1 The measurements in column (a) are taken from the American Statistical Association Publication, Vol. 7, p. 60. The original data are in a monograph by Dr. Francis H. Herrick on "The American Lobster in the United States," Fish Commission Bulletin for 1895.

Groups should invariably be of equal widths. Where this rule is violated false conclusions are likely to be drawn by comparing frequencies.1 Not only is error likely to result from hasty comparisons of this character but through the employment of unequal sized groups subsequent analysis by approved statistical methods is rendered difficult, if not impossible. The force of this generalization will be more apparent after we have discussed Dispersion and Skewness. If for any reason it is desired to change the size of groups, in order to distribute the number of frequencies more in detail, as for instance, in statistics of ages, summaries of the detailed groups should be made and all successive ones be framed in terms of multiples of the narrower ones chosen. The table on the next page showing the distribution of wage-rates of operators in woolen and worsted mills in the United States serves as an illustration of the use of unequal groups and is suggestive of the errors into which one may be led through their use.

Ignoring the widths of the groups and assuming them as equidistant a very usual thing to do unless one is accustomed to studying such data-it appears that the regular descending order of the frequencies for both male and the total, beginning at the group 10 to 11.99 cents, is abruptly broken at the frequency 2604 for the total, and at 2109 for the males, thus giving a second point of concentration of the wage earners. Of course, the rapid rise of these two instances as well as the retarded decrease in the case of the females is explained by the size of the groups. This table may only rightly be interpreted if full cognizance is taken

1 See the discussion of this point by Falkner, R. P., in connection with an analysis of "Income Tax Statistics," Publications of the American Statistical Association, N. S. No. 110, Vol. XIV, June, 1915, pp. 521-550, at pp. 422, 523, 537. See also the controversy over the meaning of the income tax statistics, published by the Department of Internal Revenue, in The Annalist, December 18, 1915, by Carl Snyder, and January 8, 1917, by William P. Malburn, Assistant Secretary of the Treasury.

of the fact that the distribution applies to groups with limits of 2, 5, 6, 10, and 15 cents, as well as to one group which is open at the upper side. If the table had been properly constructed the order of the units hourly rates of wages would have been inverted and uniform size groups employed, or groups used which were reducible to multiples of each other. Where it is impossible to use uniform groupings, breaks should be made in the body of the table to call attention to this fact.

TABLE R

FREQUENCY TABLE.SHOWING the Number oF THE OPERATIVES IN WOOLEN AND WORSTED MILLS IN THE UNITED STATES, BY SEX AND BY HOURLY RATES OF WAGES

(Report of the Tariff Board on Schedule K, Vol. IV, part 5. House Document No. 342, 62d Congress, 2d session, p. 997)

In writing the limits of groups it is generally well to use no smaller fraction of the whole unit than was employed in the actual process of measurement. For instance, wages expressed in cents would not ordinarily call for a fractional part of a cent being employed to designate the widths of the groups. Likewise, if measurements are made to the nearest half inch the limits of the groups would not normally be indicated by quarter inches. It is generally desirable, in order to guard against confusing the upper limits of a lower group with the lower limits of an upper group, to avoid writing the two in the same form. For instance, the group 30 to 40 may, for convenience, be written 30 to 39.9. In this form it is clear that a frequency of 40 belongs in the group 40 to 49.9. It may not always be so clear in case the limits are expressed in duplicate form.

TABLE S

TABLE SHOWING THE PERCENTAGE RELATION OF THE ASSESSMENT OF PERSONAL PROPERTY TO TOTAL ASSESSMENT

(Report of the Joint Legislative Committee of the State of New York, Albany, 1916, p. 260)

The preceding example is illustrative of some of the occasions for confusion resulting from a violation of this principle. In this brief table the second and ninth groups are indefinite in their upper boundaries. According to the way in which they are stated, items of three and twenty-one per cent, respectively, are not to be included, yet it is certain from the succeeding groups that they are included. If they are, the order is an exception to that which characterizes the majority of the groups. As a result, one is left in doubt as to what is intended. Moreover, the groups are so different in size that discredit is thrown upon the whole table.

'VIII. CONCLUSION

A detailed summary of this chapter seems unnecessary. The aim has been to consider only the most important aspects of the subject. The more general phases of classification and their bearing upon scientific method have for the most part been taken for granted. They need no extended consideration in this connection. We have striven only to show the application of classification to statistical facts.

The technique of tabulation has been approached with the problem of the statistician in view, the aim being to call attention to and to warn against certain indefensible practices commonly followed, and, at the same time to formulate as nearly as can be done, rules of general application. Attention is drawn to the characteristic differences in statistical data and to the proper means of bringing them out in tabular form. A logical background is always assumed for the existence of tables, and the reciprocal relation of a point of view and its tabular presentation taken for granted. Tabula

1 These are admirably treated in Venn, John, Empirical Logic, and in The Logic of Chance, as well as in Jevons, W. S., The Principles of Science.