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where
x
i
and
y
i
are the coordinates for feature
i
, and
n
is equal to the total
number of features. The Weighted Mean Center becomes
Σ
Σ
n
w x
w
Σ
Σ
n
w y
w
i
=
1
i
i
i
=
1
i
i
x
=
,
y
=
w
w
n
n
i
=
1
i
i
=
1
i
where
w
i
is the weight at feature
i
.
For example, the mean geographic center for the state of Michigan could be
constructed from the centroids, or the mean centers, of each of the state's
83 counties. The mean center can be thought of as the point at the tip of a
pencil that “balances” whatever data are on a “plane” surface. A weighted
mean center “weights” the geographic mean center by influencing its loca-
tion based on the spatial distribution of some phenomenon, such as popula-
tion, sirens, or water wells, for example. If that variable is more numerous
on one side of the plane, the pencil point, or mean center, must be moved
to accommodate the extra weight on one side of the plane to keep the plane
in balance.
Another way to represent the distribution of data is through a directional
distribution, or a standard deviational ellipse. This ellipse wonderfully sum-
marizes the spatial characteristics of geographic features, including central
tendency, dispersion, and directional trends. This ellipse is created by calcu-
lating the standard distance, or the distance within one standard deviation of
the mean center of a set of data, first from the
x
-coordinates, and then from
the
y
-coordinates; in other words, separately in the
x
-direction and then in
the
y
-direction. These two measures define the axes of the ellipse. The size
of the ellipse indicates how spatially diffuse or concentrated a set of data are
spatially, and if the distribution of features is elongated, or oriented in a par-
ticular direction, that too will be evident in the standard deviational ellipse.
For example, a specific disease such as schistosomiasis in the Philippines
may display a standard deviational ellipse that reflects the wetlands and the
vulnerable populations near those wetlands, whereas the same type of ellipse
for Acquired Immuno Deficiency Syndrome (AIDS) might be larger in shape
and less elliptical, reflecting the broader range of affected people in rural and
urban areas.
Mean centers, weighted mean centers, and standard deviational ellipses
make excellent instructional tools. A common example shown in most geog-
raphy textbooks is the mean center of the population in the USA from 1790
to 2010. Over those 220 years, this mean center moved from Maryland to
southwestern Missouri, reflecting the migration and the settlement pat-
terns diffusing from the east coast of the country to the west and, later,
to the south. A population mean center for Michigan lies to the south and
east of the geographic center, reflecting the larger population in cities such
as Lansing and Detroit, which lie in the southern part of the state. These
population centers and higher rural population density surrounding them
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