tion given to the latter applies with equal force to the former. No significance, except that of judging the successive additions or subtractions, as the case may be, depending whether the curve is read "up to and including" or "after and including," can be attributed to the heights of the successive ordinates. No meaning can be attached to the lines connecting them, whether straight or smoothed, except as indicating general direction of change as cumulation proceeds. As is the case with all discrete series, whether simple or cumulative and whether frequency or historical, the lines connecting successive ordinates must be regarded only as aids to the eye and not as characterizations of ideal distributions. For historical series of this type and treated in this manner, smoothing is not generally necessary, and when employed often has a tendency to smother the truth and to suggest license in the use of graphic methods. Neither should be cultivated. IV. CONCLUSION Both diagrammatic and graphic presentation of statistical data rightly viewed constitutes the art of statistical expression. Neither is necessary, although both are significant as preliminary to comparison, the goal of statistical studies. The aim in this chapter has been to call attention to the most important considerations bearing upon the science connected with both, and not to the infinite uses which they may legitimately and illegitimately serve. It is their appeal, their smug finality, which suggest their virtues and at the same time conceal their weaknesses. Our purpose has not been to detract from their function, nor to agitate against their use, but solely to point out the cautions and conditions which make their employment scientific and their position secure. This much, it is felt, it is necessary to do in view of the marked tendency to popularize them and to regard them as ends. REFERENCES Bowley, A. L.-Elements of Statistics, Ch. VII, Sections I, II, III, Brinton, W. C.- Graphic Methods for Presenting Facts, Chs. IX, King, W. I. - Elements of Statistical Methods, Ch. XI, pp. 97-120. Thorndike, E. L. — An Introduction to the Theory of Mental and Social Measurements, Ch. III, pp. 28–41. Yule, G. U. An Introduction to the Theory of Statistics, Ch. VI, pp. 75-105. Marshall, Alfred "On the Graphic Method of Statistics" in the Jubilee Volume, Journal of the Royal Statistical Society, 1885. Fisher, Irving "The 'Ratio' Chart for Plotting Statistics," in Quarterly Publications of the American Statistical Association, June, 1917, pp. 577-601. CHAPTER VIII AVERAGES AS TYPES I. INTRODUCTION GENERAL STATEMENT THE progress of the treatment has carried us toward a single goal that of comparison. Step by step the conditions and limitations which must be imposed in the collection of primary and in the use of secondary data have been considered. In their various aspects, the collection and classification of data and the devices currently in use and advocated for use in diagrammatic and graphic presentation have been discussed. The limits of the latter have been emphasized and the purposes and the consequences of the former considered. Throughout all stages of the treatment the limitations of statistical method, when used alone, have been acknowledged and emphasis placed particularly upon the difficulties of reducing to numerical bases the vital considerations connected with economic phenomena. The complexity of economic problems, and the many angles from which they must be considered before weight can be attributed to conclusions based alone upon statistical data, ought to stand out distinctly as one of the net results of the discussion. If the collection, classification, and arrangement of statistical data present problems, i.e. if the processes involved offer difficulties, how much more serious must be the problems when, in order to explain, describe, or establish the causal relationships between phenomena, the results or con clusions arising out of the use of these processes become the tools with which we operate. It is then not only necessary that the conditions surrounding enumeration, observation, and summarization of statistical data be appropriate, but that the conclusions which are deduced from them be logically sound and properly employed! And yet, statistically, comparison is the end toward which all previous steps are but preparatory. The data of economics are highly complex. They relate to conditions, evidences of which are not reducible to absolute uniformity of expression. They exhibit themselves in varying and changing proportions. Economic phenomena exist as cause and effect of other phenomena, and not independently. They must be dealt with as related forces. If they are inherently complex, so likewise are the methods by which they are described or measured. Simple units will not often suffice. Definitions are difficult to formulate, and to adhere to them strictly in all stages of work is frequently impossible. Care, judgment, insight, and caution are eternally necessary to guard against mistaken views, the assignment of cause for effect, the omission of qualifying or significant facts, the formation of false judgments, etc. For the focusing of judgment which comparison requires, concentrated or summary expressions are necessary. We seek for units of analysis here as we sought for units of collection earlier. Data in all their inclusiveness and in all their detail cannot readily be compared as between periods, times, or conditions. Some single expression which gathers into itself all the significant characteristics of complex data is required. We seek in actual life for an average performance, an average load, an average student or clerk, an average day, an average market, average conditions, etc., in order to bring things into relation. In general discussions such con cepts are used loosely, and frequently important matters are settled by employing no more definite or restricted terms. The willingness to be content with a single expression as a substitute for complex detail is often an evidence of ignorance either of the difficulties in making comparison or of the limitations of summarizing expressions.1 Invariably to speak and write of economic problems in terms of averages connotes a willingness either to be content with general notions often so general as to be meaningless or indifferently to employ tabloid expressions as accurate characterizations of complex things. Short cuts to the goal of comparison are too often preferred to the circuitous but more certain paths. Attempts are too frequently made to compare or contrast economic phenomena by appeal to averages in the form of median, mode, or arithmetic mean, where in reality not only are comparisons invalid but the data themselves do not admit of so being summarized. That fundamental canon which cautions against relating things to conditions incapable of producing them is flagrantly violated, and assurance of correct thinking found in the belief that we are dealing with "average conditions." This complacent belief may suffice to lull the ignorant into a state of blind indifference, but to those who are unwilling to allow themselves thus to be beguiled it offers little guaranty of intellectual repose. Rarely, if ever, does a summary expression carry with it. the same amount of truth as do detailed data.2 An average often suffices to give one a more convenient and more easily 1 Watkins speaks of averages as "representative numbers" and as containing the gist, if not the substance, of statistics." G. P. Watkins, "Theory of Statistical Tabulation," Publication of the American Statistical Association, December, 1915, p. 752. 2 Venn, Dr. John, "On the Nature and Use of Averages," Journal of the Royal Statistical Society (London), Vol. LIV, 1891, pp. 429-448, at page 433. |