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 OCCASIONAL PUBLICATIONS OF THE DEPARTMENT OF GEOGRAPHY 
 
 METHODS AND MEASURES OF 
 CENTROGRAPHY 
 
 A Critical Survey of Geographic Applications 
 
 SUM SOOT 
 
 PAPER NUMBER 8 
 APRIL 1975 
 
 
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 LUIS E. ORTIZ and SUSAN GROSS, editors 
 
 GEOGRAPHY GRADUATE STUDENT ASSOCIATION 
 UNIVERSITY OF ILLINOIS at URBANA CHAMPAIGN 
 
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 which it was withdrawn on or before the 
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 are reasons for disciplinary action and may 
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 UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN 
 
METHODS AND MEASURES OF CENTROGRAPHY 
 A Critical Survey of Geographic Applications 
 
 Si im Soot" 
 
 ABSTRACT 
 
 Centrographic measures have been utilized for decades to draw gen- 
 eralizations about areal distributions. They have proven useful in tem- 
 poral and comparative analyses in discerning trends and contrasting popu- 
 lation distributions. Misconceptions relating to the mathematical deri- 
 vations, and analytic and descriptive properties of many of the basically 
 simply centrographic techniques which have arisen are explicated in this 
 paper. This is done in a framework of a geographic literature review of 
 three measures of central tendency (mean center, median center and point 
 of minimum aggregate travel) and four measures of dispersion (standard 
 deviational ellipse, bicircular quartic and sectogram) . The merits of 
 each measure are specified and contrasted with those of similar measures. 
 
 INTRODUCTION 
 
 Centrographic measures have been utilized for analytic and descrip- 
 tive purposes in a wide range of studies. For example, the center of 
 gravity has frequently been employed to illustrate longitudinal trends 
 in areal distributions (Waterman 1969; Stephenson 1972; Larson and Soot 
 1973a; Larson 1973b), while Koch (19^2) and Murphy and Spittal (19^5) 
 have used it to compare centers of functionally related variables. Also 
 many have used the standard radius or a modification thereof to summarize 
 the dispersion about a point (Yui 1 1 1971; Buttimer 1972; Caprlo 1969; 
 Stephenson 1972). Due to their basic simplicity, these measures have a 
 high degree of appeal, but they are analytically limited. Moreover, as 
 generalizing methods they reduce distributions to the most basic charac- 
 teristics, while overlooking potentially Important features. 
 
 * Dr. S86t is Assistant Professor of Geography at the University of Il- 
 linois, Chicago Circle Campus. 
 
The fundamental principles underpinning centrographic measures are 
 elementary, but their applications are marked with misuse and general 
 misinterpretation. Thus, it is necessary to promote proper use of these 
 measures by bringing attention to procedural errors. This paper, then, 
 has a two-fold purpose: (1) to review the centrographic literature with 
 an emphasis on the correction of misconceptions, and (2) to evaluate 
 several centrographic measures. 
 
 MEASURES OF CENTROGRAPHY 
 
 Centrographic measures can be used to describe two features of dis- 
 crete distributions: (1) central tendency, and (2) dispersion. The 
 former is a point which is found by applying at least one averaging criter- 
 ion. Since there exist several averaging criteria, there are several such 
 points. The most common are the center of gravity (mean center), median 
 center, and the point of minimum aggregate travel. The latter feature, 
 dispersion, depicts the degree of scatter from a point of central tendency. 
 In the two dimensional case, there are numerous techniques and figures 
 which demonstrate the scatter of points: the standard radius, the standard 
 deviational ellipse, the standard deviational bicircular quartic and 
 several versions of the sectogram. 
 
 MEASURES OF CENTRAL TENDENCY 
 
 Three univariate measures of central tendency are commonly recognized: 
 mean, median, and mode. The univariate mean has an easily computed and 
 interpreted bivariate counterpart, the center of gravity. The bivariate 
 
 It is the belief of the author that centrographic measures also provide 
 indirect information about pattern, but cannot be classified as useful 
 measures for evaluating the same. 
 
mode is also easily interpreted; it signifies the greatest concentration 
 at one point or within a given area. The univariate median, however, 
 does not have a two dimensional equivalent possessing analogous attributes. 
 The median, as illustrated by Porter (I963), is the point of minimum 
 aggregate travel (PMAT) in the univariate case. But the bivariate PMAT 
 and the (orthogonal) median points are not the same. Each of these bi- 
 variate measures will be discussed in greater detail below. 
 
 Mean Center . The early prominence of the mean center (center of 
 
 gravity) is due partially to the fallacious assumption that it represented 
 
 2 
 the PMAT. This confusion was cleared up by Eel Is (1930). The use of 
 
 this averaging measure has continued, however, based predominantly on its 
 sensitivity to the location of every point. That is, a shift in any data 
 point will induce at least a minor shift in the mean center. Yet as Hay- 
 ford (1902, p. 50) points out, there is some objection to the undue weight 
 ascribed to distant points, since by increasing or decreasing in number, 
 they have more effect on the mean center location than close-in points. 
 Still, the measure is attractive since it is computationally simple; 
 
 /--\,-Ewx -Zwy /,\ 
 
 (x,y) where X = -^^ , Y = -^ (0 
 
 and w = weight at each point 
 
 X = value on an arbitrarily chosen x-axis 
 y = value of an axis orthogonal to the x-axis. 
 The center of gravity is particularly valuable when used in a compara- 
 tive analysis with other centers. These other centers may represent either 
 the same variable for a different point in time (Figure 1) or a temporally 
 invariate set of variables (Figure 2). For time variant data sets, the 
 
 2 
 
 The U.S. Bureau of the Census (1921, p. 32) equated the center of popula- 
 tion with the point of minimum aggregate travel. 
 
Figure 1. Illinois Centers of Population, I83O-I97O 
 
 Chicago 
 
 1870 
 I86O9 
 
 1970 
 
 1960j 
 1930 1950 
 
 ^^^^^001940 
 1900«1910 
 
 ©1890 
 ^olSSO 
 
 1850 
 
 ©center of 
 
 Springfield- ^ 1340 
 
 and 
 
 mass 
 
Figure 2. Nine Illinois Mean Centers, I970. 
 
 \ 
 
 1— 
 
 
 miles 
 
 50 
 
 / 
 
 A Population \ 
 
 
 
 
 / 
 
 B Non-white ' 
 population 
 
 C Foreign born 
 population 
 
 { 
 
 .r- 
 
 -T 
 
 H Social security 
 recipients 
 
 D Farm population 
 
 F Education (median school 
 
 years completed) 
 G Population with four or more years 
 
 of college 
 
 K Families below 
 poverty level 
 
 M Land area 
 
 uicc 
 
migration of the centers can be utilized to interpret the sum effect of 
 
 3 
 factors influencing the distribution. Centers of temporally invariate 
 
 data sets can demonstrate regional imbalances in the respective variables. 
 
 In each case, one point generalizes an entire weighted distribution. 
 
 Figure 3 depicts the population distribution summarized by the 1970 mean 
 
 center on Figure 1 . 
 
 In analyzing the movement of mean centers, the direction and rate of 
 migration are particularly important. Janelle (1971) in examining resi- 
 dential, public, commercial, and industrial land use surfaces, goes one 
 step further. He suggests the concepts of velocity, acceleration, and 
 momentum to describe the shifts in the mean center and thereby the data 
 they represent. The velocity is the rate of shift of the mean center in 
 a given direction, the acceleration is the rate of increase in velocity, 
 and the momentum is a product of the mass and velocity. Of the three, 
 velocity and momentum are the most practical. 
 
 Velocity is useful In that it indicates the distance and direction 
 moved per unit time period. The velocity vectors, as shown for Illinois 
 in Figure 4, designate the utility of this statistic. Note that the post- 
 19^0 vectors are easily distinguishable from the 1900-1930 vectors. 
 
 The advantage of momentum Is that it compensates for the deceleration 
 occurring with population increases. But since momentum is a product of 
 
 3 
 
 Figure 1 captures exceptionally well the almost relentless movement of 
 
 the center toward Chicago. The only reversal, during the 1930-19^0 
 
 decade, was precipitated by a period of economic insecurity, as the 
 
 migration from the farm to the city reversed directions. This decrease 
 
 in urban migration was significant enough to cause the only southward 
 
 movement of the center over a lAO year period. The population centers 
 
 also suggest why Springfield was selected the State Capital, as it was 
 
 the urban place closest to the 18^0 center. 
 
 For example. Figure 2 illustrates the degree to which the distribution 
 of foreign born population (C) is biased toward the northeast. 
 
Figure 3- I97O Illinois Population by Counties 
 
Figure ^. Velocity Vectors of Population Centers, Illinois 1850-1970 
 
 
 
 
 
 
 1890 
 
 n 
 
 
 
 
 I860 
 
 
 / 
 
 15 
 
 
 (N) 
 
 / 1930 
 
 / 4 1900 
 
 / 
 
 / 
 
 lO 
 
 
 
 /920// 
 
 1 A / / 
 
 
 ' 
 
 5 
 
 1970/ 
 
 I960 /> 
 
 i 
 
 117/ 
 
 //Zl910 > 
 
 j^1880 
 ^1870 
 
 
 O 
 
 E 
 
 1940 
 
 ^ 
 
 
 
 
 
 1 
 
 
 
 
 uicc 
 
 O miles 
 
 lO 
 
 The vector origins are placed at a common point to illustrate the pattern 
 of movement over the last 110 years. Two irregular trends can be observed: 
 the movement is approaching the north-south axis, and the speed is 
 decreasing. 
 
mass and velocity, it can Increase even if speed decreases. It, there- 
 fore, permits comparisons between the total impacts of population redis- 
 tribution In the early growth of Illinois and the most recent decades. 
 It Is essentially a means of "standardizing" the data for comparative 
 purposes. Notice that the momentum increased from I83O to I88O while 
 the speed was decreasing (Table 1). In short, the momentum shows that 
 the greatest thrust of population redistribution toward Chicago took place 
 during the prosperous 1920*s while the I88O-I89O decade was second, 
 followed by the rural-urban migration of the 1950's. 
 
 The acceleration of the mean centers can also be computed but it is 
 felt that such an exercise in this problem would add little to what Is al- 
 ready apparent in Figure k and it would only over-quantify the population 
 redistribution data. A cursory observation of Figure k reveals that 
 acceleration (deceleration) was great before the turn of the century and 
 relatively small since 1930. This is not unexpected since centers of grow- 
 ing populations are characterized by deceleration. 
 
 Although Janelle's methods are mathematically sound they have not re- 
 ceived widespread use. Perhaps the primary reason is the lack of reliable 
 temporal data In the consistent geographic units to examine these trends. 
 Aside from population figures how many data sets span one hundred years? 
 Also the measures such as acceleration are frequently too abstract to be 
 useful in a wide variety of circumstances. 
 
 Median Center . The median center suffers from a major deficiency: 
 it is not a unique point. This deficiency arises from the method of 
 determination. The point is defined as the intersection of two orthogonal 
 axes, each of which divides the distribution into equal halves. As the 
 
 The measure Is, therefore, also known as the orthogonal median point. 
 
10 
 
 Table 1. Movement of the Illinois Population Center, I83O-I97O 
 Mass, Speed, Acceleration, and Momentum 
 
 
 Decade 
 
 Mass^ 
 
 Distance or speed 
 
 Acceleration 
 
 Momentum 
 
 1830- 18^0 
 
 316 
 
 56.2 
 
 
 1776 
 
 18^0-1850 
 
 663 
 
 33.0 
 
 -23.0 
 
 2187 
 
 1 850- 1860 
 
 1281 
 
 16.6 
 
 - e.k 
 
 2120 
 
 1860-1 870 
 
 2125 
 
 S.k 
 
 - 8.2 
 
 1787 
 
 1870-1880 
 
 2808 
 
 9.5 
 
 1.1 
 
 2675 
 
 1880-1 890 
 
 3^52 
 
 20.3 
 
 10.8 
 
 7015 
 
 1890-1900 
 
 ^32^ 
 
 12.^ 
 
 - 7.9 
 
 5361 
 
 1900-1910 
 
 5230 
 
 7.1 
 
 - 5.3 
 
 3696 
 
 1910-1920 
 
 6062 
 
 7.7 
 
 0.6 
 
 4695 
 
 1920-1930 
 
 7058 
 
 11.9 
 
 4.2 
 
 8381 
 
 1930-19^0 
 
 7763 
 
 0.9 
 
 -10.0 
 
 699 
 
 I9AO-I95O 
 
 830^ 
 
 4.7 
 
 3.8 
 
 3873 
 
 1950-1960 
 
 9397 
 
 6.2 
 
 1.5 
 
 5805 
 
 1960-1970 
 
 10598 
 
 2.8 
 
 - 3.4 
 
 2919 
 
 Average population in thousands. 
 
 Distance in miles per decade and speed are the same. 
 
 Per decade change in speed. 
 
 Mass times distance expressed in tens of thousands of population miles. 
 
11 
 
 orthogonal axis system is rotated, a shift is necessary to maintain the 
 median quality of the axes (Figure 5). Since generally no one axis orien- 
 tation can be justified over another, there is no rationale for selecting 
 one median center. However, the median center is less influenced by ex- 
 treme data points than the mean center (Prunty, 1951, p. 202) such that 
 the points may be moved anywhere within their quadrant (Figure 5) without 
 affecting the median center. The median center also is much more easily 
 determined than the mean center, but it is not the point of minimum aggre- 
 gate travel, as is true in the univariate case (Porter, 1963). 
 
 A city with a grid transportation network, however, represents an 
 important exception. If all the movement in an area is restricted to 
 streets parallel to one of two orthogonal axes (grid pattern), then these 
 axes can define a unique center which the author contends is also the PMAT. 
 In the bivariate case if we collapse the two dimensional distribution to 
 points on two perpendicular axes, then the median can be computed individ- 
 ually on each axis (Figure 6). One axis can then be moved, without dis- 
 turbing orthogonality, so that the two median points coincide. This point 
 then is the PMAT, since all travel can be interpreted as being confined 
 to the x and y axes. By the procedure outlined above, travel on these 
 axes has been minimized. Thus, in any system in which movement is confined 
 » to a grid network, the PMAT is also the orthogonal median point. Although 
 this is a special case, several cities and some agricultural areas approxi- 
 mate the perfect grid system. The spacing of the streets, of course, is 
 not important but parallelism and orthogonality must be maintained. 
 
 Point of Minimum Aggregate Travel (PMAT) . The PMAT has significant 
 practical value as it designates the point to which the total travel by 
 all in the study area is minimized. For example, in locating a state 
 
12 
 
 Figure 5. The Median Centers and Axes Rotation 
 
 t 
 
 median centers A and B 
 mean center 
 
 UICC 
 
13 
 
 Figure 6. The Median Center and the Point of Minimum Aggregate Travel 
 
 •- 
 
 I 
 
 I 
 
 \< • 
 
 < • 
 
 A. 
 
 Original hypothetical distribution 
 
 The data points are assigned to 
 their respective X and Y axes values 
 
 -• — •- 
 
 median point- 
 
 median point 
 
 e • 
 
 The median points (the points of 
 minimum aggregate travel on each 
 axis) are identified. 
 
 e e 
 
 PMAT 
 
 ^ — •- 
 
 Preserving orthogonality, one axis is 
 shifted until the median points coin- 
 cide, becoming the PMAT. Each origi- 
 nal point is shown twice, one shows 
 the movement parallel to the X axis 
 and the other the Y movement. 
 
14 
 
 capital, if the objective is to minimize total travel time, it should be 
 located at that point. 
 
 The precise determination of this point had perplexed mathematicians 
 for centuries, but it is now accepted as having no mathematical solution. 
 To date, the best methods remain interatlon techniques. Seymour (1965) 
 suggests placing a uniform lattice of points (intersections of a regular- 
 ly spaced grid system) over the study area and identifying the point with 
 the smallest aggregate travel (Figure 7). This point becomes the center 
 of a finer gride of points, extending to a perimeter delimited by the 
 closest points in the original lattice. Successive reappl icat ions of 
 this technique produce a close approximation of the PMAT. 
 
 Despite the lack of a mathematical formula, computer programs can 
 easily provide iterative solutions, and rather accurate estimates can be 
 computed. In practical applications, only approximations are necessary 
 since the solution (location) is often not suited for the point in ques- 
 tion. For example, if a supermarket can estimate the scope of its trade 
 area and frequency of business, the PMAT can be determined for a unit 
 time period. That point may be located In a park, cemetery, or any of a 
 list of areas not zoned for commercial activity or merely not desirable 
 for large scale retail trade. Moreover, judgment Is necessary in esti- 
 mating future shifts in the trade area. All this negates the necessity 
 of precise computations. 
 
 The State of Illinois may be utilized to illustrate the location of 
 the PMAT and its practical implications. With nearly two-thirds of the 
 state's population, the Chicago area has already been demonstrated to 
 
 Porter (1963) proposed a geometric solution, but Court (1964) showed 
 Porter's method to be spurious. 
 
15 
 
 Figure 7. One Iterative Step in the Derivation of the PMAT. 
 
 
 ^^ 
 
 
 
 
 
 
 
 • 
 
 1 
 
 1 
 
 
 
 
 \ 
 
 
 
 
 1 
 1 
 / 
 
 / 
 
 • 
 
 • 
 
 • 
 
 \ 
 \ 
 \ 
 
 1 
 
 
 
 • 
 
 / 
 / 
 / 
 / 
 
 / 
 
 
 • 
 
 / 
 
 \ 
 \ 
 \ 
 
 
 
 
 / 
 
 
 
 • 
 
 I 
 1 
 
 
 
 / 
 
 
 
 
 
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 / 
 / 
 
 • 
 
 
 • 
 
 
 • \ 
 
 
 
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 1 
 
 
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 \ 
 
 
 t 
 
 
 
 
 
 
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 1 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 \ 
 
 
 r 
 
 
 
 
 
 
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 % 
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 t 
 
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 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 • 
 
 
 
 
 
 
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 • 
 
 
 
 
 ( 
 
 \ 
 
 
 
 
 
 
 
 
 
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 } 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ( 
 
 ' 
 
 
 
 
 A hypothetical dis- 
 tribution of potential 
 customers. A regular 
 grid is superimposed. 
 e is found to be the 
 grid intersect wi th 
 the lowest aggregate 
 travel . 
 
 A finer regular grid 
 is used to calculate 
 new aggregate travel 
 values, and a new 
 minimum is identified 
 (not marked on this 
 figure) . The grid 
 surrounding that point 
 can again be magnified 
 and the procedure re- 
 peated unt i 1 the de- 
 si red accuracy is 
 achieved. 
 
16 
 
 have a powerful influence on the mean center. This influence is even 
 stronger in the case of the PMAT; it is located within the City of Chi- 
 cago. Is it then advisable to locate all state-wide services in Chicago? 
 Not necessarily, since this statistic is relatively insensitive to the 
 locations of the most distant points. The fact that Cairo is four hun- 
 dred miles to the south is of little consequence. In this context a 
 point further south would probably be more suitable. The mean center is 
 one such point. 
 
 The three bivariate measures of central tendency represent distinc- 
 tive features of the distributions as well as unique computational pro- 
 cedures. The mean center minimizes the sum of the squares of the devia- 
 tions to each point, the PMAT minimizes the sum of the absolute deviations 
 and since the median center is not unique it is not a point of minimiza- 
 tion (except in circumstances when the movement is confined to a grid net- 
 work). Computationally, the median center is the most readily determined 
 since it is the only measure which does not require conversion to cartesian 
 coordinates. On the other hand, the PMAT is the most difficult to compute, 
 but it is the point with the most practical and theoretical value. Since 
 these points have different characteristics, they generally are not found 
 at the same location--the bivariate normal distribution being the major 
 exception. 
 
 MEASURES OF DISPERSION 
 
 A measure of central tendency is commonly, though not always, the 
 reference point for measures of dispersion. Dispersion measures are use- 
 ful descriptive devices for expressing the degree of concentration or 
 scatter and in some cases for specifying directional orientations of the 
 distribution. 
 
17 
 
 The use of these measures involves two basic decisions: (1) which 
 point of central tendency should be selected as a reference, and (2) which 
 measure of dispersion should be utilized. Neither is easily answered, but 
 a few fundamental principles can be deduced. The critical factor in each 
 case appears to be the desired mix of analysis and description. If de- 
 scription is the principal objective then a multitude of measures of 
 central tendency and dispersion may be suitable, depending upon the prob- 
 lem. On the other hand, for strictly analytic purposes the standard radius 
 based on the mean center should be used. In limited cases the standard 
 deviational ellipse may also be considered an analytic tool. 
 
 Although choosing the appropriate reference point for the dispersion 
 measure is often an elementary process, there is no universal agreement 
 regarding the selection of the appropriate point. Hurst and Seller (1969, 
 p. 184), for example, disagree with Blount's (1964) selection of the 
 
 "center of gravity over which to place the normal curve" (circular normal 
 
 o 
 probability surface). Since Blount is examining the distribution of 
 
 shoppers. Hurst contends that the shopping center should be the point from 
 which to measure dispersion. If the purpose is description then the shop- 
 ping center would seem more appropriate. But Blount's test for circular 
 normality requires the mean center, although the reason for testing for 
 circular normality of shoppers is not clarified. Hurst also tests for 
 circular normality, but, inappropriately, from the shopping site instead 
 of from the center of gravity. If the shopping site deviates significantly 
 
 An analytic technique is herein considered to be a method which permits 
 the testing of hypotheses. Other techniques which merely convey informa- 
 tional characteristics about a distribution are descriptive. 
 g 
 
 The circular normal probability surface is defined in the section on 
 standard radius. 
 
18 
 
 from the mean center, one may more quickly reject circular normality. 
 However, if the two points coincide, one would have to test for circular 
 normality by determining the distribution by distance zones. 
 
 The selection of the reference for the dispersion measure is then 
 mainly dependent upon the objective of the study. Tests for bivariate 
 normality require the mean center. Descriptive studies may employ the 
 median center, the point of minimum aggregate travel or any data point or 
 non-data point location within the study area. 
 
 There is also disagreement regarding the utility and interpretation 
 of measures of dispersion. Four methods, (1) standard radius, (2) stand- 
 ard deviational ellipse, (3) standard deviational bicircular quartic, and 
 (4) sectogram, are discussed and evaluated. 
 
 Standard Radius (SR) . The two dimensional counterpoint of the uni- 
 variate standard deviation is the standard radius. As such it is a wide- 
 ly used analytic and descriptive measure of scatter. It may be used: 
 (1) to test for circular normality; (2) to express degrees of dispersion 
 for a circular normal distribution; and (3) to express dispersion for non- 
 normal distributions. For example, Shachar (196?) and V/aterman (1969) 
 used the SR to depict scatters of central place functions in Tel Aviv and 
 to describe the population distribution of Ireland from 18^1 to 19^6, re- 
 spectively, although these distributions deviated significantly from the 
 bivariate normal. Also, Stephenson (1972) uses the standard radius to 
 
 9 
 
 The standard radius is also referred to as the standard distance, and 
 
 by Yulll (1971) and Caprio (1969) as Bachi's standard distance. Al- 
 though Bachi is credited with reviving this measure it was defined by 
 Furfey in 1927 and probably by statisticians before him. Hultquist 
 et. al . (1971) and Lee (1966) call the standard deviation along an axis 
 the standard distance. This can be a useful distinction when distances 
 are referenced to a line, but it is also indicative of the lack of agree- 
 ment in centrographic nomenclature. 
 
19 
 
 illustrate crime patterns in Phoenix. Each of these three applications 
 are legitimate descriptional uses, since no analytic inferences are im- 
 plied. 
 
 If a distribution is normal, then incremental standard radius bands 
 encompass easily determined percentages of data points (similar to the 
 manner in which the univariate standard deviation is used). The percent- 
 age of points in each band is used as a test for circular normality. Bur- 
 ington and May (1953, p. 97) demonstrate that 63.2^ of the data points lie 
 within one standard radius. Neft and Warntz (I960) provide a complete 
 probability table of 0.01 increments of the SR. The probabilities approxi- 
 mate, but do not equal, those of the univariate normal distribution. 
 
 Mathematically, the standard radius is expressed by the relationship 
 
 / 2 2 
 SR= V^ •" % ' (2) 
 
 where a = the standard deviation along the x-axis, and a = the standard 
 X y 
 
 deviation along the y-axis. It is, in essence, the sum of two orthogonal 
 vectors representing the standard deviations along their respective axes 
 (Figure 8). In the case illustrated, the distribution is symmetric (x=y) 
 and the standard radius (OC) is the sum of vectors OB and BC (or OA) . The 
 SR is also defined as the square root of the mean of the squared radial 
 distances from the origin: 
 
 SR = /^^L. ' (3) 
 
 n 
 
 where d. = the radial distance from the origin to any data point i, and 
 
 n = the number of data points. 
 
 Like the center of gravity, the standard radius is insensitive to 
 
 the orientation of an orthogonal axis system (see also Waterman 1969, p. 5^) 
 
20 
 
 Figure 8. Standard Radius and the Bivariate (Circular) Normal Distribution 
 
 +2SDy 
 
 +1SD, 
 
 -ISD^ 
 
 -2SDy 
 
 OB Standard deviation on the X axis 
 OA Standard deviation on the Yaxis 
 OC Standard radius 
 OD Two standard radii 
 
 UICC 
 
21 
 
 This property is illustrated in Figure 9- Regardless of the axis orienta- 
 tion, defined by AOB, COD or FOG the SR remains constant (OR = OD) . 
 
 Although numerous tables for standard radii are available, the prob- 
 ability of a point falling within a standard radius can also be derived 
 by integral calculus. A series of procedural steps outlined below summar- 
 ize the mathematical logic. The sequence proceeds from the most general 
 equation to a specific form, without deserting generality. The bivariate 
 normal surface will be translated to the special case in which the means 
 of X and Y equal zero, their standard deviations are unity, and there is 
 
 no correlation between x and y (i.e., y =y =0, a =o =1, and p = 
 
 ' X y ' X y ' xy 
 
 0) . This describes the circular normal distribution with unit variance 
 centered on the origin. 
 
 The general form for the bivariate normal (Gaussian) surface is (Bur- 
 ington and May 1953, p. 97): 
 
 f(x,y) = 
 
 1 
 
 -Q.i>^>y)y 
 
 2710 
 
 J]-p 
 
 X Y \ 
 
 where 
 
 Q(x,y) = 
 
 x(l-p^) 
 
 (x-u)2 2p(x-u)(y-u) (y-p )2 
 
 ^ - ^1 )L- + y 
 
 a 
 
 a a 
 X y 
 
 w 
 
 (5) 
 
 Since we are mainly concerned with the circular normal distribution, 
 
 the equivalence of the two standard deviations (a = a = o) can be utilized 
 
 X y 
 
 and 
 
 f(x,y) = 
 
 2(l-p^) 
 
 (x-y )^ - 2p(x-y )(y-y ) + iyu) ] 
 X ^ y y 
 
 2TTa 
 
 2.n — 2 
 
 VT^ 
 
 . (6) 
 
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23 
 
 Further, the independence of the two axes (p = 0) can be Introduced 
 
 xy 
 
 since the two variables (x,y) are normally distributed: 
 
 f (x,y) = r- e 
 
 27ra 
 
 [ (x-y^) + (y-Py) 
 
 (7) 
 
 The equation may be further simplified by translating the system to 
 the center of gravity (y = y =0): 
 
 .(x.v)=^e-'/^(^ 
 
 (8) 
 
 Lastly, the variance Is reduced to unity (a = 1 ) and 
 
 ., X _ 1 -l/2(x^ + y^) 
 f (x,y) = 2? e 
 
 (9) 
 
 The probability that a point falls within a circle, C, Is given by 
 
 (c) 
 
 - /J ^<- 
 
 y) dx dy 
 
 (10) 
 
 By changing to polar coordinates (r, 0) and integrating over a distance of 
 one standard radius through one revolution we have: 
 
 2Tr SR 2 
 -r 
 
 fi 
 
 p(C) = 2^ I I e ^ dr de 
 
 (11) 
 
 
 = 0.632. 
 
 This procedure demonstrates that for a circular normal distribution 63.2^ 
 of the points lie within one standard radius of the mean center. Similar 
 
 10 
 
 This is not true for all random variables (x,y) , but applied to normally 
 distributed variables. See Birnbaum (1962, p. 153) for a formal proof. 
 
2k 
 
 Integration shows that a circle with the univariate standard deviation 
 as a radius only encompasses approximately kO% of the points (a circle of 
 radius OA in Figure 8). However, when a bivariate normal distribution 
 
 is not circular, the analytic test of normality requires considerably modi- 
 
 12 
 fied tables. Table 2 lists probabilities for such distributions. It 
 
 is important to understand that the bivariate normal distribution does 
 not exhibit some of the properties of the univariate normal distribution; 
 therefore, they must be treated as two different entities. If all the 
 points in a bivariate normal distribution are converted to vector dis- 
 tances, the resulting distances are not equivalent to one tail of the 
 normal distribution. A bivariate distribution with a standard radius en- 
 compassing 63.2^ of the points is normally distributed, whereas a univari- 
 ate distribution with a standard deviation encompassing 63.2^ of the points 
 is not normal. Similarly, in a three dimensional case, only one of the 
 characteristics of a series of spheres--the diameter, the surface area or 
 the volume--can be normally distributed. Thus, the univariate test cannot 
 be employed to determine circular normality of a two dimensional distribu- 
 tion. This point has evaded Blount (1964), Hurst and Sellers (I969), 
 Fitzgerald (197^), and Davis (197^). 
 
 The fundamental disadvantage of the SR is that It is primarily a test 
 of circular normality. As a descriptive tool It requires a circular dis- 
 tribution. For Instance, the Illinois SR extends beyond the boundaries of 
 the state, and for this reason, it Is obviously not a good measure of 
 scatter (Figure 10). A measure of scatter should delimit the area of 
 
 There is, of course, no such thing as a standard deviation In a two di- 
 mensional distribution. 
 
 1 2 
 
 Groenewood et al . (1967) provide an extensive table of bivariate normal 
 
 probabi 1 i t ies. 
 
25 
 
 TABLE 2 
 
 PROBABILITY OF A POINT FALLING WITHIN ONE STANDARD DEVIATION AND ONE 
 STANDARD RADIUS OF THE MEAN CENTER IN A BIVARIATE NORMAL DISTRIBUTION 
 
 Ratio : 
 
 Probat 
 
 )i 1 1 ty 
 
 
 one SD : 
 
 one SR 
 
 
 
 
 0.0 : 
 
 .6826 : 
 
 .6826 
 
 0.1 
 
 .6802 : 
 
 .6824 
 
 0.2 : 
 
 .6724 
 
 .6818 
 
 0.3 
 
 .6568 : 
 
 .6785 
 
 O.it : 
 
 .6291 
 
 .6732 
 
 0.5 
 
 : .5901 
 
 .6634 
 
 0.6 
 
 : .5461 
 
 .6541 
 
 0.7 
 
 : .5026 
 
 .6424 
 
 0.8 
 
 .4621 
 
 .6362 
 
 0.9 
 
 : .4258 
 
 : .6335 
 
 1.0 
 
 : .3935 
 
 : .6320 
 
 Source: Groenewoud (1967). 
 
26 
 
 Figure 10. The 1970 Illinois Mean Center and the Standard Radius 
 
 PMAT 
 
 'mean 
 center 
 
 Peoria^ 
 
 iSR 
 
 Springfield 
 
 / 
 
 O 
 
 2SR 
 
 East 
 St. Louis 
 
 r 
 
 
 nniles 
 
 -1 
 50 
 
 UICC 
 
27 
 
 highest concentration, while not extending beyond the study area; however, 
 such an occurrence is difficult to avoid when the distribution is confined 
 largely to a limited area near the perimeter. In excluding most of Il- 
 linois, the circle illustrates the dominance of Chicago in the state's 
 population distribution. 
 
 This disadvantage has led to modifications of this measure to Improve 
 Its descriptive qualities. All these modifications represent departures 
 
 from the analytic realm, and are inherently descriptive in nature, thus 
 
 13 
 they must be judged on their descriptive merits. 
 
 Standard Deviatlonal Ellipse (SDE) . Due to the insensi t I vi ty of the 
 SR to strong directional trends in distributions, Lee (1966) and Yuill 
 (1971) recommend an ellipse for description. The ellipse is a well under- 
 stood geometric form which is easily described and readily computed. The 
 SDE is centered on the mean center with the major axis of the ellipse being 
 the principal axis least squares line which minimizes the sum of the 
 squared perpendicular deviations. The minor axis is perpendicular to the 
 major axis at the mean center. The respective standard deviations along 
 these two axes define the standard deviatlonal ellipse (Figure 11). 
 
 Lee (1966) further justifies Its use by showing that a blvariate 
 normal surface, cut by a plane parallel to the (x,y) plane traces an 
 ellipse. Indicating that. In some circumstances, the ellipse can be a 
 test for a blvariate normal distribution. Since the blvariate normal is 
 an uncommon distribution, the ellipse Is almost always used as a descrip- 
 tive tool . 
 
 13 
 
 The descriptive qualities may be determined by how well the measure 
 
 communicates the distribution given only the measure as an indicator. 
 
 When the major and minor axes are twice the respective univariate 
 standard deviations, the ellipse Is known as the standard ellipse. 
 
Figure 1 1 . 
 
 The 1970 
 El 1 Ipse. 
 
 28 
 
 linols Standard Radius and the Standard Deviational 
 
 Peoria. 
 
 Springfield 
 
 / 
 
 
 H; 
 
 "^t 
 
 ^*i 
 
 PMAT 
 
 '«/mean 
 center 
 
 ^^0 
 
 y^ 
 
 
 # 
 
 ISR 
 
 X' 
 
 East 
 St. Louis 
 
 
 
 miles 
 
 50 
 
 UICC 
 
29 
 
 Yulll (197J, pp. 30-31), although acknowledging some of the method- 
 ological shortcomings, also extols the virtues of the ellipse. He pre- 
 sents an intricate trigonometric procedure for determining the nature of 
 the ellipse and specifically the axis orientation. He fails to point out, 
 however, that the axis which minimizes the standard deviation is the 
 principal axis. As such, no trigonometric procedure, although interest- 
 ing, is essential. His advocacy of the SDE is accepted, but Yuill recom- 
 mends placing the ellipse on the median center rather than on the mean 
 center. The use of the median center raises the question of orientation. 
 Since the location of the median center varies with the orientation of the 
 reference axes, an ellipse placed on the median center, rather than on the 
 mean center, will have no unique orientation. Although the same orienta- 
 tion as used with the mean center was implied, it was not so expressed nor 
 was a justification for maintaining this orientation provided. 
 
 Ellipses have numerous propert ies--ci rcular i ty , area, and orientation-- 
 by which they can be compared (Hultqulst, et al . , 1971) • Nevertheless, the 
 ellipse has certain shortcomings such as its inability to capture specific 
 directional biases, and its limited analytic utility. Using Table 2, on a 
 more complete version, the ellipse may be used to test for a bivariate 
 normal distribution. This is done by producing successive ellipses with 
 their major and minor axes being fractional multiples of their respective 
 standard deviations. Although the figure is different, the procedure 
 parallels that of the test for circular normality. As a descriptive tool, 
 its value is completely dependent upon the shape of the distribution. 
 If a distribution is (1) mul t i -nnodal , (2) uniform, (3) random, or (A) 
 pear-shaped, then an ellipse is not a good descriptive form. When the 
 
 Several examples may be found in Yuill (1971) 
 
30 
 
 distribution approaches an elliptical form, it is a good descriptive 
 figure. Presented alone, however, the ellipse may disguise the shape of 
 the distribution, in which case it may not truly represent the underly- 
 ing distribution. When the purpose is to indicate the general shift over 
 time, or to compare several variables, then the ellipse proves useful in 
 contrasting the distributions in question. 
 
 Perhaps if the SDE were to receive universal adoption, researchers 
 could develop an intuitive feel for what the ellipse Is describing. Such 
 an adoption is not warranted due to the limited descriptive capabilities 
 of the ellipse. The weak link, then, is the limitation of the ellipse in 
 describing the point scatter, rather than finding ways to describe the 
 ellipse. Used with this limitation In mind, it still may prove a practical 
 measure for contrasting spatial distribution. 
 
 Standard Deviational Biclrcular Quart I c (BCQ) . The basic procedure 
 in delimiting the ellipse Is slightly modified to produce the BCQ. Rather 
 than defining the figure with two axes and four points, the BCQ consists 
 of a locus of points (Figure 12). These points are derived by computing 
 the standard deviations of each of two orthogonal axes anchored at the 
 mean center and incrementally rotated by 90^ (or rotating one axis 180°). 
 This figure is then designed to be responsive to specific directional biases 
 in the distribution, but as will be evident shortly. It fails to be a 
 significant improvement over the SDE. 
 
 1 6 
 
 The ellipse is frequently used In this manner, for example see: Caprlo 
 
 (1969), Buttimer (1972), and Hyland (1970). 
 
 The figure eight in Figure 9 has been produced by such a procedure. 
 Notice that the standard deviation of the axis perpendicular to the line 
 connecting the data points Is zero. 
 
31 
 
 Figure 12. Three Measures of 1970 Illinois Population Dispersion 
 
 PMAT 
 
 'mean 
 center 
 
 PeoriaJ 
 
 Springfield 
 
 / 
 
 O 
 
 East 
 St. Louis 
 
 UICC 
 
32 
 
 Lefever (1926) first recommended this technique but mistakenly called 
 the resulting figure an ellipse. Furfey (1927) corrected this misconcep- 
 tion and provided a mathematical argument to demonstrate that the figure 
 is, in fact, a bicircular quartic (a fourth degree equation consisting 
 mainly of two circles). One limiting case for such a geometric form is 
 illustrated in Figure 9; the other limit is a circle. 
 
 The mathematical derivation of this figure can best be understood by 
 
 assigning each point a new set of coordinate values produced by an axes 
 
 1 8 
 rotation of angle 6. By rotating the axes and computing the respective 
 
 standard deviations the figure is delimited. The transformation of co- 
 ordinate values is performed with the following equations: 
 
 x' = y sine + x cosG (12) 
 
 y' = y cose + X sine, (13) 
 
 where (x,y) is the original point and (x' , y') is the transformed point. 
 For the rotated axis system the ax' becomes: 
 
 ax' 
 
 k U'f - (-.^ Ji U')^ (,,,) 
 
 The average term, (X') , drops out since the system is centered on the 
 center of gravity (x' = y' = O) . This center is a unique point, independent 
 of axis rotations. 
 
 Substituting (12) into (1^) we have: 
 
 / 2 2 2 2 
 
 /Zy sin 9 + Z2xy sin0 cosB + Zx cos 6 
 
 where r is the radial distance in polar coordinates. 
 
 1 8 
 
 Much of this argument follows from a derivation by Furfey (1927) and 
 
 from a general discussion of the BCQ by Salmon (1879)- 
 
33 
 
 Introducing Pearson's product moment correlation coefficient (p) ; 
 
 Zxy 2 2 2 2 
 
 P ^ „^ I o*" ^xy = PriG o and Zy = na and Ex = no ; ma^ 
 na^a ' "^ x y ' y x ' Ubj 
 
 / 2 2 2 2~ 
 
 thus, r =./ a sin e + 2pa a sinGcose + a^ cos 6. (ly) 
 
 If p = 0, as In the case of a circular normal distribution, (a = 
 a ) then the relationship collapses to: 
 
 r =/a/ (cos^e + sin^e) = a (18) 
 
 The circle, of course, is the only figure which satisfies this condition. 
 
 Returning to the general case, the relationship may be transformed 
 back to the cartesian system by first squaring both sides (19), and then 
 introducing enough 'r' terms (20) so as to permit utilization of the 
 polar coordinate to cartesian coordinate transformation formulas (21) and 
 (22): 
 
 2 2 2 2 2 
 
 r = (a^ cos e + 2pa o cosGsine + a sin 0) (19) 
 
 X X y y 
 
 r = (a ^r^cos^e + 2pa a r^cose sinG + a ^r^sin^e) (20) 
 X X y y 
 
 X = r cosO (21) 
 
 y = r sine (22) 
 
 2 2 2 2 2 
 Using the relationship (x + y ) = r (cos 6 + sin 6) equation (20) 
 
 may be expressed in cartesian coordinates as: 
 
 (x^ + y^)^ = o ^x^ + 2po o axy + a ^y^. (23) 
 
 / ^ "^ X y y ' 
 
 Equation (23) is the standard form of the bicircular quartic. 
 
 Like the ellipse, the BCQ is a member of the family of curves gener- 
 ated by conic sections. The quartic is produced by three conies. The 
 
3^ 
 
 Figure 13. Comparison of the Standard Deviational Ellipse and the 
 Bici rcular Quartic 
 
 M N 
 
 Blcircular Quartic with local maxima at M, N, and 8/ and local minima at 
 M' , N' , and B. 
 
 Ellipse with local maximum at B and local minimum at B'. 
 
35 
 
 BCQ and SDE have four points in common. These are the intersections of 
 the curves with the major and minoraxes (Figure 13, points A, A', B, and 
 B'). Their shapes are clearly different, however, the quartic having 
 three local maxima and minima and the ellipse having only one of each. 
 
 In the geographic literature, the bicircular quartic has been de- 
 scribed by Caprio (1969; 1970) and Hultquist and others (1970), but they 
 utilized computerized programs which followed dissimilar procedural logic. 
 As a result, they produced two different figures, yet each program made a 
 distinct contribution to centrographic research. Caprio iteratively com- 
 puted the standard deviations along axes rotated by increments of five 
 degrees. This procedure yielded a locus of points sufficient to adequately 
 approximate the BCQ. Hultquist also set out to computerize the BCQ, but 
 utilized an ellipse functional subroutine, thereby producing the SDE and 
 not the BCQ as described in the report. Hultquist's title, "standard 
 ellipse", matches the computerized figure but not the verbal description. 
 This program should thus be used when the SDE is desired. 
 
 Geographers have utilized the BCQ to study areal population patterns. 
 Buttimer (1972) employed the BCQ to examine social contacts for residents 
 
 ]Q 
 
 of two planned and two unplanned communities. The BCQ's, called standard 
 ellipses, are shown to be effective figures for contrasting the respective 
 fields of social contacts. Not only is the scope of each field succinctly 
 summarized, but also the directional biases are revealed. 
 
 19 
 
 Buttimer (1972) describes in detail the utility of using the ellipse. 
 
 Reference is made, for Instance, to the area of the ellipse, which is 
 
 based on the major and minor axes. Although the area of the BCQ is 
 
 often similar, the BCQ is larger. Indeed, in Figure 9 all of the points 
 
 are on a straight line and thus the ellipse is not shown because its 
 
 area is equal to zero. The area of the BCQ is 1/2 tt a , where o is the 
 
 standard radius. 
 
36 
 
 Hyland (1970, p. 78) employed the BCQ to graphically display social 
 
 20 
 interaction. The BCQ's of social trips to friends, organizations, and 
 
 local relatives identified the spatial scope of each of these Interaction 
 fields. The method depicts well the marked contrasts in extent and 
 orientation of the three fields. For example, local relatives appear to 
 be scattered throughout the central city, while friends are confined main- 
 ly to the neighborhood. 
 
 Brown and Holmes (1971) employed Hultquist's program to analyse intra- 
 urban residential movements. However, they were not dealing with a dot 
 distribution, but rather with a series of migration flows. By transfer- 
 ring all the migration orgins to a common point, centrographi c measures 
 may be computed. For example, the BCQ Identified the primary directional 
 component as well as the distances traveled. 
 
 In summary, since the BCQ and SDE indicate the principal trend in 
 the distribution, they represent improvements over the SR. However, they 
 add only one directional component and rarely is the SDE used to test for 
 bivariate normality. Perhaps their basic advantage lies in providing sum- 
 mary generalizations of complex distributions. Comparing the two, the SDE 
 is a simpler form, but the BCQ is intuitively more satisfying. One would 
 expect the figure which is traced by a multitude of standard deviations 
 along a series of incrementally rotated axes (the BCQ) to be more repre- 
 sentative of the original distribution. In reality, with some exceptions. 
 
 20 
 
 Hyland called his figure a "standard deviational ellipse". Clearly, how- 
 ever, he calculated the BCQ: "Employing an arbitrary cartesian grid, the 
 mean center of the points is calculated. A new grid, composed of ortho- 
 gonal axes in the same scale as the first, is centered on the mean center. 
 The deviation of each point, from the x-axIs, is employed to calculate the 
 standard deviation . The axes are then rotated, say by ten degrees, and 
 the procedure repeated. If all the standard deviations along the x-axes 
 are joined, an el 1 ipse will be described." (Hyland 1970, p. 78) 
 
37 
 
 there is little visual difference between the BCQ and the SDE (Figure 
 13). Still, it must be recognized that the two figures are mathematical- 
 ly distinct, and the appropriate derivation should be used. 
 
 Furthermore, the BCQ does not indicate whether the principal devia- 
 tion is in one or many directions. That distinction can only be made by 
 a measure that is based on directional deviations from a central point. 
 
 Sectogram . The figure sensitive to the deviations in all radial 
 directions is the sectogram. It is produced by connecting the locus of 
 points representing the degrees of deviation from the mean center within 
 radial sectors (Figure 1^). Schneider (I968), an early proponent of this 
 technique, computed the deviations in eight sectors. He noticed, however, 
 that the within sector deviation was significantly affected by rotating 
 the eight sectors by a few degrees. Thus, depending on the original 
 orientation of the sectors, varying results could occur. 
 
 A SECTOGRAM 
 
 WITH EIGHT ^^^ 
 
 RADIAL SECTORS /^ 
 
 • iTiean center (\ 
 
 standard distance 7 Nv 
 
 • in each sector / \ 
 
 > 
 
 [ 
 
 ukc 
 
 ^ 
 
 I 
 
 Figure 1^ 
 
38 
 
 Caprio (I969) modified Schneider's sectogram by decreasing the sector 
 size to twenty degrees. The sectors were rotated by increments of five 
 degrees to produce a new sector overlapping fifteen degrees with the pre- 
 ceeding sector. In this manner seventy-two sectorial deviations were com- 
 puted and placed on the respective bisectors of each sector, producing 
 the sectogram. Smaller sector sizes and incremental axes rotation both 
 serve to reduce the effects of the original axes orientation in addition 
 to providing a measure with a finer degree of resolution. Caprio also in- 
 cludes a computer program listing of his modified sectogram, used exten- 
 sively in a study of Newark's population and housing characteristics. The 
 listing is more an explanation of method rather than a program intended 
 for general use. 
 
 Unlike the SR, the SDE, and the BCQ, the sectogram does not have a 
 unique solution for a given distribution. The solution is determined in 
 part by the nature of the distribution, but more importantly, it is a 
 function of several free parameters, primarily the axis orientation and 
 the sector size. But basically, it is purely a descriptive figure which 
 is responsive to unusual distributional shapes. In contrast to the BCQ. 
 and SDE it is a more effective geometric form when more detail is desired. 
 But this property contributes to its major disadvantage: a lack of 
 characteristics which permit comparisons of sectograms. For example, 
 there is no such thing as a radius or major axis of a sectogram. Cer- 
 tainly, only one's imagination limits the number of indices which may be 
 computed, such as the range of the sector deviations, the area of the 
 sectogram and the autocorrelation of sector deviations. These indices 
 describe the properties of the sectogram, but few are as obvious as the 
 major and minor axes of the ellipse, and none have been universally ac- 
 cepted. 
 
39 
 
 SUMMARY AND EVALUATION 
 
 It is evident at this point that each centrographic measure character- 
 izes a particular property of a distribution. The mean center has a dis- 
 tinctly different derivation than the PMAT and, with the exception of the 
 circular normal distribution, is not located at the same point. The mean 
 center is pulled in the direction of the most distant points, while the 
 PMAT is influenced by the clustering of points. Therefore, the mean center 
 Is biased by the extreme distance values of those at the periphery of the 
 study area, whereas the PMAT is not. 
 
 The shape of the distribution is also significant when a phenomena is 
 studied through time. If the objective is, for example, to illustrate the 
 historical migration of people to Chicago, then the PMAT would hardly be of 
 any value. The PMAT has moved very little, being in the Chicago area for 
 the last fifty years, belying the actual population shifts that have 
 occurred. Given a different population scatter, the PMAT may have repre- 
 sented extremely well the temporal shifts. 
 
 In summary, the measure of central tendency to be utilized Is dependent 
 upon (1) the objective of the study, and (2) the nature of the distribution. 
 In most instances, the choice is between the mean center and the PMAT. The 
 orthogonal median point can be used as a quick approximation of the other 
 two measures with the understanding that its accuracy is uncertain. Un- 
 like the other measures, the orthogonal median point can be quickly deter- 
 mined without converting the points to coordinate values. 
 
 The conclusions regarding the measures of dispersion are also general, 
 but some specific observations can be made. One characteristic of the 
 standard radius that distinguishes it from the other three measures--the 
 SDE, the BCQ, and the sectogram-- i s that it is primarily used for 
 
ko 
 
 analytical purposes. As descriptive measures all have distinct disadvan- 
 tages. Since considerable detail is lost, the SR, the SDE, and the BCQ 
 are not good descriptive figures. Each has traits which permit compari- 
 sons among similar figures. To the contrary, comparisons among sectograms 
 prove to be awkward. But as a detailed descriptive form, the sectogram 
 is superior. Yet, if only a simple geometric figure is desired then the 
 standard deviational ellipse is appropriate. 
 
 We may conclude that of the four measures, the SR is analytic, the 
 sectogram is purely descriptive, and while the SDE may be used as an 
 analytic measure it is predominantly used for descriptive purposes. The 
 BCQ is an adaptation of the SDE, and represents no improvement over the 
 SDE, and perhaps leads only to confusion. Geometrically, their shapes 
 are similar and only diverge in cases where dispersion is principally in 
 one dimension. In such cases, the BCQ is clearly misleading and should 
 not be employed (Figure 9). Its use stems from an intuitive appeal. It 
 would seem that connecting many standard deviations is better than connect- 
 ing only four as in the SDE. This proves not to be true. For these rea- 
 sons, the use of the BCQ should be abandoned , leaving us wuth three basic 
 choices--the analytic SR , the "simple descriptive" SDE, and the "detailed 
 descriptive" sectogram. 
 
 Judgment must be exercised in the selection of centrographic mea- 
 sures. Their limitations must be heeded, but in general their use is 
 encouraged. Few methods can convey as much information as a few measures 
 of centrography . Although detail may be sacrificed, centrographic mea- 
 sures are valuable geographic generalizations worthy of wide-spread use. 
 
k] 
 
 REFERENCES 
 
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 Spatial Analysis. Papers Regional Science Association ^ vol. 10, 
 pp. 83-132. 
 
 Birnbaum, Z., I962. Introduction to Probability and Mathematical Statistics 
 New York, N.Y.: Harper & Bros. 
 
 Blount, S., 196^. The Spatial Distribution of Shoppers Around the Valley 
 Shopping Center, St. Charles, 111. The Professional Geographer , vol. 
 16, pp. 8-17. 
 
 Brown, L. , and Holmes J., 1971. Intra-Urban Migrant Lifelines: A Spatial 
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 Burlngton, R. , and May, D. , 1953. Handbook of Probability and Statistics 
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 Buttimer, A., 1972. Social Space and the Planning of Residential Areas. 
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 Caprio, R. , 1969. A Geostatistical An;^lvsis of Population and Housing 
 
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 Geography, University of Cincinatti. 
 
 , 1970. Centrography and Geostat i st ic. Professional Geographer , 
 
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 Court, A., 196^. The Elusive Point of Minimum Travel. Annals , The Associa- 
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 Davis, P., 197^. Science in Geographer, 3» Data Description and Presenta- 
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 Eells, W. , 1930. A Mistaken Conception of the Center of Population. 
 
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 FitzGerald, B., 197^. Science in Geography, 1, Developments in Geographical 
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 Furfey, P., 1927. A Note on Lefever's Standard Deviational Ellipse. Amer- 
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 Groenewoud, C, Hoaglin, D.C., and Vital is, J.V., I967. Bivariate Normal 
 
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 Hart, J., 195^. Central Tendency in Areal Distributions. Economic Geog- 
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 Hayford, J., 1902. What is the Center of an Area, or the Center of Popula- 
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k2 
 
 Hultqulst, J., Brown, L., and Holmes, J., 1971. CENTRO: A Program for 
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 Hurst, M., and Sellers, J., 1869. An Analysis of the Spatial Distribu- 
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 Hyland, G., 1970. Social Interaction and Urban Opportunity: The Appala- 
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DEPARTMENT OF GEOGRAPHY 
 UNIVERSITY OF ILLINOIS at Urbana-Champaign 
 
 PUBLISHED PAPERS 
 
 April, 1972 
 
 Paper No. 1 A Theoretical Framework for Discussion of CI imatological 
 Geomorphology , by Dag Nummedal. 
 
 *Paper No. 2 Social Areas and Spatial Change in the Black Community 
 
 of Chicago: 1950-1960, by Charles M. Christian. 
 
 October, 1972 
 
 "Paper No. 3 Regional Components for the Recognition of Historic Places, 
 
 by Richard W. Travis 
 
 "Paper No. k Matrix and Graphic Solutions to the Traveling Salesman 
 
 Problem, by Ross Mullner. 
 
 April, 1973 
 
 Paper No. 5 Regional Changes in Petroleum Supply, Demand and Flow in 
 the United States: I966-I98O, by Ronald J. Swager. 
 
 "Paper No. 6 Social Problems in a Small Jamaican Town, by Curtis C. 
 
 Roseman, Henry W. Bullamore, Jill M. Price, Ronald 
 W. Snow, Gordon L. Bower. 
 
 Apri l, 197^ 
 
 Paper No. / Some Observations on the Late Pleistocene and Holocene 
 
 History of the Lower Ohio Valley, by Charles S. Alexander. 
 
 April, 1975 
 
 Paper No. 8 Methods and Measures of Centrography : A Critical Survey of 
 Geographic Applications, by Siim Soot. 
 
 Paper No, 9 A Re-Evaluation of the Extraterrestrial Origin of the Carolina 
 Bays, by J. Ronald Eyton and Judith I. Parkhurst. 
 
 " Papers out of Stock 
 
WJ 
 

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