kingdom the work is done by the facts which are contained implicitly in the original " primaries." registrars-general who are in office when the period for taking the Complex Tables. - Suppose now we have another table precisely census comes round. On the Continent the work is carrie out similar in form to the first, and also relating to the counties of d E by the statistical bureaus of each country, - except France, where England, but giving the number of houses existing in each of them it is under the supervision of the minister of the interior. For at the same two dates. A combination of the two would form a c further information on this subject reference may be made to the complex table, and an application of the processes of arithmetic would make evident a number of fresh facts, all of which would be excellent chapter in M. Maurice Block's Traild entitled " Recensement." See also " Instructions to the Superintendent Registrar implied in the table, but would not be obvious to most people until of Births and Deaths as to his duties in taking the Census," 1871 ; explicitly stated.

also CENSUS, vol. V. p. 334 sq. The technical work of the statistician consists largely in operaB. The primary statistical quantities for which individuals or tions of which the processes just referred to are types.

corporations are responsible may be divided into three categories. Proportions. - The most usual and the best mode of expressing Among those which are compiled in obedience to the law of the proportion borne by one statistical quantity to another is to the land are the accounts furnished by municipal corporations, by state it as a percentage. In some cases another method is adopted railway, gas, water, banking, insurance, and other public companies - namely, that of stating the proportion in the form "one iu so making returns to the Board of Trade, by trades unions, and by many." This method is generally a bad one, and its use should other bodies which are obliged to make returns to the registrar of be discouraged as much as possible, the chief reason being that friendly societies. The information thus obtained is published in the changing portion of this kind of proportional figure becomes full by the departments receiving it, and is also furnished by the greater or less inversely, and not directly, as the phenomenon it companies themselves to their proprietors or members. represents increases or diminishes.

There are cases in which individuals have devoted themselves - namely, prices. Tho price of a given article is the approximate with more or less success to obtaining original statistics on special mathematical expression of the rates, in terms of money, at which points. The great work done by Messrs Bann and Wagner in exchanges of the article for money were actually made at or about arriving at an approximate estimate of the population of the earth a given hour on a given day. A quotation of price such as appears does not belong to this category, though its results are really in a daily price list is, if there has been much fluctuation, only a primary statistical quantities. Many of these results have not been very rough guide to the actual rates of exchange that have been arrived at by a direct process of enumeration at all, but by ingeni- the basis of the successive bargains making up the day's business. ous processes of inference. It need hardly be said that it is not But let us suppose that the closing price each day may be accepted easy for individuals to obtain the materials for any primary as a fair representative of the day's transactions, and let us further statistical quantity of importance, but it has been done in some suppose that we desire to obtain the average price for thirty days. cases with success. Now the sum of the prices in'question divided by thirty would be Operations Performed on Primary Statistical Quantities. - Only the arithmetical mean, and its weak point would be that it made a brief description of matters connected with the technique of the no allowance for the fact that the business done on some days is statistical method can be given in this article. In order to form much larger than that done on others; in other words, it treats statistics properly so called the primary statistical quantities must them as being all of equal weight. Now if, as is actually the case be formed into tables, and in the formation of these tables lies the in some markets, we have a daily account of the total quantities, art of the statistician. It is not a very difficult art when the prin- sold we can weight the members accurately, and can then obtain ciples relating to it have been properly grasped, but those who are their geometrical mean. There are cases in which the careless use unfamiliar with the subject arc apt to underrate the difficulty of of arithmetical means misleads the student of the social organism correctly practising it. seriously. It is often comparatively easy to obtain arithmetical Simple Tables. - The first thing to be done in the construction means, but difficult to obtain geometncal means. Inferences based a table is to form a clear idea of what the table is to show, and to on the former class of average should be subjected to the most express that idea in accurate language. This is a matter which is rigid investigation.

often neglected, and it is a source of much waste of time and Before closing this short survey of the very important subject of inches, and if they had been collected by chance. The extent of the divergence of the items composing an average from the average itself may be accurately measured and expressed in percentages of the average, the algebraic signs + and - being employed to indicate the direction of the variation from the mean. Au average may, therefore, advantageously be supplemented - (1) by a figure showing what proportion of the members from which it is derived differ from the average by a relatively small quantity, and (2) by figures showing the maximum and minimum deviations from the average. The meaning of the term "relatively small" must be considered independently in each investigation. Further remarks on averages will be found in the works mentioned at the conclusion of this article.

Prices. - Reference has already been made to the peculiar class of statistical quantities known as prices. Prices in their widest sense include all figures expressing ratios of exchange. In modern society the terms of exchange are always expressed in money, and the things for which money is exchanged are - (1) concrete entities with physical attributes, such as iron or wheat ; (2) immediate rights, such as those given by interest-bearing securities of all kinds, by bills of exchange, by railway or steamship contracts to carry either passengers or goods, and by bargains relative to the foreign exchanges ; (3) contingent rights, such as those implied in policies of insurance. All these rates of exchange belong to the same category, whether they are fixed within certain limits by law, as in the 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 of late years attempts have been made to obtain what may be called natural index numbers, the most successful so far being that of Mr Robert Giffen, whose index numbers are obtained from the declared values of the imports or exports into or from the United Kingdom of the articles whose prices are dealt with. In the case of both imports and exports Mr Giffen worked out the proportion borne by the value of each article to the total value for a series of years. Deducting the " unenumerated " articles, a series of numbers was thus obtained which could be used as the means of weighting the prices of the articles in an investigation of a movement of prices. This procedure is no doubt susceptible of further improvement, like its predecessors, but it is a great advance on the arbitrary systems of index numbers employed in them.

The Desirability of Increased Uniformity in Statistics. - One of the most serious difficulties in connexion with statistical investigations is the variety of the modes in which primaries of the same order are obtained, as regards dates and periods. This is a matter of which all persons who have occasion to use statistics are made painfully aware from timo to time. Some attempts have lately been made to introduce more harmony into the official statistics of the United Kingdom, and some years ago a committee of the Treasury sat to inquire into the matter. The committee received a good deal of evidence, and presented a report, from which, 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 report and evidence were published in the June number of the Journal of the Statistical Society for 1881, as well as in the usual official form.

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

Literature. - Manriec Block, Traia Therorique et Pralique de Statistique, Paris, 1878; Luigi Bodlo, Della Statistica nei suoi Rapporti colt' Economia Politica, ac., Milan, 1869; Antonin.Gabaglio, Storia e Teoria Generale della Statistica, Milan, 1880; Max Hanshofer, Lehr- u. Handbuch der Statislik, 2d ed., Vienna, 1882; K. Knees, Die Statistik ale selbsteindige Wissenschaft, Cassel, 1800; Georg Mayr, Die Gesetztniissigkeit ins Gesellschaftsleben, Munich, 1877 (abridged translation in Journ. Stat. Sec., Sept. 1883; the work has also been translated into Italian with valuable notes by G. B. Salvionl, Turin, 1886) ; Adolphe Quetelet, various works, but especially that entitled Sur l'Homme et le Developpentent de see Facultes, ou Essai de Physique Sedate. 2 vols., Paris, 1835, and Letters on the Theory of Probabilities, already referred to; Albert. C. F. Sehtiftle, lieu and Leben des socialen Korpers, Tiibingen, 1881 ; Herbert Spencer, Principles of Sociology, especially part B. pp. 465 sq.; Adolf Wagner, article " Statistik" in Buntsclill-Brater's Staatswitrterbuch, vol, x. (W. 110.)