The maps discussed up to this point have been general purpose maps or aerial photographs representing the largely physical character of the earth's surface. As we have noted already, in addition to representing the topography, the Canadian N.T.S. maps also show a considerable amount of information about the physical environment in general and about human activity as well. For example, lakes are shown as enclosed blue areas, forests by a green pattern of tree symbols, rivers by blue lines, and houses, churches, post offices, roads, bridges, water towers, quarries, and so on, are depicted by a variety of other special syols.
We must now turn our attention to maps which depict just a single theme. These thematic maps, as they are called, use a variety of cartographic symbolization to depict the spatial pattern of a particular geographic quality or quantity rather than a variety of information of different sorts. The purpose of a thematic map is to give exclusive emphasis to a particular subject; its success is measured by the clarity, directness, accuracy and impact of the particular pattern being communicated to the map reader.
As you might suspect, most maps used in the study of spatial information are thematic maps.
Because they vary so widely in type and application, it will be useful to organize our discussion of them by the kind of data they show (qualitative or quantitative) and by the kinds of cartographic devices they employ.
Qualitative thematic maps display descriptive (nominal) data such as the locations of particular features on a base map. Examples include maps showing the locations of public facilities in a municipality, the locations of Provincial capital cities, maps of ski resorts, road maps, maps showing the principal rivers of a region, geology maps, soil maps, vegetation maps, and so on.
If the features in question are at precise points, at the scale of the base map (such as the location of a house on a 1:50 000 map), they can be shown by a nominal point symbol . For example, a house might appear as a round or square dot, Christian churches in a region might be depicted by small crosses, locations of fishing resorts along the coast might be shown by a small fish symbol while campgrounds might be marked by a small tent symbol. In Figure 9.1 nominal point symbols are used to locate particular types of mineral resource developments in Canada .
Linear features such as roads, rivers, railroads, pipelines, powerlines, and administrative boundaries, are depicted on thematic maps by a nominal linear symbol (Figure 9.1) while areas such

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9.2: A soil map using nominal area symbols (Williams, B.J. and Ortiz-Solorio, C.A., 1981, Middle American folk soil taxonomy. Annals of the Association of American Geographers , 71 (3) 354).
as those characterized by a particular geology, soil type, vegetation or crops, are depicted by a nominal area symbol (Figure 9.2). Examples of the former are discussed in the context of legend symbols in
Chapter 1 and area symbols typically involve the use of shading or colour to distinguish bounded areas of a given quality from others that are different. It is very important that nominal linear and area symbols are drawn so that they do not imply quantity. For example, line pattern might be varied (solid, broken, alternating dots and dashes) but thickness of a given line pattern should not vary systematically across nominal classes because it may convey some sense of unimplied increasing importance to the symbols.
Similarly, area symbols should not take the form of graded shading (increasing density) within the same shading pattern; they should be randomly rather than sequentially patterned. It also is important to
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Chapter 9: Thematic mapping remember that, invariably, some degree of averaging or generalization is involved in applying areal symbols. For example, a map showing the principal wheat-growing areas of Canada might include areas with many farms which do not grow wheat. Such a map simply generalizes at a regional level to include areas in which wheat-growing is the principal, but almost certainly not the only, crop. Similarly, locations mapped as not being in a wheat-growing area may well have some farms involved in the landuse.
Quantitative thematic maps depict ordinal, interval, or ratio data on a base map. Ordinal data involve ranked values rather than absolute measurements. For example, highways might be distinguished from secondary roads by using a bolder linear symbol for the former. National boundaries might similarly be distinguished from Provincial boundaries to reflect the administrative hierarchy. Ordinal-scale symbols convey relative importance rather than strictly mathematically comparable measures.
Interval-scale and ratio-scale data are similar in the sense that both refer measurements to some standard scale. A ratio scale is a measurement scale such as length in metres, mass in kilograms, areas in hectares, or value in dollars. These and all other ratio scales have a non-arbitrary zero corresponding to a complete absence of the quantity being measured. Consequently, ratio-scale measurements can be compared directly with each other as a true ratio of quantity. Thus, 200 m is twice as long as 100 m, 50 kg is one quarter of 200 kg, 1000 hectares is ten times the area of one hundred hectares, and so on.
Interval-scale data refer to measurements on a scale with a non-zero arbitrary datum. A good example of such a scale is the commonly used Celsius and Fahrenheit temperature scales. Temperature is a measure of heat but each of these two scales employs an arbitrary non-zero datum. In the case of
Celsius, zero is arbitrarily defined as the temperature at which water freezes and not to the condition corresponding to an absence of heat. Indeed, absolute zero temperature (no heat) on the Celsius scale is
-273oC. Similarly, the Fahrenheit scale arbitrarily starts at a zero which gives the freezing temperature of water as 32oF and the boiling point as 212oF (100oC). Only the Kelvin scale, which has a non-arbitrary zero (at -273oC), is an absolute or ratio scale of temperature. Therefore, temperature comparisons on the commonly used scales are not true ratios of heat. In other words, it is not valid to say that when the temperature is 30oC, it is twice as hot as when the temperature is 15o!
Although the distinction between interval and ratio scales is important, it turns out that almost all quantitative data in geography are represented on ratio scales. Indeed, temperature constitutes one of the few examples of a commonly used interval scale measure.
The main classes of quantitative thematic maps are dot-distribution maps, isoline maps, choropleth maps, cartograms, flow maps, and maps using a variety of proportional symbols.
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9.3: A dot-distribution map of farm produce sales in the United States. The plotting basis of this map is the county-level statistical unit (from the United States Department of Commerce, based on data of the
Bureau of The Census).
Dot-distribution maps
Using dots to locate features is the most straightforward type of quantitative point symbolization and it gives rise to a dot-distribution map , or simply, a dot map (Figures 9.3 and 9.4). Dot maps can be employed to map the locations of discrete occurrences (for example, locations of debris flows or motor vehicle accidents or people) and the variation in the density of dots indicates the geographic pattern of the phenomenon. It should be noted, however, that most dot-distribution maps do not employ a one-to-one ratio for location and dot. Some generalization is often necessary to avoid cluttering the map and therefore obscuring the spatial distribution. Thus it is common for one dot to represent a number of occurrences in a given area. For example, a dot-distribution map showing the population of Canada would require over 20 000 000 dots at a one-to-one person/dot ratio! A more practical solution might involve each dot representing 500 or 1 000 people. Such grouping of data is a common practice necessitated by the size of the data set relative to the scale of the base map. The appropriate degree of generalization is that which best shows the spatial pattern. Obviously, it is not helpful to have large areas on such a map in which the dot pattern is so dense that dot-merging and blackout occurs. It is usually helpful to test map small areas in the most and least densely patterned areas in order to select the best dot/data ratio.
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Dot distribution maps of an area, such as a country, usually are based on data from a set of smaller statistical units, such as Provinces or counties or census tracts. Within the county boundaries, for example, dots may be distributed uniformly or they may be distributed in such a way as to reflect other geographic factors. Applying a uniform distribution of dots within the counties has the advantage of simplicity but it also has the disadvantage of creating unlikely discontinuities in dot density patterns at the common boundaries of counties with distinctly different dot densities (Figure 9.4A). One way of removing this artifact of averaging is to arrange the dots so that their distribution densities blend with those of adjacent counties (Figure 9.4B). Of course we must only employ this smoothing device if we judge that gradually varied dot patterns are more representative of reality than are discontinuities. Yet another way of achieving a more realistic pattern of dots than shown in the uniform case, is to place the dots in their most geographically appropriate positions. For example, dots in a population map would cluster about cities and urban areas and thin out in rural communities and remote areas. This non-uniform but geographically informed mapping of dots is sometimes referred to as dasymetric plotting (see Figure
9.4C).
9.4: Schematic dot map illustrating three methods of varying dot density within statistical mapping units. The small dots represent retail commercial activity and the two larger dots and connecting line represent two towns linked by a major transportation route.
Isoline maps
We already have encountered one important kind of isoline map: the contoured topographic map.
Isolines simply are lines joining equal quantities measured above some common and arbitrary datum. On a topographic map, a given contour joins points which are the same height above sea level. The contours describe the three-dimensional surface that is the ground. This concept easily can be extended to describe any three-dimensional surface. For example, a map of meteorological stations for which annual
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9.5: Part of an isohyet map showing the mean annual precipitation in Canada (from the Hydrologic
Atlas of Canada ; Atmospheric Environment Service and the Department of Fisheries and the Environment, Canada). average rainfall data are available, provides the basis for an isoline map of a statistical surface which is completely analogous to our ground surface example. Instead of heights above sea level we have rainfall in mm (above zero datum - no rainfall) and the isolines are not contours but isohyets (Figure 9.5).
Similarly, a barometric surface representing atmospheric pressure might be described by isobars and an air temperature surface by isotherms.
Unfortunately, far too many different terms and definitions have been used in the past to describe the various kinds of isolines and the terminology is quite confusing. For example, some cartographers use the terms isoline , isogram , and isopleth as synonyms, while others consider an isopleth to be one of two types of isogram , the other being the isometric line . Still others insist that the term isopleth be restricted to isolines joining relative quantities (ratios) rather than absolute values.
Some prefer the term isarithm to isoline while others reserve it for regularly spaced isolines. Furthermore,
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9.6: An isopleth map of flow density of illegal Mexican immigrants entering the United States, per million population of constituent INS districts in 1973 (from Jones, R.C., 1982, Undocumented migration from Mexico: some geographical questions. Annals of the Association of American Geographers , 72 (1) 80).
additional terms have been invented for maps with these various kinds of isolines with and without intervening graded shading! For our purposes, we might do best to avoid these complications of nomenclature and consider all isolines to be of just two basic types: isometric lines and isopleths.
Isometric lines are isolines along which all points represent equal quantities on a continuous surface Thus, contours, isohyets, isobars and isotherms are isometric lines. The real surface of the ground and the statistical surface of atmospheric properties are continuous and interpolation between isohyets to estimate rainfall, for example, is just as valid as interpolation between contours to estimate heights.
In this simplified terminology isopleths are similar to isometric lines in the sense that they are lines joining equal quantities but they differ in that they are drawn on the basis of spatially averaged control points on a discontinuous surface. For example, one could construct an isopleth map by plotting isolines through the county population-density (population/area) values plotted at the centre of the counties. In Figure 9.6 is shown an isopleth map of the number of illegal Mexican immigrants arriving in statistical districts of the U.S. Immigration and Naturalization Service. Each district centre has been assigned a number of alien entries per year (flow density) and isolines are used to summarize the spatial pattern across the United States. But these isolines do not join actual point values on a continuous surface; they simply pass through areas of equal flow density. For example, there may be a district of high
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Chapter 9: Thematic mapping flow density immediately adjacent to one receiving no entries. For this reason, values cannot be interpolated between isopleths as they can between isometric lines. You may detect a logical inconsistency in this discussion! If it is invalid to interpolate between isopleths, surely then it is also not valid to interpolate between the original control points to draw them in the first place! Indeed, many cartographers feel that isopleth maps are not a valid means of representing spatial averages and that some other technique such as the choropleth mapping should be used to show such data instead. These arguments against the use of isopleths apparently are not accepted by everyone, however, because you will find isopleth maps in common use. Nevertheless, you must keep in mind the concerns raised when using such maps for your own purposes.
Using linear interpolation to construct an isoline map through a field of data points (see
Chapter 4 for a discussion of contouring principles) is a relatively straightforward task once the isoline interval has been selected. This interval should be small enough to depict all significant undulations in the surface while at the same time effectively conveying a picture of the surface overall. Small and simple surfaces may yield readily to manual contouring although the effort required to contour a large and complex surface may be sufficient to justify digitizing the three-dimensional reference points (x,y,z) for computer mapping.
Choropleth maps
Choropleth maps are commonly employed to display spatially averaged ratio data such as population density (population/km
2
), crop yields (kilograms/hectare), and average incomes ($ per capita) rather than the corresponding absolute values (total population, total weight or volume of production, and total income). They are maps which employ shading of relatively small statistical divisions such as census tracts or counties to depict geographic patterns in a larger region; choropleth maps of entire countries might be based on the correspondingly larger statistical divisions of state or province (Figures
9.7 and 9.8). Once the range of the data for the area to be mapped has been determined, divisions are established to partition the range into about six to eight classes. The shading scheme used is graded so that low values appear lightly shaded and high values are densely shaded. When each statistical division of a region is mapped with its appropriate shading, the general pattern of spatial variability for the entire region should become apparent. Using much fewer than six classes results in too much information loss and employing many more than this number makes it difficult to distinguish between the shading of adjacent classes.
There is a great deal of judgment and subjectivity involved in selecting the appropriate shading patterns. If the classes of grouped data are formed as a linear division of the range, then it is important that the shading is perceived to increase in a linear fashion through the classes. An example of a set of linear classes is 0-9; 10-19; 20-29; 30-39; and so on; the ratio of the mid-point values of any adjacent pair of classes is constant throughout the range. It should be noted that, in this regard, the eye is easily
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9.7: Choropleth map of the Canadian Prairies showing wheat acreage as a percentage of all cereals grown in the region in 1966 (from Found, W.C. and McDonald, G.T., 1972, The spatial structure of agriculture in Canada - a review. The Canadian Geographer , XVI (2) 175).
deceived; linear perception does not necessarily imply a linear increase in shading density. Trial and error is an important route to success here. Pitfalls include the use of solid black (which overemphasizes the highest class with respect to the others) and unfilled (white) classes which are underemphasized.
Fortunately, computer-graphics packages now have made the task of experimentation with shadings a distinctly less tedious task than it used to be.
If the data are not subdivided into linear classes, choice of an appropriate shading becomes a much more complex task. For example, sometimes the range of the data is so large, perhaps because of the occurrence of a few very high values, that a geometric scale might better be suited to the task of choropleth mapping than is a linear scheme. An example of a set of geometric classes is 10-19; 20-39;
40-79; 80-159; and so on; the ratio of the mid-point values of adjacent classes is changing at a geometric rate throughout the range (by a factor 2.0 in this case). In other cases the distribution of values might conform to a normal distribution and class boundaries might be established at one, two and three standard deviations above and below the mean score. Clearly, the scope for invention and creativity here is considerable! Nevertheless, whatever basis is chosen for establishing the choropleth classes, the overriding goal must remain the pursuit of effective and honest communication to the map user. In general this goal is best met through the use of the simplest and most direct methods of data processing.
Figures 9.7 and 9.8 show two examples of a common use of choropleth maps: displaying agricultural
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9.8: A choropleth map of relative changes in rice yields in India, 1966-71
(from Chakravarti, A.K., 1976, The impact of the high-yielding varieties program on foodgrain production in India. The Canadian Geographer , XX (2) 208.
statistics. There is a considerable literature on the effectiveness of shading used in choropleth maps and the interested reader is encouraged to explore this topic in the reference material listed for this chapter at the end of the manual.
Choropleth maps should not be used to depict absolute quantities because such data are not area-controlled. For example, a large statistical division with the lowest crop yield in a region may still produce the largest amount of crops and mapping this high absolute value over such a large area would produce a quite distorted view of the spatial pattern of production.
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Cartograms
Isopleth and choropleth maps are used to map quantitative areal data onto base maps showing true relative areas. Cartograms provide an ingenious twist to the representation of areal data; these maps distort the true boundaries of the base map so that distances or areas of the statistical divisions are proportional to the values being represented. In Figure 9.9 a world cartogram depicts countries with their areas redrawn so that they are directly proportional to the amount of international trade each completed in
1957-8. The relative importance of each country in this regard is readily apparent.
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In another example, a map of per capita income (1971) in Canada is constructed on a cartogram base in which the provinces are drawn so that their areas are proportional to population (Figure 9.10). Thus the area of Ontario is distorted to 3.37 million km2 (from an actual value of 1.07 million km2) because it contains 34% of the total population of Canada. The huge area of the Yukon and Northwest Territories
(3.86 million km2) would be reduced to 0.01 million km2 because it includes less than 0.3% of Canada's population! Interestingly, British Columbia is little changed on such a cartogram because the ratio of provincial area to national area is similar to its share of the total population.
9.10: A cartogram-based choropleth map showing per-capita income in Canada (1971). The schematic provincial and statistical boundaries enclose areas proportional to population (from Simmons,
J.W., 1976, Short-term income growth in the Canadian urban system. The Canadian Geographer , XX (4) 425).
Another common basis for constructing cartograms is the substitution of travel time for actual distance.
For example, in city maps of travel-to-work times, areas serviced by major roads or public transit are drawn into the centre in proportion to their shorter travel times while less well serviced areas correspondingly move outward from the city centre.
Thus cartograms result from the substitution of the normal map-distance measurement by a completely different dimension. Although such substitution often results in a highly 'distorted' view of a region it also can lead to significant geographic insight because we are able to see the world from an unconventional perspective. The reader interested in taking a closer look at how map users perceive cartograms might like to consult the work of Dent (1975).
Flow-line maps
Flow-line maps are used to depict the quantity of goods or services moving between centres.
They are ideal for representing such flows as the volume of vehicle traffic between centres, the volume of airline traffic between cities, the amount of trade between commercial centres, or the number of telephone calls between particular locations. The links in question are depicted by quantitative linear symbols, lines with the width scaled to the volume carried (Figures 9.11 and 9.12). For example, 1 mm of line thickness might correspond to 10 000 passengers or to one million of metric tonnes of exported ore ingots, and so
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9.11: A flow-line map depicting the internal cargo tonnage entering Port Kembla, Australia, 1959-
60 (from Briton, J.N.H.,1962, The transport functions of the Port of Port Kembla. Economic Geography , 38 (4)351).
9.12: Part of a flow-map depicting average daily traffic flow on the U.S. interstate system (1967)
(from the U.S. National Atlas , U.S. Government Printing Office, Washington, 235).
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Chapter 9: Thematic mapping on. The scale used to set the thickness of the flow line depends on the range of values to be shown. An appropriate scale will show all values from maximum to minimum without cluttering the map to the point of obscurity. As in most decisions relating to the choice of symbols in thematic mapping, this one requires judgment, and often trial and error, in reaching a satisfactory conclusion. Line widths might also be varied to show ordinal traffic flows (heavy, medium, and light, for example). In any case, flow-line maps should always include a key so that the user can relate the symbol to the actual volume.
Flow lines may follow actual transportation routes such as roads or railroads or they may simply indicate the destination/origin of flows by straight or curved lines. Some may specify the direction of floby arrow heads while in others direction of the flow is unimportant. In still others, flows in each direction may be shown by splitting the flow line into two sub-flow-lines.
Maps employing proportional symbols, graphs, and statistical summaries
Figures 9.13-9.17 illustrate a few of the many different types of statistical map that use symbols to convey quantity. Symbols may be repeated to show increasing magnitude (Figure 9.13) or the size of the symbol may be varied to achieve the same end. Symbol repetition is ideally suited to showing spatial variation in numbers of units such as nuclear power plants by country or the size of arsenals depicted as so many tanks or missiles, for example. In the most graphic cases the symbol shape is appropriate to
9.13: Distribution of geographical-research grants in the United States, 1979-1981 (Moriaty, B.M.,
1982, A geography of research-grant getting in geography. Professional Geographer , 34 (3) 326). Grant type is shown by particular symbols and the number of grants is depicted by symbol repetition.
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Chapter 9: Thematic mapping the data being shown. This type of symbolization is best suited to data with a relatively small range, however, because the statistical division is linear in character. If the data range is large, statistical divisions with small inventories will be shown clearly by one or two symbols but those divisions with the largest inventories will have so many symbols that overlap and confusion become a significant problem in some areas of the map. For this reason symbols often are constructed so that their area is proportional to the data; simple symbols such as squares and circles are the most useful in this regard because the eye finds it less easy to differentiate small size differences of more complex shapes. If still greater contraction of the data is necessary, sphere symbols can be used to depict quantities as volumes. The advantage of using these non-linear proportional symbols is illustrated by the data shown below in Table 9.1. In order to map the population of these six cities using a linear symbol, the largest city by population would have to be represented by a symbol which is 500 times larger than the one representing the smallest city by population. For example, if a silhouette figure of a person were used at each city on the map, even if city
A's figure were only 1.0 mm high, city F's symbol would have to be a half metre long to be correctly proportioned! Clearly, a linear symbol scale is inappropriate for displaying these data.
Length of Side length or radius of Diameter of
City population linear scale proportional circles or squares proportional spheres
A
B
C
10 000
50 000
100 000
D 500 000
E 1 000 000
F 5 000 000
1.000
5.000
10.000
50.000
100.000
500.000
1.000
2.236
3.162
7.071
10.000
22.361
1.000
1.710
2.154
3.684
4.642
7.937
Table 9.1: Variations in the size of symbols representing the population of six cities based on a linear and on two non-linear methods. City A is arbitrarily set at unit length.
This extreme range problem is considerably reduced if these data were plotted as proportional squares or circles. Because the data are scaled now to area rather than length (that is, symbol length is proportional to the square root of population), symbol size from the smallest to largest city varies only by a factor of about 20. Thus, if the population of city A were represented by a 1 x 1 mm square, the population of city F would have to be represented by a 2.24 x 2.24 cm square. Even with this amount of data transformation the squares on the map may be judged to be too dissimilar in the case of the extremes. Further contraction in the size range of symbols can be achieved by using cubes or spheres in which the respective lengths and diameters are drawn proportional to the cube root of the population. In
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9.14: A map employing bar graphs to depict the geography of per capita industrial output, provision of doctors, and university places, in China (from Chu, D.K.Y., 1982, Some analyses of recent
Chinese provincial data. The Professional Geographer , 34 (4) 436).
this case symbol size from the smallest to largest city varies only by a factor of about 8.
Calculating the mathematically correct lengths or radii for squares or circles and cubes or spheres to represent data such as those in Table 9.1 is quite straightforward enough but unfortunately this solution creates another problem! When comparing symbols of varying size, our perception of relative symbol size does not always correspond with the actual symbol geometry. This phenomenon has encouraged some cartographers to suggest a subjective adjustment be made to the diameters of geometrically correct proportional circles, for example, to give perceptually correct diameters. Another important distortion of perceived symbol size on thematic maps is caused by clustering of symbols of different sizes. One of these effects, known as the Ebbinghaus illusion, is illustrated in Figure 9.18. Here the size of a circle is seen to depend on the size and cluster pattern of surrounding symbols. These and other related effects relating to the perception of symbol size and shape are discussed in the selected references listed at the end of the chapter. Because these tricks of perception are very complex, adjusting for them on a map also is a very complicated business. Although you should be aware of these perception problems in
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9.15: A map employing a variety of symbolization, including proportional circles, to depict transport facilities and urban development on the Sydney coalfield, eastern Australia ( from Millward,
H., 1985, A model of coalfield development: six stages exemplified by the Sydney field. The Canadian Geographer ,
29 (3) 241).
general, at our introductory level of cartographic analysis, it is not recommendthat you attempt such adjustments to your own maps. Nevertheless, you may encounter thematic maps where such adjustments have been made although it likely will not be obvious nor explicitly stated on the map. For this reason thematic maps should never be used as a source of quantitative data for other analyses.
Quite apart from the fact that map measurements of symbols are bound not to be very precise, there also may be a quite complex and perhaps subjective relation between symbol size and the quantity being represented.
Statistical summary graphics can be depicted in a variety of ways on thematic maps to show spatial variability. Bar graphs may be simple or divided to show both quantity and category (Figure 9.14).
The same type of data might also be represented by a pie diagram in which categories appear as proportional segments of divided circles. The circles in turn may be proportional circles so that both total quantity and component quantity may be shown (Figure 9.16). Data of the frequency-distribution type might be represented as a histogram.
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9.16: A map employing a variety of symbolization, including proportional pie diagrams, to depict forest resource harvesting potential in Canada, 1970-71 ( from Chapman, J.D., 1976, Natural resource developments in Canada, 1970-75.
The Canadian Geographer , XX (1) 21) .
9.17: Part of a population distribution map employing dots and proportional spheres. Although the use of volumetric symbols solves the problem of representing very large data ranges on a single map, they are difficult to interpret accurately (from Smith, G.H., 1928, A population map of Ohio for 1920. Geographical
Review , 18 (3) 422-27).
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B
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9.19: Computer-generated graphics depicting the 1981 population and area of the provinces of Canada: A. threedimensional bar graphs and B. threedimensional pie graphs. The output from such computer-graphics packages, combined with appropriate base maps, constitutes a powerful and very flexible cartographic tool for thematic mapping
(Cricket Graph (1986) by Cricket Software).

Chapter 9: Thematic mapping
In conclusion we might remind ourselves that, although data on thematic maps may be presented in a great many different ways, the choice must always meet the fundamental criteria of good communication: clarity and honesty. With so much computer-power and so many statistical and graphical options now at our fingertips and keyboards, we may be tempted to overindulge our creativity and artistic inclinations in the map-making enterprise! But we must never let our enthusiasm in this regard compromise the effectiveness of the map as a means of communication.
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