Graphic presentation represents a highly developed body of techniques for elucidating, interpreting, and analyzing numerical facts by means of points, lines, areas, and other geometric forms and symbols. Graphic techniques are especially valuable in presenting quantitative data in a simple, clear, and effective manner, as well as facilitating comparisons of values, trends, and relationships. They have the additional advantages of succinctness and popular appeal; the comprehensive pictures they provide can bring out hidden facts and relationships and contribute to a more balanced understanding of a problem.
The choice of a particular graphic technique to present a given set of data is a difficult one, and no hard and fast rules can be made to cover all circumstances. There are, however, certain general goals that should always be kept in mind. These include completeness, clarity, and honesty; but there is often conflict between the goals. For instance, completeness demands that all data points be included in a chart, but often this can be done only at some sacrifice of clarity. Such problems can be mitigated by the practice (highly desirable on other grounds as well) of indicating the source of the data from which the chart was constructed so that the reader himself can investigate further. Another problem occurs when it is necessary to break an axis in order to fit all the data in a reasonable space; clarity is then served, but honesty demands that attention be strongly called to the break.
A choice among graphic techniques also depends upon the proposed use to which the chart will be put. As Schmid (1954) has pointed out, graphs that are satisfactory in memoranda for private circulation may be inappropriate for a published book or paper.
In classifying charts and graphs, criteria of purpose, of circumstances of use, of type of comparison, and of form have been used. On the basis of form, charts and graphs may be classified as (1) rectilinear coordinate graphs; (2) semilogarithmic charts; (3) bar and column charts; (4) frequency graphs and related charts; (5) maps; (6) miscellaneous charts, including pie diagrams, scattergrams, fan charts, ranking charts, etc.; (7) pictorial charts; and (8) three-dimensional projection charts.
In graphic presentation, three different basic geometrical forms can be utilized for purposes of comparing magnitudes of coordinate items: (1) linear or one-dimensional, (2) areal or two-dimensional, and (3) cubic or three-dimensional. The simplest and most exact comparisons are those made on a linear basis; comparison of relative sizes of areas is more difficult and of volumes most difficult. Accordingly, where possible, the use of areal and cubic forms should be avoided in graphic presentation.
Rectilinear coordinate graphs . Perhaps the best known form of graphic presentation, and certainly one of the most frequently used, is the simple arithmetic line chart, one of several types of the rectilinear (or Cartesian) coordinate graph.
The basic form of this type of graph is derived by plotting one or more series of figures on a coordinate surface in which the successive plotting points are joined together in the form of a continuous line, customarily referred to as a “curve.” A curve on a graph of this kind is not necessarily smooth and regular but instead may be straight and angular. Figure 1 indicates the basic structural characteristics of a rectilinear coordinate system.
Figure 2 portrays the characteristic features and basic standards of the rectilinear coordinate graph. Many of the essential elements and specifications of the rectilinear coordinate graph are also applicable to other graphic forms.
Semilogarithmic charts . The semilogarithmic chart is often superior to the arithmetic chart (as used in Figure 1), for the former can show relative changes clearly; hence it is sometimes referred to as a ratio chart. Sometimes the semilogarithmic chart has the merit of representing as nearly straight lines functions that otherwise would be appreciably curved.
The essential feature of the semilogarithmic chart is that one scale is logarithmic and the other arithmetic, so that the chart effectively is a convenient device for plotting log y against x, or log x against y. There is no zero line on the logarithmic scale, since the logarithm of zero is minus infinity. The logarithmic scale consists of one or more sets of rulings calibrated in terms of logarithmic values; each complete set of rulings is referred to as a “deck,” “cycle,” “bank,” or “tier.” The rulings for each deck are the same, but the scale values change from one to the other. For example, if one deck runs from 1 to 10, the adjacent deck above will vary from 10 to 100, the third from 100 to 1,000, and so forth. On the other hand, the adjacent deck below the one from 1 to 10 would vary from 0.1 to 1.0. The logarithmic scale can thus be extended upward or downward indefinitely.
In a semilogarithmic chart, relative rate-of-change comparisons can be made readily between different parts of a single series or between two or more series. The relative rate of change of y with respect to x is the slope of y as a function of x, divided by x; equivalently it is the slope of log y except for a constant depending on the logarithmic base. The slope of the logarithmic scale variable is the relative rate of change of the variable. If the slope is steep, the relative rate of change is great. It makes no difference on what part of a semi logarithmic chart a curve is located; the same slope
means the same relative rate of change. This is particularly convenient if the two series being compared have very different ranges of values.
This situation is illustrated in Figure 3, where the same data have been plotted on both arithmetic and semilogarithmic grids, placed injuxta position. On the arithmetic grid, the relative rate of growth of the city in comparison to that of the entire state cannot be easily determined; by contrast, the curves on the semilogarithmic chart portray relative rate of growth clearly and correctly. Note, however, that this clarity is achieved only at the expense of some distortion; careful labeling is imperative to prevent the reader from mistakenly concluding that the absolute growth of Dallas is greater than that of the state as a whole.
Other special graph papers exist, such as double logarithmic, normal, and hyperbolic.
* Comparison clearly demonstrates the superiority of the semilogarithmic graph in portraying relative rates of change.
Source: Data from U.S. Bureau of the Census 1963, pp. 45-19, 45–22.
Bar and column charts . Bar and column charts are simple, flexible, and effective techniques for comparing the size of coordinate values or of parts of a total. The basis of comparison is linear or one-dimensional; the length of each bar or column is proportional to the value portrayed. Bar and column charts are very much alike; they differ mainly in that the bars are arranged horizontally in a bar chart and the columns are arranged vertically in a column chart. In addition, the bar chart is seldom used for depicting time series, whereas the column chart is often used for that purpose.
There are several different kinds of bar and column charts. Four of them are illustrated in Figure 4.
Frequency graphs and related charts . The most common graphic forms for portraying simple frequency distributions are the frequency polygon, the histogram, and the smoothed frequency curve. Frequency graphs are usually drawn on rectilinear coordinates, with the Y axis representing frequencies and the X axis representing the class intervals. The Y axis always begins with zero, and under no circumstances is it broken. The horizontal scale does not have to begin with zero unless, of course, the lower limit of the first class interval is zero. In laying out a frequency polygon the appropriate frequency of each class customarily is located at the midpoint of the interval, and the plotting points are then connected by straight lines. The typical histogram is constructed by erecting vertical lines at the limits of the class intervals and forming a series of contiguous rectangles or columns. The area of each rectangle represents the respective class frequencies. (If the class intervals are unequal, special care must be taken.) Smoothed frequency curves may be constructed by either mathematical or graphic techniques. The main purpose of smoothing a frequency graph is to remove accidental irregularities resulting mainly from sampling errors; the data from which the smoothed curve was obtained should always be presented as should some indication of the method of smoothing employed.
For some purposes, the cumulative-frequency curve, or ogive, is more useful than the simple frequency graph. In a cumulative-frequency distribution, the frequencies of the successive class intervals are accumulated, beginning at one end of the distribution. If the cumulation process is from the lesser to the greater, it is referred to as a “less than” type of distribution; if from the greater to the lesser, it is known as a “more than” type of distribution. In constructing an ogive, the cumulative frequencies are represented by the vertical axis and the class intervals by the horizontal axis. The cumulated frequencies are plotted either at the lower or upper end of the respective class intervals, depending on whether the cumulation is of the “less than” or “more than” type. A common configuration of the cumulative-frequency curve is that of an elongated S.
Concentration curves. A special kind of graph, related to cumulative frequency graphs, is known as a concentration curve or a Lorenz curve. Such a graph is used to portray the nature of non-uniformity in the distributions of inherently positive quantities like wealth, income, amount of retail sales, etc. The graph shows the proportions of total wealth (to be definite) held by various proportions of the relevant population; one reads from the graph that the least wealthy 10 per cent of the population holds, say, 1 per cent of total wealth, that the most wealthy 5 per cent of the population holds, say, 30 per cent of total wealth, etc. Thus the graph is a curve running through points given parametrically by cumulative relative frequencies and cumulative relative totals; it is usually presented in square form, with both axes taking values from 0 to 1 (for proportions) or 0 to 100 (for per cents). If wealth, or whatever, is uniformly (equally) distributed in the population, the concentration curve is a straight line, the diagonal of the square from (0,0) to (1,1). The more the concentration curve deviates from the straight line the greater is inequality.
Statistical maps . There are many varieties of maps used in portraying statistical data. They can be grouped under the following basic types: (1) crosshatched or shaded maps; (2) spot or point-symbol maps; (3) isoline maps; (4) maps with one or more types of graphs superimposed, such as the bar, column, line, flow, or pictorial forms (see Figure 7); and (5) a combination of two or more of the preceding types. [For an overview of map making, seeCartography.]
Crosshatched maps. The crosshatched or shaded map, characteristically, is used to portray rates and ratios that are based on clearly delineated areal units such as regions, nations, states, counties,
or census tracts. Value ranges of rates and percentages are represented by a graded series of crosshatchings. Figure 5 is an illustration of a crosshatched map. Since this figure is for illustration only, the medians were computed by the simplest formula, with no attempt made to correct for the fact that most school dropouts occur at the end of a school year.
Spot maps. In spot or point-symbol maps emphasis is placed on frequencies or absolute amounts rather than on rates or proportions as in the crosshatched map. Although there may be some overlapping, spot maps may be differentiated into five types on the basis of the symbols used. Symbols may stress (1) size, (2) number, (3) density, (4) shading, or (5) form. In the first type of map the size of each symbol is proportional to the frequency or magnitude of the phenomena represented. Symbols may be either two-dimensional or three-dimensional and are normally in the form of circles or spheres rather than rectangles, cubes, or irregular forms. In the second type of spot map the basic criterion is not size but number or frequency of spots or point symbols. The spots are uniform in size, each representing a specific value. The spots are designed and arranged to make them as readily countable as possible. The third type of map is also a multiple-spot variety; but instead of emphasizing countable frequencies, comparative density and distribution are emphasized. Figure 6 is an example of the third type of spot map, in which over-all density patterns are stressed. In the fourth type of spot map the criterion is shading. The size of the symbols is uniform, but the amount of shading is indicative of the magnitude or value represented. The form of the symbol in the fifth type of map represents certain qualities or attributes rather than quantities, as in the previous types. For example, if the dichotomy male-female is to be portrayed on a map, one type of symbol would represent male and another symbol would represent female.
Isoline maps. There are two fundamental types of isoline (from the Greek isos, meaning equal) maps: the isometric map, in which the lines are drawn through points of equal value or intensity, and the isopleth map, in which the lines connect equal rates or ratios for specific areas. The isopleth map is particularly valuable in the social sciences.
Miscellaneous graphic forms . Although not as basic or as widely used as the graphs and charts discussed in the preceding sections, there are cer tain other graphic forms that possess advantages for certain problems.
Pie charts. The pie or sector chart is widely used to portray proportions of an aggregate or total. Given the several proportions of the total, a pie chart is made by dividing a circle into pieces, one for each separate proportion, by boundary radii. Each proportion corresponds to a single slice, or sector, thus formed; and the central angles (equivalently, the circumference arcs, or again the sector areas) are proportional to the magnitudes of the proportions. Frequently, shading or coloring is used to help distinguish the sectors, and of course proper labeling is essential. Although a pie chart can be very effective in simple situations, it becomes difficult to use if there are more than four or five sectors or if one wishes to compare corresponding proportions for several pie charts.
Trilinear charts. The trilinear chart (or barymetric coordinate system) is used to portray simultaneously three nonnegative variables with a fixed sum. Usually they are percentages and the sum is 100. It is drawn in the form of an equilateral triangle, each side of which is calibrated in equal percentage divisions ranging from 0 to 100.
Scatter diagrams. The scatter diagram (scattergram) and other types of correlation charts portray in graphic form the degree and type of relationship or covariation between two series of data. The scatter diagram shows a two-way or bivariate frequency distribution. Customarily, arithmetic scales are used in the construction of scatter diagrams, although semilogarithmic or double-logarithmic scales sometimes may be more appropriate.
Pictorial charts . Pictorial graphs are used mainly because of their popular appeal, although they rarely convey more information than do more conventional graphic forms. In general, there are four basic types of charts in which pictorial symbols are used, distinguished by criteria of purpose and emphasis.
The four types of pictorial charts are (1) charts in which the size of the pictorial symbol is proportional to the values portrayed; (2) pictorial unit graphs, in which each symbol represents a definite and uniform value; (3) cartoon and sketch charts, in which the basic graphic form, such as a curve or bar, is portrayed as a picture; and (4) charts with pictorial embellishments ranging from a single pictorial filler to elaborate and detailed pictorial backgrounds.
Three-dimensional projection charts . In recent years it has become common practice to portray various kinds of graphs and charts in axonometric, oblique, and perspective projection. Charts in three-dimensional form, with depth and other picturelike qualities, unquestionably possess definite popular appeal.
The design of charts and graphs in three-dimensional
form should be based on technically acceptable principles of axonometric, oblique, and perspective projection. Axonometric and oblique projections are the most satisfactory for three-dimensional graphs. Although perspective projection is perhaps the most realistic of the three, it possesses serious limitations as a technique in graphic presentation. Charts constructed in perspective projection are generally distorted; they do not portray exact distance, shape, or size. Figure 7 illustrates a map in three-dimensional form.
Calvin F. Schmid
[See alsoTabular Presentation.]
REFERENCES ON GRAPHIC PRESENTATION
Brinton, Willard C. 1939 Graphic Presentation. New York: Brinton.
Funkhouser, H. G. 1937 Historical Development of the Graphical Representation of Statistical Data. Osiris 3:269–404.
Huff, Darrel; and Geis, Irving 1954 How to Lie With Statistics. New York: Norton. → Contains a discussion of graphical fallacies. Also published in paperback.
Jenks, G. F.; and Brown, D. A. 1966 Three-dimensional Map Construction. Science 154:857–864.
Modley, Rudolf; and Lowenstein, Dyno 1952 Picto-graphs and Graphs: How to Make and Use Them. New York: Harper.
Pepe, Paul 1959 Presentation des statistiques. Paris: Dunod.
Royston, Erica 1956 Studies in the History of Probability and Statistics. III. A Note on the History of the Graphical Representation of Data. Biometrika 43:241–247.
Schmid, Calvin F. 1954 Handbook of Graphic Presentation. New York: Ronald.
Schmdd, Calvin F. 1956 What Price Pictorial Charts? Estadistica: Journal of the Inter-American Statistical Institute 15:12–25.
Schmid, Calvin F.; and Maccannell, Earle H. 1955 Basic Problems, Techniques, and Theory of Isopleth Mapping. Journal of the American Statistical Association 50:220–239.
SOURCES OF FIGURES AND DATA
Schmid, Calvin F.; and Mcvey, Wayne W. Jr. 1964 Growth and Distribution of Minority Races in Seattle, Washington. Seattle Public Schools.
Schmid, Calvin F. et al. 1962 Enrollment Statistics, Colleges and Universities: State of Washington, Fall Term, 1961. Seattle: Washington State Census Board.
Schmid, Calvin F. et al. 1966 Studies in Enrollment Trends and Patterns. Part 1: Regular Academic Year: 1930 to 1964. Seattle: Univ. of Washington.
U.S. Bureau Of The Census 1962 U.S. Censuses of Population and Housing: 1960. Census Tracts, Final Report Phc (1) 142: Seattle, Wash. Washington: Government Printing Office. → See especially pages 15-34, Table P-l, “General Characteristics of the Population, by Census Tracts: 1960.”
U.S. Bureau Of The Census 1963 U.S. Census of Population: 1960. Vol. 1, Part 45: Texas. Washington: Government Printing Office.
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