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definitely limited magnitudes which form parts of a larger con- ceptual whole — for example, series showing the population of the several states of the Union. Averages computed from such series are usually arithmetic means ; for ordinarily the conditions required for the application of other forms of averages are not present. But, whatever their form, the averages commonly possess small sig- nificance, because the number of members in the series, which serves as divisor of the total, is arbitrary. Zizek's third group of series is characterized by the fact that the members are ratios, instead of measurements. These ratios do not themselves reveal the actual size of the magnitudes from which they have been computed, and hence commonly require to be "weighted" in striking averages. Of this classification of series the writer makes frequent use; for in most of his subsequent discussion he treats series of these three types in turn.
After a brief discussion of the inferiority of estimated averages to averages computed from ascertained measurements or ratios, Dr. Zizek proceeds to treat in detail the requirement of uniformity of data, the postulate of the homogeneity of series, and the consti- tution of groups from which averages are to be struck. The first, or general, part of the book closes with a chapter upon the ends for which averages are used — as possessing independent interest, as devices for making comparisons, as standards for judging individual cases, or as bases for measuring the dispersion of series. The second part is devoted to the characteristic merits and defects of various forms of averages — arithmetic means, simple and weighted, geo- metric means, medians, and modes. In view of the writer's wide reading, it is surprising to find him unacquainted with Mr. C. M. Walsh's elaborate treatment of the same range of problems in the special case of index-numbers. The concluding part deals with the dispersion of statistical series about their averages. Here the framework is provided by the threefold classification of series.
Judged from the standpoint of usefulness, the book is in danger of falling between two stools. Since it deals with only one stage of statistical work — the striking of averages from data already col- lected — it cannot compete with works of wider scope, like Bowley's Elements. And since the writer confesses his inability to employ refined statistical methods, he cannot appeal strongly to the specialists who read the papers of Edgeworth, Pearson, Lexis, and their peers. But there may be others, never or no longer on inti-
