superiority over the old form of cross-hatching for the uses which are common is not proved. In many respects they are at a disadvantage in any such comparison. For other purposes, such as giving a notion of absolute frequency, they add little to the tabular form.

The

The third type of dot maps has decided merits and at the same time certain limitations. The size of the dot is immaterial; the relative frequency with which it occurs is everything. (See Plate 11.) Absolute frequency is secondary, though in theory it may be approximated, as in the other types of dot maps, by considering the number of dots in connection with the value assigned them. Such approximations are generally as unnecessary as they are impossible. Where frequency is great, the number cannot be determined, the individual dots losing their identity in the group. value assigned to the dot is largely arbitrary, since the purpose of the map is not to record absolute magnitude but to reveal relative abundance and scarcity in relation to position. The densities of the shaded areas are the important facts. Areas of uniform density are not political jurisdictions, as in colored and cross-hatched maps, but actual positions, so far as the sizes of maps will allow these to be shown. This form of illustration gives the impression of gradual changes from scarceness to abundance, from "highs" to "lows," and it seems to smooth out the breaks which would prevail were cross-hatching used. Geographical barriers are ignored in the drawing, but may be inserted for purposes of study and interpretation. It is easy to visualize places and degrees of concentration and "scatteration"; to get a continuous view of distribution. Dot maps of the third type suggest "continuous" rather than "discrete" series.

The technique of diagram and map construction is not here discussed nor even an attempt made to enumerate the

multitude of functions which diagrams serve in the hands of statisticians, publicists, advertisers, manufacturers, financial houses, etc. Numerous examples of well- and ill-drawn illustrations taken from these fields together with a discussion of free-hand and mechanical cross-hatching, the uses of pins in map making, preparation of copy for duplicating whether by photographing or otherwise, etc., are given in Brinton: Graphic Methods for Presenting Facts. Our interest is more in describing the functions, discovering and defining the limitations of diagrammatic presentation in statistical studies than in describing the processes of drawing and reproducing diagrams, and in indicating for various businesses the precise functions which they might have in exhibit or other work. Such matters are important but they are treated elsewhere very much more fully than we could hope to do at this time and with all the fullness that they merit.

If the reader understands the psychological bases upon which diagrammatic illustration rests, if he appreciates the position which it occupies with respect to tabulation and other steps in statistical analysis, and feels the warning, which it has been the purpose of much of the above to sound, against too free a use of or too complete a reliance in pictorial figures, he is in the proper attitude to use the process. Execution may be left to those who have acquired the requisite skill; the determination to use should be in the hands of those who have a correct attitude toward the problem. It is necessary that diagrams should be well drawn and that those who prepare them should have knowledge of the mechanical aids for drawing, duplicating, etc. Such a knowledge constitutes the art; knowledge of the principles underlying the use of diagrams constitutes the

1 Brinton, Willard C., Graphic Methods for Presenting Facts, The Engineering Magazine, New York, 1914.

science, and it is the latter in which we are more vitally interested.

It may be helpful in closing the discussion of the principles and forms of Diagrammatic Presentation to outline a few suggestions to be followed in its use.

IV. SUGGESTIONS TO BE FOLLOWED IN THE USE OF STATISTICAL DIAGRAMS

1. Choose illustrations which are least liable to be misunderstood, and which most faithfully and correctly interpret the facts.

2. See that fact and representation agree and that all diagrams are provided with concise, clearly stated, and appropriate titles.

3. Avoid figures which must be read according to more than one dimension.

4. Indicate on diagrams the scales of values used, and where necessary to avoid confusion, the dimension or dimensions which are significant in interpretation.

5. Include as a component or as an accompanying part of diagrams the concrete data which they illustrate.

6. In expressing the different parts of a total, use lines or bars or sectors of circles.

7. In statistical maps representing a series, divide the range of frequencies and not the number of districts or divisions into equal parts.

8. In statistical maps representing a series, incorporate as a part of the legend the frequency with which the units of measurements occur, thus indicating the distribution by map and by legend.