Centrography: Types of Centers

Although there are many different types of centers, only two are widely used. These are the so-called center of gravity, which is used mostly for the analysis of the movement of a population over a period of time, and the median center, which is used primarily for planning the location of future facilities.
 
Center of Gravity

The geographical concept of center of gravity is derived from the concept of the same name used by physicists to denote the balancing point of a material body. The U. S. Bureau of the Census first applied this concept to human populations in the first Statistical Atlas of the United States in 1874, and in every subsequent major census bureau publication. The definition of the center of population used by the census bureau is "that point upon which the United States would balance if it were a rigid plane, without weight, with the population distributed thereon with each individual having equal weight." An alternative definition of this center is that point at which the sum of the squared distances from the point to each member of the population is a minimum. If one were working with a set of numbers, instead of populations distributed over an area, the second definition would be the definition of the arithmetic mean—the familiar "average"—and so the center of gravity is also called the arithmetic mean center.

There are two main reasons for the usefulness of the center of gravity. First, it is an extremely sensitive measure and will change, even though the change may be minute, with any movement within the population. This sensitivity makes the center of gravity a very useful device for studying general trends in the movement of population over long periods of time. In addition, since it is the most often used centrographic measure, there is a vast library of previously computed centers of gravity of populations of all kinds. For example, the center of gravity of population for each state in the United States has been computed for each census since 1880. Since 1920, centers of gravity have been computed for wage earners and for foreign-born, native white, native black, urban, and rural populations. In addition, centers of gravity have been computed for the distribution of farms, farm acreage, and the production of a wide variety of manufactured and agricultural goods.
 
Second, the center of gravity is useful because it possesses the statistical properties of the arithmetic mean, which is particularly important when the entire population cannot be studied and sampling techniques must be used.
 
Median Center

The first applications of the concept of a median center were based on the property of a median to divide a distribution in half. Thus, the median point was defined as the intersection of two perpendicular lines, each of which divides the population in half. In 1902, however, J. F. Hayford recognized that the location of this intersection depends on the direction in which the lines lie and that the differences in the positions of the various median centers were not small enough to be ignored. Hayford, therefore, redefined the median center as the point at which the sum of the distances from the point to each of the members of the population is a minimum. This is similar to the definition of the center of gravity, except that for the median center the distances are not squared.
 
If each member of the population were to travel in a straight line from his or her home to some point, the total amount of travel would be least if that point were the median point. Therefore the median point is sometimes also referred to as either the point of minimum aggregate travel or the point of minimum average travel. It is this minimum travel property that makes the median center a useful device in determining locations for schools and offices as well as warehouses and other facilities.
 
Modal Center and Harmonic Mean Center

Two other types of centers have found increasing use in recent years. These are the modal center and the harmonic mean center. The modal center is defined as the location of the unit of area that has the greatest density of population. The practical difficulty in working with this concept is that a center is defined as a point, but density must be defined in terms of an area. To avoid this difficulty many centrographers have adopted the harmonic mean center, often called the peak of potential of population; although more difficult to define and compute, it is more precise mathematically and is usually located at the same place as the modal center.

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