Table Of ContentSPATIAL
ANALYSIS
A READER IN STATISTICAL GEOGRAPHY
BRIAN J. L. BERRY
University of Chicago
DUANE F. MARBLE
N orthweatern University
PRENTICE-HALL. INC., ENGLEWOOD CLIFFS, NEW JERSEY
2
A QUANTITATIVE EXPRESSION
OF THE PATTERN OF URBAN SETTLEMENTS
IN SELECTED AREAS OF THE UNITED STATES
LESLIE J. KING
The agglomerated settlement in which there is another. The widely spaced, irregular pattern
a concentration of nonfarm activities is a ubi of settlements characteristic of much of the
quitous feature of the American landscape. semiarid West, for example, contrasts strongly
Throughout the country these urban settle with the more compact and rectangular arrange
ments range in population size from that of the ment of settlements to be found in the Midwest.
small lowly hamlet of 100 or so inhabitants up Many comparisons and contrasts along these
to the large metropolitan area of several lines have already been pointed up within the
million people. Functionally they also vary in body of geographic literature.1 However, the
character, although the majority of them might greater part of this descriptive work to date
be considered to have basically the same raison has been characterized by an almost unquestion- ·
d'etre in that they provide some form of service ing reliance upon [such] qualitative terms as
to the inhabitants of the surrounding rural "sparse," "dispersed," "agglomerated," or
areas. However, the demand for goods and "dense." While these terms have meaning with
services originating within the farming sector reference to the context in which they are used,
of any large area can seldom be satisfied by a they lack objectivity for more extended com
single urban center. On the contrary, the parative purposes. It appears desirable, there
physical and economic limitations to move fore, that a more precise connotation be given
ment over the surface of the earth generally to these descriptive terms as a means of facili
necessitate the existence of several urban tating comparative analysis and the discovery
settlements within any large area. Furthermore, of more universal generalizations concerning
the fact that these limitations are not constant the nature of settlement patterns.
in their effects throughout the continent is
Reprinted from Tijdschrift Voor Econ. en Soc.
reflected in the character of the distribution
Geographie, 113 (1962), 1-7, by permiesion of the author
pattern of urban settlements from one area to and editor.
1159
160 ANALYSIS OF SPATIAi. DISTRIBUTIONS
The problem of deriving a ma.thema.tica.l probability• 1 it can then be demonstrated that
expression of the pattern of settlement dis the mean distance (r.E) between each point and
tribution has already received considerable its nearest neighbor which could be expected
attention from European geographers. In this in such a random distribution is equal to 111p-111,
regard, the works of Berna.rd, 2 Cola.s, 1 Debou where p is the observed density of points in the
verie4 Demangeon° Mcynier,1 and Zierho.ffer7 area. under consideration. The ratio of the
a.re especially worthy of mention. observed mean distance (rA) to this expected
However, these earlier formulations were value (r.E), is termed the near-neighbor statistic
based on certain critical ratios--for example, (R). This ratio has a. range in value from zero,
the average area per dwelling or the number of when there is maximum aggregation of all the
settlements in the commune-and generally points at one location (Fig. la), through 1, which
no analysis was ma.de of the actual linear dis represents a random distribution, up to 2.1511
tances separating the settlements within a.n which is expressive of a pattern of maximum
area..1 These attempts a.t quantitative analysis spacing analagous to the hexagonal arrangement
have therefore been criticized for their gener discussed by Christa.lier and others" (Fig. le).
ality and for the fact that they' are frequently In this sense the statistic R provides a mean
insensitive to important variations in settle ingful and precise expression of the distribution
ment pattern.• pattern of points within any area. Furthermore,
In this study there is outlined a new approach the ratios for different areas can be compared
to the problem of expressing the character of directly with one another, and the significance
settlement patterns in mathematical terms. of the differences between them can be assessed
The approach is based upon modern statistical by standard statistical procedures. 16 For any
theory and the notions of probability. In con single area it is possible to test not only the
trast to earlier work in this field, greater em significance of the departure of the observed
phasis is given to the actual distance separating mean distance (rA) from that expected in a
settlements, and the statistical techniques random distribution (r.E), but also the signifi
which a.re used in this study are in fact known cance of the deviation of the observed mean
as the "near-neighbor analysis." distance from that which would be expected if
Statistical analysis of the near-neighbor the points within the area were either ag
measure, which is, a.s the name suggests, a gregated at one location or dispersed in a. hexag
straight-line measurement of the distance onal pattern. In the first of these two hypo
separating any phenomenon and its nearest thetical oases tho expected value of r A would
neighbor in space, was originally developed by be zero, while in the second it would be equal to
plant ecologists who were concerned with the 2111/(3''~p'fl). where pis defined as before.11
distribution patterns of various plant species In this s~udy an attempt is ma.de, using the
over the surface of the ea.rth.10 Near-neighbor techniques outlined above, to describe some of
analysis indicates the degree to which any the empirical characteristics of the distribution
observed distribution of points deviates from of urban settlements in different areas of the
what might be expected if the points were United States. Whereas in the preceding dis
distributed in a random manner within the same cussion the emphasis has been on a set of points
area. A random distribution of points is defined distributed in space, the focal point of attention
as a set of points on a given area. for which now becomes a number of urban places which
"any point has had the same chance of occurring are distributed over the surface of the earth.
on any suba.rea. as any other point; that any It is assumed that within each area selected
subareo. of specified size has had the same chance for study there is an equal chance that an urban
of receiving a point as any other subarea. of that settlement will be located on any one unit area
size, and that the placement of each point has of ground as on any other area, and conversely,
not been influenced by that of any other point."11 that every location has an equal chance of
While this is essentially a mathematical con receiving an urban center. Location theory
cept, it is possible by use of a square grid with suggests that these assumptions are quite
numbered intersections and a. random numbers unrealistic, but as a theoretical norm or model
table to construct a.n artificial random distri they have considerable value.
bution (Fig. lb). From the laws of mathematical The sample areas for this study were chosen
A QUANTITATIVE EXPRESSION OF THE PATTERN OF URBAN SETTLEMENTS 161
THEORETICAL SETTLEMENT PATTERNS
..
..
......
.• ...e .... .... .... ....• ...I. . ..I ..I .. . .
·.· ·.·..·.. . ...
...
..
. .. .
::
. .
...
AGGREGATE RAl'.OOM UNIFORM
FIGURE 1
in such a way as to insure a representative area were located closer to one another than
cross section of as many of the physical and they were to any other town, in which case the
economic regions within the United States as same distance was measured twice.
was practicable. The locations of the twenty The obtained values of the near.neighbor
areas which were selected are shown in Fig. 2. statistic R are presented in Table 1. Intuitively
Within each area the term "urban settlement" it was expected that these values would vary
was taken to include all settlements listed in the in magnitude from ono area to another, for it
United States census17 together with any smaller would seem as though variations in the physical
communities shown on the United States geography, the economic base, transportation
Department of . Commerce Transportation facilities, and land-occupance history are likely
Maps,18 and published road atlases.1° to influence the spacing of urban settlements
For each area a series of straight-line measure. in any area. An examination of Table l reveals
mcnts were taken between the urban centers that the expected range in value for the near
and their respective nearest neighbors.20 There neighbor statistic R did occur, and it is apparent
was no consideration given to the population that the tendencies toward aggregated, random,
size of the towns concerned, since the emphasis and uniform spacing of towns vary considerably
was upon urban settlement as an attributal throughout the country.
phenomenon, and while it is highly desirable On the one hand, the most marked tendency
that the variable, population size, should toward an aggregated-settlement pattern ap
eventually be incorporated into the analysis, pears in two of the areas within the western
for the time being it is disregarded. Cordillera system, namely, Washington and
In many cases the nearest neighbor of a given Utah, for which the R values are .71 and .70,
town lay outside of the sample area. The respectively. These low values here reflect an
distances to these neighbors were measured and almost linear arrangement of pairs of towns
included in the computations, but no town along major rivers, the Yakima River in
lying outside of the sample area was ever used Washington and the Duchesne River and its
as a center of measurement. In addition, it tributaries in Utah (Fig. 3). In ea.ch case settle
was often the case that two towns within an ment is associated with irrigated "oases," while
162 ANALYSIS OF SPATIAL DISTRIBUTIONS
LOCATION fl' SAMPLE AREAS
--
.
FIGURE 2
TABLE 1
Near-neighbor statistic&
Sample Area Number Density Mean .Ezpecletl Near- N at.ure of pat.tern
of of tOU1718 obaerwd mean neighbor
WUlnB per square diawnce diawnce 8'at.iatio
mile (miles) rA in random R
diatribWion
(miles) rE
Co.lifornia. 96 .0243 3.46 3.21 1.08 Random
Florida. 64 .0200 3.32 3.53 .94 u
Georgia. 132 .0350 3.52 2.67 1.32 Approa.ching uniform
Iowa. 82 .0307 3.86 2.85 1.36 " u
Kansas 51 .0166 5.16 3.88 1.33 " "
Louisiana. 140 .0437 2.57 2.39 1.08 Random
Minnesota. 55 .0169 5.32 3.85 1.38 Approa.ching uniform
Mississippi 104 .0280 3.84 2.99 1.28 II u
Missouri 80 .0219 4.67 3.38 1.38 II II
New Mexico 23 .0065 6.82 6.20 1.10 Random
North Dakota. 28 .0082 6.13 5.52 1.11 II
Ohio 131 .0512 2.80 2.21 1.27 Approa.ching uniform
Oregon 128 .0317 2.86 2.81 1.02 Random
Pennsylvania. 177 .0466 2.28 2.32 1.22 Approa.ehing uniform
Texas (N.W.) 38 .0104 6.03 4.90 1.23 II II
Texas (S.E.) 61 .0182 4.29 3.70 1.16 II II
Utah 20 .0061 4.49 6.40 .70 Aggregated
Virginia. 122 .0363 3.20 2.62 1.22 Approa.ching uniform
Washington 32 .0073 4.14: 5.85 .71 Aggregated
Wisconsin 97 .0299 3.58 2.89 1.24 Approa.ching uniform
A QUANTITATIVE EXPRESSION OF THE PATTERN OF URBAN SETTLEMENTS 163
the greater part of both areas remains virtually
uninhabited. With such an arrangement of
pairs of settlements, the observed mean distance
between towns and their nearest neighbors tends
toward zero, and as a result the obtained value
AGGREGATED PATTERNS of R approaches that which would be charac
teristic of a more aggregated grouping.
At the other extreme, a. tendency toward
uniform spacing was apparent in twelve of the
areas studied, most notably in Minnesota and
= =
Missouri (R 1.38), and in Iowa (R 1.35).
At the same time, however, additional statistical
. .. tests concerning the magnitude of the r A values
.: .. . ..·. · .. .. .. .... . .. .. revealed that in no area could the distribution
' \. . ·. of towns be described as a. truly uniform one
:
(Table 2). Nevertheless, the fa.ct that the
tendency toward uniform spacing is most
1
marked in the midwestern states of Minnesota,
Missouri, and Iowa. (Fig. 4), emphasizes the
·~
descriptive value of the near-neighbor statistic
UHM
R, since the general uniformity of relief, the
FIGURE 3 method of original subdivision of the land based
upon the quarter section, the rectangular road
pattern, and the generally even spread of pop
ulation associated with a relatively intensive
feed-grain/livestock economy are characteristic
TENDENCY TOWARDS UNIFORM SPACING
.
• •
• • • • • • • • • • •• • •• • • • .•. • ••• •• • • • • • •• • •• •• • •
• • • • • •• •• • • •• • • •• • •• ••• •• •••• •• ••• ••• • • • ••• •• • ••• • • •• • ••• •• • •••• ••• ••• ••• •• •••••• •••• • • •••••• ••• ••••• • •• •• •• • •••
• • • •
• • • • •
• • • •
• •
-
2 I ...I O IO 0 I ...IO.. IO 0 I 10 ao
"
MINNESOTA MISSOURI IOWA
l"IGURE A
164 ANALYSIS OF SPATIAL DISTRIBUTIONS
TABLE 2
Tests for uniform spacing
Area Denaity of towna Mean obaerved E:z;pect.ed mean Difference
per aquare cliatance cliatanee in uniform between
mile rA (miles) diatribution of given rA and ru
clenaity ru Significant?
California. .0243 3.46 6.90 Yes
Florida .0200 3.32 7.59 "
Georgia .0350 3.52 5.74 "
Iowa. .0307 3.86 6.13 "
Kansas .0166 5.16 8.34 "
Louisiana. .0437 2.57 5.13 II
Minnesota .0169 5.32 8.27 II
Mississippi .0280 3.84 6.42 "
Missouri .0219 4.67 7.27 II
New Mexico .0065 6.82 13.28 II
North Dakota .0082 6.13 11.86 II
Ohio .0512 2.80 4.75 II
Oregon .0317 2.86 6.03 II
Pennsylvania. .0466 2.82 4.98 II
Texas (NW) .0104 6.03 10.55 "
Toxos (SE) .0182 4.29 7.97 II
Utah .0061 4.49 13.74 "
Virginia. .0363 3.20 5.64 II
Washington .0073 4.14 12.57 II
Wisconsin .0299 3.58 6.22 II
feo.tures of these areas which would favor a climate and land use it is significantly different
more even spacing of towns. That even in from the other midwestern areas. A genera.I
these areas there is not apparent a more marked east-west linearity in the arrangement of many
tendency toward a hexagonal arrangement of of the towns within this area reflects the direc
towns can in part be attributed to adverse tion of the major transportation routes which
physics.I features, such as the glacial lake. traverse the area (Fig. 5). A similar control of
strewn surfaces of Minnesota; to the linear and settlement patterns by transportation networks
dendritic patterns associated with the trans is apparent in the Mississippi area and on the
portation routes of rivers, highways, or rail Staked Plains of northwestern Texas. In both
ways; to original site features, such as river of these areas a predominantly rectangular
fords, power sites, forestry and mining centers; pattern of major highways is reflected in the
and finally to mere cha.nee. The pattern of relatively close spacing of towns a.long these
settlements in Wisconsin (R = 1.24) appears routes, while in tho interstitial areas towns a.re
to reflect the importance of these factors to a.n spaced further a.part. These conflicting patterns
even greater extent, for in this area. the tendency within the areas have tended to lower the ob
toward uniform spacing is even less pronounced tained values of R. By contrast, in Georgia. the
than in the other midwestern areas, notwith pattern of transportation routes is in very close
standing the fact that it is very akin to these agreement with the arrangement characteristic
same areas in its broad physical, economic, and of a hexagonal pattern of settlements (Fig. 5).
social characteristics. Many of the larger towns, for example, appear
Elsewhere throughout the United States the to be associated with the intersection of five or
tendency toward a uniform spacing of settle six radial transportation routes. However, the
ments is most marked in Kansas and Georgia comparatively low value of R is attributable to
and is of varying importance in a further six the irregular spacing of numerous small set
areas. Kansas might well be considered to be tlements along and between the major highways.
part of the Midwest, although in terms of Considerations such as these emphasize the
A QUANTITATIVE EXPRESSION OF THE PATTERN OF URBAN SETTLEMENTS 165
SETTLEMENTS AND ROAD NETWORKS
• •
0 I 10 ZO
0 5 10 10
lllHH
Milll
KANSAS GEORGIA
FIGURE 15
desirability of incorporating the population-size In the remaining areas of southeastern Texas,
variable into the analysis. In many areas the Virginia, and Pennsylvania, in which the
tendency toward either aggregation or uni tendency toward a uniform spacing of settle
formity would undoubtedly be more pronounced ments is also statistically significant, the results
if the distribution patterns of only certain size a.re more difficult to interpret. In each case the
towns were to be analyzed. A further inade observed mean distance (rA) is greatly exceeded
quacy of the study design is pointed up by a by that which would be expected of a uniform
consideration of the pattern of settlements in distribution. In Pennsylvania this discrepancy
Ohio (Fig. 6). In the northwestern half of this stems in part from the fa.ct that there were
area, that is, to the north and northwest of the many closely spaced settlements in the north
Maumee River, the pattern of settlements east-southwest trending valleys, while the ridges
appears more uniform than it does to the south and interfl.uves were frequently devoid of
of the river, where there is considerable linearity settlements. Similarly, on the Virginia. Pied
in the arrangement of the settlements. Possibly mont and the Texas Coastal Prairies, a. close
this is related to the fa.ct that in the northwest spacing of settlements a.long the major trans
the terrain is slightly more hilly in association portation routes with wide areas of intervening
with old morainic deposits, whereas to the territory in which nucleated settlements a.re
south and southeast a lacustrine plain surface few in number, resulted in comparatively low
predominates. Considerable care should ob values for the observed mean distance.
viously be exercised in the choice of sample What of the six areas in which the distribu
areas, since, as in this case, two distinct pat tion of settlements is apparently random
terns within an area may tend to contradict one (Table 1)1 These anomalies a.re easily explained
another and thereby render the statistical results a.way in terms of the mathematics of the near.
largely meaningless. neighbor measure. For example, the compa.ra-
RANDOM PATTERN
. .
. ..
•
. . .
• • •
• • •
•
• •
•
0 I ..1.0. . So
NORTH DAKOTA
FIGURE 7
9 IO IS
llilll a uniform spacing of settlements becomes more
pronounced.
FIGURE 8
The discussion up to this point has been
focused upon the validity of the near-neighbor
tively low value of R for North Dakota is a statistic as a descriptive statistic of the manner
function of, on the one hand, the low density in which urban settlements are distributed over
of towns per square mile which accounts for a portions of the earth's surface. Considerable
large value of rE, and, on the other hand, a emphasis has been given to the interpretation
large value of rA resulting from the fact that of the various obtained values of R, and it is
there are only a few widely spaced towns evident that the statistic is sufficiently sensitive
within the area (Fig. 7). However, the more in character to point up the differences which
important consideration is whether or not a do exist between various distribution patterns.
random distribution of settlements has any Whether or not the hypothetical norms of
real geographic significance. In the North aggregation, randomness, and uniform spacing
Dakota area, for example, the critical fact a.re accepted as meaningful reference points, the
appears to be that the settlements a.re located near-neighbor statistic provides a logically
a.long three major transportation axes which acceptable and precise expression of the charac·
have an approximate ea.st-west orientation, ter of a distribution pattern of settlements.
rather than that the distribution is statistically Furthermore, inasmuch as the statistic ranges
random. Indeed, the concept of randomness in value from 0 to 2.15, within which limits
with respect to settlement patterns might well any value is possible, it is logically consistent to
be disregarded, except for the fact that the consider the near-neighbor statistic as a con·
value of R = 1 is a convenient and useful tinuous variable over this range. Accepting this
origin from which to measure the tendencies to be the case, then a number of powerful,
toward an aggregation or uniform spacing of sophisticated, statistical techniques, such as
settlements. As the value of R decreases and correlation and regression analysis, ca.n be
approaches zero, so the tendency toward an brought to bear on the more fundamental and
aggregation of settlements might be presumed important problem of providing an explanation
to be greater, and, conversely, as the value of of the variation in the magnitude of the obtained
R increases from 1, then the tendency toward values of R. Some tentative hypotheses relevant
A QUANTITATIVE EXPRESSION OF THE PATTERN OF URBAN SETTLEMENTS 167
to this problem have already been suggested in percentage of total area. in farms is high, or,
this study, but the degree to which these finally, the percentage of the total population
hypotheses a.re valid with respect to a.11 of the classed a.s rural farm population is high. Con
areas has yet to be determined. While an ex versely, the tendency toward an aggregation of
tended analysis along these lines falls outside settlements will be greater a.s the magnitude of
of the scope of this paper, it might be noted that these ea.me variables decreases. It should be
the value of R appears to be related in a. positive stressed that the amount of explained variation
direction with the variables percentage of an in the values of R which is accounted for by the
area in cropland, percentage of land area in relationship of these variables is not particularly
farm11, and percentage of total population classed high, and there a.re obviously a. number of
as rural farm. 11 That is to ea.y, the tendency additions.I relevant variables which must be
toward a. uniform spacing of settlements ap incorpora.ted into the analysis before a satis
pears to be more pronounced in those a.rea.e in factory level of explanation is achieved. The
which the amount of cropland is high, or the problem awe.its further investigation.
FOOTNOTES
1 See, for example, J. Garland, ed., The North Vol. I: L'Habitat rural (Paris: Centre de Documenta
American Midweae (New York: John Wiley & Sons, tion Univeraitaire, 1959), pp. 72-75.
Inc., 1955), pp. 28-39. Also, H.H. McCarty, The 10 See P. J. Clark and F. C. Evans, "Distance to
Geographic Bcuia of American Economic Life (New Nearest Neighbour as a Measure of Spatial Relation
York: Harper & Row, Publishers, 1940). In his discus ships in Populations," Ecology, 85 (1954), 445-453.
sion of the Great Plains area, for example, McCarty II Ibid., 446.
states: "Cities and towns are widely spaced, since 11 The derivation of the rlil values involves con
their frequency nece&B&rily varies with the density of sideration of the Poisson exponential function. The
the open.country population" (p. 237). mathematical derivation of the formulae used in this
2 J. Bernard, "Une Formula pour la cartographie de study is presented in ibid., 451-452.
l'habitat rural aveo application au departement de 11 It is true that the mean distance between nearest
l'Yonne," Oomptu Rendua du Oongria International de neighbors is maximized in a hexagonal distribution
Geographie, a (1931), 108-117. where each point has eix equidistant nearest neighbors.
a M. R. Colas, "Repartition de }'habitat rural," In this case it can be ehown that the maximum value
Bulletin de l'Aaeociaeion Gl.ographigue Franpaia (1945), for rA is 21/1/(31/, p 1/1) and that R = 2.15.
51-56. 1' W. Christs.lier, Die zentralen Orte in 81ldcleutach
' A. H. Debouverie, "Une Methode a base numeri land (Jena, 1933); A. L6soh, Die rdmuliche Ordnunq der
que pour la cartographie de l'habitat, specialement Wirtachaft (Jena, 1944); J. E. Brush, "The Hierarchy
applicable a la Belgique," Bulletin de la Sociite Belge of Central Places in Southwestern Wisconsin," Geo
d'Etudu G8ographiguee, 13 (1943), 146-196. graphical Review, 43 (1953), 380-402.
s A. Demangeon, "Une Carte de }'habitat," Annalu 15 An analysis of variance is sufficient. For the
de Geographie, 4Z (1933), 225-232. neceaaa.ry transformations see Clark and Evans, op. cu.,
e A. Meynier, "L'Habitat rural dans lea Begs.las," 462.
Oomptea Rendua du Oongru International de Gl.ographie, 19 Ibid., 452.
a (1931), 90-102. 11 U.S. Bureau of the Census, Seventeenth Decennial
1 A. Zierhoft'er, "Sur une Formula servant a expri Oenaua of the Untied Slatu: Oenaua of Population, 1950,
mer la dispersion et la concentration absolue de Vol. I: Number of Inhabitants (Washington: U.S.G.
l'habitat rural," Oomptu Rendua du Oongr~a Interna P.O., 1962).
tional de G8ographie, I (1934), 410-415. ta U. S. Department of Commerce, United Seatu
a In a senee thie judgment is not entirely correct. Transportation Mapa (Washington: U.S.G.P.O., 1938-
For example, Zierhoffer considered dwellings to be 49).
dispersed if more than 150 to 200 meters separated 11 Principally: Road Atlas (Chicago: Rand Mc
them. Debouverie, in his study of Belgium, regarded Nally & Co., 1959).
the distance of 100 meters between dwellings as the IO The distances, measured in miles, represent the
limit of dispersion. However, no attempt was ever direct airline distances between the approximate
made to analyze these distances and to provide an geographic centers of the towns concerned.
explanation for the variation in them. In effect, the 21 These tentative conclusions were presented in
distance between dwellings (or towns) was always L. J. King, "Consideration of the Spatial Distribution
regarded as a constant and not as a variable. of Urban Places in Selected Areas of the United
1 See J.M.A. Houston, Social Geography of Europe States," unpublished manUBOript, Department of
(London: Gerald Duckworth & Co., Ltd., 1953), pp. Geography, State University of Iowa (1958).
81-85; and J. Tries.rt, Ooura de gwgraphie humaine,
