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In demographics, the **centre of population** (or **population center**) of a region is a geographical point that describes a centrepoint of the region's population. There are several different ways of defining such a "centre point", leading to different geographical locations; these are often confused.^{ [1] }

Three commonly used (but different) center points are:

- the
*mean centre*, also known as the*centroid*or*centre of gravity*; - the
*median centre*, 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 centre points are obtained depending on whether the centre is computed in three-dimensional space, or restricted to the curved surface, or computed using a flat map projection.

A **map projection** is a systematic transformation of the latitudes and longitudes of locations from the surface of a sphere or an ellipsoid into locations on a plane. Maps cannot be created without map projections. All map projections necessarily distort the surface in some fashion. Depending on the purpose of the map, some distortions are acceptable and others are not; therefore, different map projections exist in order to preserve some properties of the sphere-like body at the expense of other properties. There is no limit to the number of possible map projections.

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.

The median centre is the intersection of two perpendicular lines, each of which divides the population into two equal halves.^{ [2] } 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. Tukey called this the *cross median*.^{ [3] }

A **circle of latitude** on Earth is an abstract east–west circle connecting all locations around Earth at a given latitude.

In geography, **latitude** is a geographic coordinate that specifies the north–south position of a point on the Earth's surface. Latitude is an angle which ranges from 0° at the Equator to 90° at the poles. Lines of constant latitude, or *parallels*, run east–west as circles parallel to the equator. Latitude is used together with longitude to specify the precise location of features on the surface of the Earth. On its own, the term latitude should be taken to be the *geodetic latitude* as defined below. Briefly, geodetic latitude at a point is the angle formed by the vector perpendicular to the ellipsoidal surface from that point, and the equatorial plane. Also defined are six *auxiliary latitudes* which are used in special applications.

A (**geographic**) **meridian** is the half of an imaginary great circle on the Earth's surface, terminated by the North Pole and the South Pole, connecting points of equal longitude, as measured in angular degrees east or west of the Prime Meridian. The position of a point along the meridian is given by that longitude and its latitude, measured in angular degrees north or south of the Equator. Each meridian is perpendicular to all circles of latitude. Each is also the same length, being half of a great circle on the Earth's surface and therefore measuring 20,003.93 km.

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 mathematics, a **closed-form expression** is a mathematical expression that can be evaluated in a finite number of operations. It may contain constants, variables, certain "well-known" operations, and functions, but usually no limit. The set of operations and functions admitted in a closed-form expression may vary with author and context.

In computational mathematics, an **iterative method** is a mathematical procedure that uses an initial guess to generate a sequence of improving approximate solutions for a class of problems, in which the *n*-th approximation is derived from the previous ones. A specific implementation of an iterative method, including the termination criteria, is an algorithm of the iterative method. An iterative method is called **convergent** if the corresponding sequence converges for given initial approximations. A mathematically rigorous convergence analysis of an iterative method is usually performed; however, heuristic-based iterative methods are also common.

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 centre 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 Brasília, Nur-Sultan 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.

**Brasília** is the federal capital of Brazil and seat of government of the Federal District. The city is located atop the Brazilian highlands in the country's center-western region. It was founded on April 21, 1960, to serve as the new national capital. Brasília is estimated to be Brazil's 3rd most populous city. Among major Latin American cities, Brasília has the highest GDP per capita.

**Nur-Sultan**, previously **Astana**, is the capital city of Kazakhstan. It is located on the banks of the Ishim River in the northern portion of Kazakhstan, within the Akmola Region, though administered separately from the region as a city with special status. The 2017 official estimate reported a population of 1,029,556 within the city limits, making it the second-largest city in Kazakhstan, behind Almaty, the capital from 1991 to 1997.

**Austin** is the capital of the U.S. state of Texas and the seat of Travis County, with portions extending into Hays and Williamson counties. It is the 11th-most populous city in the United States and the 4th-most populous city in Texas. It is also the fastest growing large city in the United States, the second most populous state capital after Phoenix, Arizona, and the southernmost state capital in the contiguous United States. As of the U.S. Census Bureau's July 1, 2017 estimate, Austin had a population of 950,715 up from 790,491 at the 2010 census. The city is the cultural and economic center of the Austin–Round Rock metropolitan statistical area, which had an estimated population of 2,115,827 as of July 1, 2017. Located in Central Texas within the greater Texas Hill Country, it is home to numerous lakes, rivers, and waterways, including Lady Bird Lake and Lake Travis on the Colorado River, Barton Springs, McKinney Falls, and Lake Walter E. Long.

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),^{ [4] } 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 somewhere north of South Asia ^{ [5] } with an average distance of 5,200 kilometers (3,200 mi) to all humans.^{ [4] } 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. Another analysis utilising city level population data found that the world's center of population is located close to Almaty, Kazakhstan.^{ [6] }

The examples and perspective in this article may not represent a worldwide view of the subject.(August 2015) (Learn how and when to remove this template message) |

Australia's population centroid is in central New South Wales. By 1996 it had moved only a little to the north-west since 1911.^{ [7] }

In Canada, a 1986 study placed the point of minimum aggregate travel just north of Toronto in the city of Richmond Hill, and moving westward at a rate of approximately 2 metres per day.^{ [8] }

China's population centroid has wandered within southern Henan from 1952 to 2005. Incidentally, the two end point dates are remarkably close to each other.^{ [9] } China also plots its economic centroid or center of economy/GDP, which has also wandered, and is generally located at the eastern Henan borders.

The center of population of Estonia was on the northwestern shore of Lake Võrtsjärv in 1913 and moved an average of 6 km northwest with every decade until the 1970s. The higher immigration rates during the late Soviet occupation to mostly Tallinn and Northeastern Estonia resulted the center of population moving faster towards north and continuing urbanization has seen it move northwest towards Tallinn since the 1990s. The center of population according to the 2011 census was in Jüri, just 6 km southeast from the border of Tallinn.^{ [10] }

In Finland, the point of minimum aggregate travel is located in the former municipality of Hauho.^{ [11] } It is moving slightly to the south-west-west every year because people are moving out of the peripheral areas of northern and eastern Finland.

In Germany, the centroid of the population is located in Spangenberg, Hesse, close to Kassel.^{ [12] }

The 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.^{ [13] }^{ [14] }^{ [15] }^{[ need quotation to verify ]}

The mean center of the India lies at the Lat 22.49 N, Long 80.10 E near the town of Nainpur in the state of Madhya Pradesh.

The centre of population of the entire island of Ireland is located near Kilcock, County Kildare. This is significantly further east than the Geographical centre of Ireland, reflecting the disproportionately large cities of the east of the island (Belfast and Dublin).^{ [16] } The centre of population of the Republic of Ireland is located southwest of Edenderry, County Offaly.^{ [17] }

The centroid of population of Japan is in Gifu Prefecture, almost directly north of Nagoya city, and has been moving east-southeast for the past few decades.^{ [18] } More recently, the only large regions in Japan with significant population growth have been in Greater Nagoya and Greater Tokyo.

In June 2008, New Zealand's median centre of population was located near Taharoa, around 100 km (65 mi) southwest of Hamilton on the North Island's west coast.^{ [19] } In 1900 it was near Nelson and has been moving steadily north ever since.^{ [20] }

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.^{ [21] }

The center of population in the Russian Federation is calculated by A. K. Gogolev to be at 56°34′N53°30′E / 56.567°N 53.500°E , 42 km (26 mi) south of Izhevsk.^{ [22] }

Heping District, Taichung.^{ [23] }

The mean center of the United States population (using the centroid definition) has been calculated for each U.S. Census since 1790. Over the last two centuries, it has progressed westward and, since 1930, southwesterly, reflecting population drift. For example, in 2010, the mean center was located near Plato, Missouri, in the south-central part of the state, whereas, in 1790, it was in Kent County, Maryland, 47 miles (76 km) east-northeast of the then-new federal capital, Washington, D.C.

- Bellone F. and Cunningham R. (1993). "All Roads Lead to... Laxton, Digby and Longford." Statistics Canada 1991 Census Short Articles Series.

In mathematics and statistics, the **arithmetic mean**, or simply the mean or **average** when the context is clear, is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results of an experiment or an observational study, or frequently a set of results from a survey. The term "arithmetic mean" is preferred in some contexts in mathematics and statistics because it helps distinguish it from other means, such as the geometric mean and the harmonic mean.

In statistics, a **central tendency** is a central or typical value for a probability distribution. It may also be called a **center** or **location** of the distribution. Colloquially, measures of central tendency are often called *averages.* The term *central tendency* dates from the late 1920s.

**Longitude**, is a geographic coordinate that specifies the east–west position of a point on the Earth's surface, or the surface of a celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek letter lambda (λ). Meridians connect points with the same longitude. By convention, one of these, the Prime Meridian, which passes through the Royal Observatory, Greenwich, England, was allocated the position of 0° longitude. The longitude of other places is measured as the angle east or west from the Prime Meridian, ranging from 0° at the Prime Meridian to +180° eastward and −180° westward. Specifically, it is the angle between a plane through the Prime Meridian and a plane through both poles and the location in question.

The **Mercator projection** is a cylindrical map projection presented by the Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for nautical navigation because of its ability to represent lines of constant course, known as rhumb lines or loxodromes, as straight segments that conserve the angles with the meridians. Although the linear scale is equal in all directions around any point, thus preserving the angles and the shapes of small objects, the Mercator projection distorts the size of objects as the latitude increases from the Equator to the poles, where the scale becomes infinite. So, for example, landmasses such as Greenland and Antarctica appear much larger than they actually are, relative to landmasses near the equator such as Central Africa.

In mathematics and physics, the **centroid** or **geometric center** of a plane figure is the arithmetic mean position of all the points in the figure. Informally, it is the point at which a cutout of the shape could be perfectly balanced on the tip of a pin.

The **mean center of the United States population** is determined by the United States Census Bureau from the results of each national census. The Bureau defines it as follows:

The concept of the center of population as used by the U.S. Census Bureau is that of a balance point. The center of population is the point at which an imaginary, weightless, rigid, and flat surface representation of the 50 states and the District of Columbia would balance if weights of identical size were placed on it so that each weight represented the location on one person. More specifically, this calculation is called the mean center of population.

There has long been debate over the exact location of the **geographical centre** of the United Kingdom, and its constituent countries, due to the complexity and method of the calculation, such as whether to include offshore islands, and the fact that erosion will cause the position to change over time. There are two main methods of calculating this "centre": either as the centroid of the two-dimensional shape made by the country, or as the point farthest from the boundary of the country. These two methods give quite different answers.

The location of the **geographical centre of Europe** depends on the definition of the borders of Europe, mainly whether remote islands are included to define the extreme points of Europe, and on the method of calculating the final result. Thus, several places claim to host this hypothetical centre.

** k-means clustering** is a method of vector quantization, originally from signal processing, that is popular for cluster analysis in data mining.

**Medoids** are representative objects of a data set or a cluster with a data set whose average dissimilarity to all the objects in the cluster is minimal. Medoids are similar in concept to means or centroids, but medoids are always restricted to be members of the data set. Medoids are most commonly used on data when a mean or centroid cannot be defined, such as graphs. They are also used in contexts where the centroid is not representative of the dataset like in images and 3-D trajectories and gene expression. These are also of interest while wanting to find a representative using some distance other than squared euclidean distance.

In computer science and electrical engineering, **Lloyd's algorithm**, also known as **Voronoi iteration** or relaxation, is an algorithm named after Stuart P. Lloyd for finding evenly spaced sets of points in subsets of Euclidean spaces and partitions of these subsets into well-shaped and uniformly sized convex cells. Like the closely related *k*-means clustering algorithm, it repeatedly finds the centroid of each set in the partition and then re-partitions the input according to which of these centroids is closest. However, Lloyd's algorithm differs from *k*-means clustering in that its input is a continuous geometric region rather than a discrete set of points. Thus, when re-partitioning the input, Lloyd's algorithm uses Voronoi diagrams rather than simply determining the nearest center to each of a finite set of points as the *k*-means algorithm does.

The **geographic center of the contiguous United States** is the center of 48 U.S. states. It has been regarded as such by the U.S. National Geodetic Survey (NGS) since the 1912 additions of New Mexico and Arizona to the United States.

The **two-point equidistant projection** is a map projection first described by Hans Maurer in 1919. It is a generalization of the much simpler azimuthal equidistant projection. In this two-point form, two locus points are chosen by the mapmaker to configure the projection. Distances from the two loci to any other point on the map are correct: that is, they scale to the distances of the same points on the sphere.

**Statistical geography** is the study and practice of collecting, analysing and presenting data that has a geographic or areal dimension, such as census or demographics data. It uses techniques from spatial analysis, but also encompasses geographical activities such as the defining and naming of geographical regions for statistical purposes. For example, for the purposes of statistical geography, the Australian Bureau of Statistics uses the Australian Standard Geographical Classification, a hierarchical regionalisation that divides Australia up into states and territories, then statistical divisions, statistical subdivisions, statistical local areas, and finally census collection districts.

The **spectral centroid** is a measure used in digital signal processing to characterise a spectrum. It indicates where the "center of mass" of the spectrum is located. Perceptually, it has a robust connection with the impression of "brightness" of a sound.

In geography, the centroid of the two-dimensional shape of a region of the Earth's surface is known as its **geographical centre** or **geographical center**. There has long been debate over the methods of calculation of the geographical centres of various countries and regions, proposed methods include:

In computer vision and computer graphics, **3D reconstruction** is the process of capturing the shape and appearance of real objects. This process can be accomplished either by active or passive methods. If the model is allowed to change its shape in time, this is referred to as non-rigid or spatio-temporal reconstruction.

**Centre points of Australia** are those geographical locations that have been considered to be *centre* of Australia, as distinct from the extreme points of Australia.

- ↑ Kumler, Mark P.; Goodchild, Michael F. (1992). "The population center of Canada – Just north of Toronto?!?". In Janelle, Donald G. (ed.).
*Geographical snapshots of North America: commemorating the 27th Congress of the International Geographical Union and Assembly*. pp. 275–279. - ↑ "Centers of Population Computation for the United States, 1950-2010" (PDF). Washington, DC: Geography Division, U.S. Census Bureau. March 2011.
- ↑ Tukey, John (1977).
*Exploratory Data Analysis*. Addison-Wesley. p. 668. ISBN 9780201076165. - 1 2 Claude Grasland and Malika Madelin (May 2001). "The unequal distribution of population and wealth in the world" (PDF).
*Population Et SociétéS*. Institut national d'études démographiques.**368**: 1–4. ISSN 0184-7783. - ↑ exact phrase in the paper is "at the crossroads between China, India, Pakistan and Tajikistan"
- ↑ "Center of World Population".
*City Extremes*. 2017. Retrieved 21 August 2017. - ↑ "Figure 15: Shifts in the Australian Population Centroid*, 1911–1996".
*Parliament of Australia Parliamentary Library*. Archived from the original on 19 August 2000. Retrieved 7 January 2009. - ↑ "The Population Center of Canada – Just North of Toronto?!?" (PDF). Retrieved 21 April 2012.
- ↑ http://sourcedb.igsnrr.cas.cn/zw/lw/201007/P020100706529106697457.pdf
- ↑ Haav, Mihkel (2010) - "Eesti dünaamika 1913-1999"
- ↑ Uusirauma.fi
^{[ permanent dead link ]}Kaupunkilehti Uusi Rauma 03.08.2009 Päivän kysymys? Missä Rauman keskipiste? (in Finnish) - ↑ Dradio.de Archived 24 October 2007 at the Wayback Machine (in 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. - ↑ http://imgur.com/q439KW0
- ↑ http://imgur.com/prTTQOK
- ↑ "Our Country's Center of Population (我が国の人口重心)". Stat.go.jp. Retrieved 21 April 2012.
- ↑ "Subnational Population Estimates: At 30 June 2008 -- Commentary". Statistics New Zealand. Retrieved 11 November 2014.
- ↑ http://nzbooks.org.nz/tag/bridget-williams-books/
- ↑ "Sweden's demographic centre, SCB.se, 2008-03-18". Scb.se. 18 March 2008. Archived from the original on 29 March 2012. Retrieved 21 April 2012.
- ↑ Сайт "Встарь, или Как жили люди"
- ↑ https://www.ptt.cc/man/Geography/D1F0/DD1E/M.1105123514.A.427.html

- Australian parliamentary research paper with map of population centroids.
- U.S. Census Bureau: Centers of Population

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