Center of population
In demographics, the center of population (or population center) of a region is a geographical point that describes a centerpoint of the region's population. There are several different ways of defining such a "center point", leading to different geographical locations; these are often confused.
Three commonly used (but different) center points are:
- the mean center, also known as the centroid or center of gravity;
- the median center, which is the intersection of the median longitude and median latitude;
- 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 the 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.
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.
In practical computation, decisions are also made on the granularity (coarseness) of the population data, depending on population density patterns or other factors. For instance, the center of population of all the cities in a country may be different from the center of population of all the states (or provinces, or other subdivisions) in the same country. Different methods may yield different results.
Practical uses for finding the center of population include locating possible sites for forward capitals, such as Brasilia, Astana or Austin. Practical selection of a new site for a capital is a complex problem that depends also on population density patterns and transportation networks.
It is important to use a method that does not depend on a two-dimensional projection when dealing with the entire world. As described in INED (Institut national d'études démographiques), the solution methodology deals only with the globe. As a result, the answer is independent of which map projection is used or where it is centered. As described above, the exact location of the center of population will depend on both the granularity of the population data used, and the distance metric. With geodesic distances as the metric, and a granularity of 1,000 kilometers (600 mi), meaning that two population centers within 1000 km of each other are treated as part of a larger common population center of intermediate location, the world's center of population is found to lie "at the crossroads between China, India, Pakistan and Tajikistan", essentially located in Kashmir, with an average distance of 5,200 kilometers (3,200 mi) to all humans. The data used in the reference support this result to a precision of only a few hundred kilometers, hence the exact location is not known.
By country 
Australia has not seen its population centroid move drastically since the creation of the country. In 1911, the centroid was in central New South Wales; in 1996, it was only slightly further northwest.
In Finland, the point of minimum aggregate travel is located in the municipality of Hauho. It is moving slightly to the south-west-west every year because people are moving out of the periphery areas of northern and eastern Finland.
Great Britain 
A centre of population in Great Britain did not move much in the 20th century. In 1901, it was in Rodsley, Derbyshire and in 1911 in Longford. In 1971 it was at Newhall, South Derbyshire and in 2000, it was in Appleby Parva, Leicestershire.[need quotation to verify]
The centroid of population of Japan is in Gifu Prefecture, almost directly north of Nagoya city, and has been moving south-east-east for the past few decades. More recently, the only large regions in Japan with significant population growth have been in Greater Nagoya and Greater Tokyo.
The demographical center of Sweden (using the median center definition) is Hjortkvarn in Hallsberg Municipality, Örebro county. Between the 1989 and 2007 census the point moved a few kilometres to the south, due to a decreasing population in northern Sweden and immigration to the south.
United States 
The mean center of United States population (using the centroid definition) has been calculated for each U.S. Census since 1790. In 2010 this point was located near Plato, Missouri, in the central part of the state. However, when Washington, D.C. was chosen as the federal capital of the United States in 1790, the center of the U.S. population was in Kent County, Maryland, a mere 47 miles (76 km) east-northeast of the new capital. Over the last two centuries, the mean center of United States population has progressed westward and, since 1930, southwesterly, reflecting population drift.
- Bellone F. and Cunningham R. (1993). "All Roads Lead to... Laxton, Digby and Longford." Statistics Canada 1991 Census Short Articles Series.
- Kumler, Mark P.; Goodchild, Michael F. (1992). "The population center of Canada – Just north of Toronto?!?". In Janelle, Donald G. Geographical snapshots of North America: commemorating the 27th Congress of the International Geographical Union and Assembly. pp. 275–279.
- Claude Grasland and Malika Madelin (May 2001). "The unequal distribution of population and wealth in the world". Population Et SociétéS (Institut national d’études démographiques) 368: 1–4. ISSN 0184-7783.
- "Figure 15: Shifts in the Australian Population Centroid*, 1911–1996". Parliament of Australia Parliamentary Library. Retrieved 7 January 2009.
- "The Population Center of Canada – Just North of Toronto?!?" (PDF). Retrieved 21 April 2012.
- Uusirauma.fi Kaupunkilehti Uusi Rauma 03.08.2009 Päivän kysymys? Missä Rauman keskipiste? (Finnish)
- Dradio.de (German)
- "News Item:". University of Leeds. Retrieved 25 November 2007.
- "Population Centre". Appleby Magna & Appleby Parva. Archived from the original on 23 November 2007. Retrieved 25 November 2007.
- "Coffee Break: The movable Midlands; ANSWERS TO CORRESPONDENTS". The Daily Mail (London). 7 February 2002. p. 64.
- "Our Country's Center of Population (我が国の人口重心)". Stat.go.jp. Retrieved 21 April 2012.
- "Sweden's demographic centre, SCB.se, 2008-03-18". Scb.se. 18 March 2008. Retrieved 21 April 2012.