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Center of population

Geographical point

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Definitions

Three commonly used (but different) center points are:

  1. the mean center, also known as the centroid or center of gravity;
  2. the median center, which is the intersection of the median longitude and median latitude;
  3. the geometric median, also known as Weber point, Fermat–Weber point, or point of minimum aggregate travel.

A further complication is caused by the curved shape of the Earth. Different center points are obtained depending on whether the center is computed in three-dimensional space, or restricted to the curved surface, or computed using a flat map projection.

Mean center

The mean center, or centroid, is the point on which a rigid, weightless map would balance perfectly, if the population members are represented as points of equal mass.

Mathematically, the centroid is the point to which the population has the smallest possible sum of squared distances. It is easily found by taking the arithmetic mean of each coordinate. If defined in three-dimensional space, the centroid of points on the Earth's surface is actually inside the Earth. This point could then be projected back to the surface. Alternatively, one could define the centroid directly on a flat map projection; this is, for example, the definition that the US Census Bureau uses.

Contrary to a common misconception, the centroid does not minimize the average distance to the population. That property belongs to the geometric median.

Median center

The median center is the intersection of two perpendicular lines, each of which divides the population into two equal halves. Typically these two lines are chosen to be a parallel (a line of latitude) and a meridian (a line of longitude). In that case, this center is easily found by taking separately the medians of the population's latitude and longitude coordinates. John Tukey called this the cross median.

Geometric median

The geometric median is the point to which the population has the smallest possible sum of distances (or equivalently, the smallest average distance). Because of this property, it is also known as the point of minimum aggregate travel. Unfortunately, there is no direct closed-form expression for the geometric median; it is typically computed using iterative methods.

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