Proportional symbol map
A Proportional symbol map is a type of Thematic map that uses map symbols that vary in size to represent a quantitative variable. Symbols vary in height, length, area or volume from place to place depending on the variable that they represent.  For example, discs may be used to show the location of cities within the map where the size of the disc is proportional to the population of the city.
This type of map is useful for visualization when raw data cannot be dealt with as a ratio or proportion. Although circles are the most typical symbol because they are more compact due to their low perimeter to area ratio, studies show that it is easier for the reader to estimate the size of the symbol if it is a square or a bar.  The best way to increase the reader's ability to correctly estimate the size of a circle is through effective legend design, including providing examples of different sized circles. Proportional symbol maps are effective because they allow the reader to understand large quantities of data in a fast and simple way.
Choosing a Symbol
There are two different types of proportional symbols: geometric and pictographic. Geometric are symbols of shapes typically circles, triangles or squares. Pictoraphic symbols are those of pictures. Typically the picture symbol is in very similar relation to the value, such as a house or a field of wheat represented by small bushels of wheat. Three-dimensional versions of geometric and pictographic symbols can also be used.
It is important to keep in mind that it should be easy for the user to estimate the value of each proportional symbol. Research has been conducted to determine how map users interpret symbols and it was discovered that users can interpret values of bars and rectangle symbols better than circle proportional symbols. However, the rectangle and bar symbols can also make a map more cluttered. Pictographic symbols can be difficult to interpret because of their complexity, especially if they are in a congested area. Three-dimensional symbols, though fascinating to look at, are also more difficult to interpret. Choosing the right symbol can be difficult, but it is critical step in making an effective proportional symbol map. 
Non-point Proportional Symbols
The above mentioned proportional symbols are examples of point symbols. Proportional data can be represented in other ways, such as on flow maps and cartograms. A flow map uses arrows or lines of varying thickness to represent movement of objects. A cartogram is an abstract map that scales geographic areas (i.e. countries) proportionally to the value of an attribute.
Arguably the most important part of proportional symbol maps is the scaling technique used to represent data. Though under some debate, absolute scaling, apparent magnitude scaling, and range grading are the three most common types of scaling.  Scaling is important to ensure that the map is legible by making it possible for the reader to estimate the value of the symbol.
This method makes each symbol fit it's data value on the scale being used proportionately. For example, if circles are being used to represent GDP on a global map, then a country with a value of 58 would have a circle with twice the area as a country with a value of 29.
If circles are being used, the sizes of all symbols are calculated based on a chosen size for any one of the symbols. Say the cartographer decides that a value v0 will have a circle of radius r0. Then for any other value v, the radius r is determined by setting the areas in direct proportion to the values:
v / v0 = a / a0 = πr2 / πr02
This can then be solved for r:
r = r0 (v / v0)0.5
Apparent Magnitude (Flannery) Scaling
Apparent magnitude scaling is a technique used to compensate for human error in interpreting scale. While estimating length is not an issue, people tend to underestimate the areas and volumes of objects. When creating a proportional symbol map for example, a cartographer would use a circle with an area of 2 square cm to represent 100 people and a circle with an area of 4 square cm to represent 200 people. The second circle is exactly twice the size in area than the first to represent exactly twice the amount of people. Studies have found that the second circle will look less than twice the size of the original circle, leading to interpretation error. Apparent magnitude scaling takes this natural error into account by increasing the area difference by more than the proportional amount.
This methods classifies the data being represented into different groups or classes. These groups will be a range of values, and all cases that have values within a given range will all have the same-sized symbol. Range Grading makes it relatively easy to match a symbol to its proportional values.
A common problem with proportional symbol maps is that there can be overlap and congestion of different symbols. A way fix this problem would to make the symbols on top of the overlapped symbols transparent so that the map viewer can see the symbols beneath. Another way to combat this problem is to manually move the symbols apart from each other. However, that brings up a problem with disassociating the feature with its symbol. Moving the symbols apart from each other would only work if the features were still associated with their symbol after they were moved. 
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