When data are arranged in frequency groups and distributions are irregular, showing no tendency to be dispersed in a definite order around a central norm, it is frequently desirable successively to widen the groups, at the same time altering the frequencies to correspond, until regularity appears. However, there is always the danger of so concealing the individual peculiarities of the data, when dealing with discrete series particularly, as to negative any real value which they may possess. Frequently, the desire for regularity of distribution is so strong that its securing is made an end. Group adjustment should properly be looked upon as a means of correcting a false impression, as for instance, when data clearly of the continuous type have been distorted, by the limitations of the units in which they are expressed or by inadequacy of sampling, from the order which they should properly assume.1 It is always a problem to know how far to carry this synthesizing process. There is no rule-of-thumb principle which will answer the question. In effect, it is a process of smoothing and therefore, in discrete series, sacrifices individual characteristics in order to secure general impressions. The peculiarities of the whole series dominate the peculiarities of the parts. It should be remembered that for most data, particularly discrete, group widening results in a real sacrifice unless through it error is eliminated. This topic was discussed for both types of series in Chapter V, and can, therefore, be disposed of with this word of caution, and with brief reference to the following table and the corresponding graphs.

1 See the Table showing the measurements of lengths of lobsters, Chapter V, p. 152.

TABLE Q

TABLE SHOWING THE FREQUENCY OF RATIOS OF BUILDING VALUES TO LAND VALUES FOR BUILDINGS TEN STORIES OR MORE IN HEIGHT, NEW YORK CITY, 1914

By successively widening the groups in which the ratios of building to land values are expressed in Table Q, it is possible to reduce the frequencies to a gradually ascending and descending order but not without destroying somewhat the peculiarities of the distribution as revealed in the column marked 1. Graphically, the result of widening the groups is shown on Plate 19.

VI. THE PROPERTIES OF AVERAGES OR THE AVERAGE TO USE 1

Probably the properties of the different averages discussed above can more clearly be seen if the conditions are formulated which help to determine which average to use for a number of widely different cases.

Suppose we were interested in the experience of a salesman as a basis for promotion to a new territory or to an advanced wage or salary scale. The sales record of this man is given over a sufficient period, the sales being listed by territory, by grade of commodity, by prices of the article sold, by profits realized by the firm, by the length of time utilized in making them, by cost to the firm in present salary and expenses, etc., the supposition being that the sales are in the detail that is current with the best appointed sales records. Without making an elaborate judgment on the basis of all the data listed above and such other as may be available, could one employ an average of the sales for the purpose in mind, and if so in which one could he place most reliance? Is the arithmetic mean, an average of good and bad days, of sales among all classes of buyers, of those requiring one call and those requiring close following up, of small and large sales, of those upon which little as well as

1 This topic is further considered in Chapter IX.

large profits are realized, etc., to be taken as a measure of a salesman's activity, test of fitness, or worth to a company? Or are we interested in that average which takes account of the bad days and the small sales, of the good days and the large sales, but which gives no more importance to one of them than to another, realizing that the best of salesmen occasionally have off days and poor territory and that these will have to be reckoned with? Such a line of thought suggests the advisability of using the median. But, comes the retort from one who approaches the problem from another angle: "This man has had a consistent record of a high order and it is neither fair to the man nor to the company to give weight to his misfortunes. The facts show that we can expect him to make such and such a record the overwhelming percentage of his sales are of this character; or, in other words, the percentage of the time in which he fell below a high standard is negligible and should be given no weight. If his mistakes and failures are counted, we shall be putting a premium upon mediocrity and not be giving sufficient recognition to real merit." Such an argument suggests the wisdom of using the mode as a test of fitness.

It may be argued that it is unwise to let any one set of circumstances govern, no matter from what angle the problem is approached, and, undoubtedly, this is true. However, no matter how carefully the promotion is considered, if the facts above indicated are held to be germane, it is necessary to decide upon the weight to be assigned to the approaches indicated in the various averages. It is, of course, conceivable that the various averages would not be materially different. If this is true, the case for using one at all is strengthened. As to whether averages can be used is one question: which one to use, in case they are allowable, is quite another. It is the latter question which is now