CrimeStat IV 

Part II: Spatial Description 
 

 

Chapter 4: 

Centrographic Statistics 

Ned Levine 

Ned Levine & Associates 

Houston, TX 
 

 

 

 

 

 

 

 

Table of Contents 

Centrographic Statistics 4.1 

Mean Center 4.1 

Weighted Mean Center 4.4 

Median Center 4.12 

Center of Minimum Distance 4.12 

Standard Deviations of the X and Y Coordinates 4.14 

Standard Distance Deviation 4.16 

Standard Deviational Ellipse 4.17 

Geometric Mean 4.20

 Uses 4.23 

Harmonic Mean 4.23

 Uses 4.24 

Average Density 4.26 

Output Files 4.26

 Calculating the Statistics 4.26 

 Tabular Output 4.27 

 Graphical Objects 4.27 

Statistical Testing 4.30

 Decision-making Without Formal Tests 4.30 

Examples of Centrographic Statistics 4.30 

Example 1: June and July Auto Thefts in Precinct 11 4.30 

Example 2: Serial Burglaries in Baltimore City and Baltimore County 4.32 

Directional Mean and Variance 4.39

 First Quadrant 4.40 

 Third Quadrant 4.40 

 Second and Fourth Quadrants 4.40 

 Mean Angle 4.41 

 Circular Variance 4.42 

 Mean Distance 4.42 

 Directional Mean 4.42 

 Triangulated Mean 4.43 

 Directional Mean Output 4.43 

Convex Hull 4.45

 Uses and Limitations of the Convex Hull 4.46 
 

 

 

 

Table of Contents (continued) 

References 4.49 

Endnotes 4.51 

Attachments 4.54 

A. Using Spatial Measures of Central Tendency with Network 

 Analyst to Identify Routes Used by Motor Vehicle Thieves 

By Philip R. Canter 4.55 

B. Centrographic Analysis: Man With a Gun Calls for Service 

By James L. LeBeau 4.56 
Chapter 4: 

Centrographic Statistics 

In this chapter, the spatial distribution of crime incidents will be discussed. The statistics 

that are used in describing the spatial distribution of crime incidents will be explained and will be 

illustrated with examples from CrimeStat7 III. For the examples, crime incident data from 

Baltimore County and Baltimore City will be used. Figure 4.1 shows the user interface for the 

spatial distribution statistics in CrimeStat. For each of these, the statistics will first be presented 

followed by examples of their use in crime analysis. 

Centrographic Statistics 

The most basic type of descriptors for the spatial distribution of crime incidents are 

centrographic statistics. These are indices which estimate basic parameters about the 

distribution (Lefever, 1926; Furfey, 1927; Bachi, 1957; Neft, 1962, Hultquist, Brown and 

Holmes, 1971; Ebdon, 1988). They include: 

1. Mean center 

2. Median center 

3. Center of minimum distance 

4. Standard deviation of X and Y coordinates 

5. Standard distance deviation 

6. Standard deviational ellipse 

They are called centrographic in that they are two dimensional correlates to the basic 

statistical moments of a single-variable distribution - mean, standard deviation, skewness, and 

kurtosis (see Bachi, 1957). They have been applied to crime analysis by Stephenson (1980) and 

by Langworthy and Jefferis (1998). Because two dimensions add complexity not seen in one 

dimension, these statistical moments have been modified to be appropriate. Figure 4.2 shows 

how the centrographic statistics are selected in CrimeStat. 

Mean Center 

The simplest descriptor of a distribution is the mean center. This is merely the mean of 

the X and Y coordinates. It is sometimes called a center of gravity in that it represents the point 

in a distribution where all other points are balanced if they existed on a plane and the mean 

center was a fulcrum (Ebdon, 1988; Burt and Barber, 1996). 

4.1 
Figure 4.1: 

Spatial Distribution Screen 
Figure 4.2: 

Selecting Centrographic Statistics 
 

 (4.2) 

 

 (4.4)

For a single variable, the mean is the point at which the sum of all differences between 

the mean and all other points is zero. Unfortunately, for two variables, such as the location of 

crime incidents, the mean center is not necessarily the point at which the sum of all distances to 

all other points is minimized. That property is attributed to the center of minimum distance (see 

below). However, the mean center can be thought of as a point where both the sum of all 

differences between the mean X coordinate and all other X coordinates is zero and the sum of all 

differences between the mean Y coordinate and all other Y coordinates is zero. 

The formula for the mean center is: 

(4.1) 

(4.3)

where Wi is the weight of observation . and Xi and Yi are as defined in equations 4.1 and 4.2. 

The advantage of a weighted mean center is that points associated with areas can have the 

characteristics of the areas included. For example, if the coordinates are the centroids of census 

4.4 

where Xi and Yi are the coordinates of individual locations and N is the total number of points. 

To take a simple example, the mean center for burglaries in Baltimore County has spherical 

coordinates of longitude -76.608482, latitude 39.348368 and for robberies longitude -76.620838, 

latitude 39.334816. Figure 4.3 illustrates these two mean centers. 

Weighted Mean Center 

A weighted mean center is produced by weighting each coordinate by another variable, 

Wi. For example, if the coordinates are the centroids of census tracts, then the weight could be 

the population within the census tract. The weights have to be a positive number greater than or 

equal to 1. The numerator is the sum of the product of the variable and the weight while the 

denominator is the sum of weights, 
Figure 4.3: Burglary and Robbery in Baltimore Coun

Comparison of Mean Centers 

0 

Miles 

2 4 

 Mean center of burglaries

 Mean center of robberies 
tracts, then the weight of each centroid could be the population within the census tract. This will 

produce a different center of gravity than the unweighted center of all census tracts. 

CrimeStat allows the mean to be weighted by either the weighting variable or by the 

intensity variable. Users should be careful, however, not to weight the mean with both the 

weighting and intensity variable unless there is an explicit distinction being made between 

weights and intensities. 

To take an example, in the six jurisdictions making up the metropolitan Baltimore area 

(Baltimore City, and Baltimore, Carroll, Harford, Howard and Anne Arundel counties), the mean 

center of all census block groups is longitude -76.619121, latitude 39.304344. This would be an 

unweighted mean center of the block groups. On the other hand, the mean center of the 1990 

population for the Baltimore metropolitan area had coordinates of longitude -76.625186 and 

latitude 39.304186, a position slightly southwest of the unweighted mean center. Weighting the 

block groups by median household income produces a mean center which is still more 

southwest. Figure 4.4 illustrates these three mean centers. 

Weighted mean centers can be useful because they describe spatial differentiation in the 

metropolitan area and factors that may correlate with crime distributions. Another example is 

the weighted mean centers of different ethnic groups in the Baltimore metropolitan area (figure 

4.5). The mean center of the White population is almost identical to the unweighted mean 

center. On the other hand, the mean center of the African-American/Black population is 

southwest of this and the mean center of the Hispanic/Latino population is considerably south of 

that for the White population. In other words, different ethnic groups tend to live in different 

parts of the Baltimore metropolitan area. Whether this has any impact on crime distributions is 

an empirical question. 

When the Mcsd box is checked, CrimeStat will run the routine. CrimeStat has a status bar 

that indicates how much of the routine has been run (Figure 4.6).1 The results of these statistics 

are shown in the Mcsd output table (figure 4.7). 

Hint. There are 40 bars indicated in the status bar while a routine is running. For long runs, users can 

estimate the calculation time by timing how long it takes for two bars to be displayed and then multiply by 

20. 

4.6 

1 
Figure 4.4: Center of Baltimore Metropolitan Populat

Mean Center of Block Groups Weighted By Selected Variables 

Baltimore County 

City of Baltimore 

 Population Unweighted HH Income 

Howard County 

Miles 

2 4 Anne Arundel County 0 
Unweighted White

Figure 4.5: Center of Baltimore Metropolitan Populat

Mean Center of Block Groups Weighted By Selected Variables 

Baltimore County 

City of Baltimore 

 

Black 

 Latino

Howard County 

Miles 

2 4 

Anne Arundel County 0 
Figure 4.6: 

CrimeStat Calculating a Routine 
Figure 4.7: 

Mean Center and Standard Distance Deviation Ou
Figure 4.8: Non-Uniqueness of a Median Center 

Lines Splitting Incident Locations Into Two Halves 

0 

Miles 

2 4 

Baltimore County 





City of Baltimore 



Area of non-uniqu







Howard County 

Anne Arundel County 
 

Median Center 

The median center is the intersection between the median of the X coordinate and the 

median of the Y coordinate. The concept is simple. However, it is not strictly a median. For a 

single variable, such as median household income, the median is that point at which 50% of the 

cases fall below and 50% fall above. On a two dimensional plane, however, there is not a single 

median because the location of a median is defined by the way that the axes are drawn. 

For example, in figure 4.8, there are eight incident points shown. Four lines have been 

drawn which divide these eight points into two groups of four each. However, the four lines do 

not identify an exact location for a median. Instead, there is an area of non-uniqueness in which 

any part of it could be considered the >median center=. This violates one of the basic properties of 

a statistic is that it be a unique value. 

Nevertheless, as long as the axes are not rotated, the median center can be a useful 

statistic. The CrimeStat routine outputs three statistics: 

1. The sample size

2. The median of X

3. The median of Y

4.12 

The tabular output can be printed and the median center can be output as a graphical 

object to ArcGIS >shp=, MapInfo >mif=, Google Earth ‘kml, or various Ascii files. A root name 

should be provided. The median center is output as a point (MdnCntr<root name>). 

Center of Minimum Distance 

Another centrographic statistic is the center of minimum distance. Unfortunately, this 

statistic is sometimes also called the median center, which can make it confusing since the above 

statistic has the same name. Nevertheless, unlike the median center above, the center of 

minimum distance is a unique statistic in that it defines the point at which the sum of the distance 

to all other points is the smallest (Burt and Barber, 1996). It is defined as: 

(4.5)

where dic is the distance between a single point, i, and C, the center of minimum distance (with 

an X and Y coordinate). Unfortunately, there is not a formula that can calculate this location.
Figure 4.9: 

Center of Minimum Distance Output 
 (4.6) 

 (4.7) 

 

Instead, an iterative algorithm is used that approximates this location (Kuhn and Kuenne, 

1962; Burt and Barber, 1996; see endnote .). Depending on whether the coordinates are 

spherical or projected, CrimeStat will calculate distance as either Great Circle (spherical) or 

Euclidean (projected), as discussed in the previous chapter. The results are shown in the Mcmd 

output table (figure 4.9). 

The importance of the center of minimum distance is that it is a location where distance 

to all the defining incidents is the smallest. Since CrimeStat only measures distances as either 

direct or indirect, actual travel time is not being calculated. But in many jurisdictions, the 

minimum distance to all points is a good approximation to the point where travel distances are 

minimized. For example, in a police precinct, a patrol car could be stationed at the center of 

minimum distance to allow it to respond quickly to calls for service. 

For example, figure 4.10 maps the center of minimum distance for 1996 auto thefts in 

both Baltimore City and Baltimore County and compares this to both the mean center and the 

median center statistic. As seen, both the center of minimum distance and the median center are 

south of the mean center, indicating that there are slightly more incidents in the southern part of 

the metropolitan area than in the northern part. However, the difference in these three statistics 

is very small, especially the median center and the center of minimum distance. 

Standard Deviations of the X and Y Coordinates 

In addition to the mean center and center of minimum distance, CrimeStat will calculate 

various measures of spatial distribution, which describe the dispersion, orientation, and shape of 

the distribution of a variable (Hammond & McCullogh 1978; Ebdon 1988). The simplest of 

these is the raw standard deviations of the X and Y coordinates, respectively. The formulas 

used are the standard ones found in most elementary statistics books: 

where Xi and Yi are the X and Y coordinates for individual points, ܺത and ܻത are the means of X 

and Y respectively, and N is the total number of points. Note that 1 is subtracted from the 

number of points to produce an unbiased estimate of the standard deviation. 

4.14 
Figure 4.10: 1996 Metropolitan Baltimore Auto Theft

Mean Center and Center of Minimum Distance for 1996 Auto The
 

 

 

Figure 4.11 shows the standard deviation of the coordinates for auto thefts and represents 

this as a rectangle. As seen, the distribution of auto thefts spreads more in an east-west direction 

than in a north-south direction. 

Standard Distance Deviation 

While the standard deviation of the X and Y coordinates provides some information 

about the dispersion of the incidents, there are two problems with it. First, it does not provide a 

single summary statistic of the dispersion in the locations and is actually two separate statistics, 

the dispersion in X and the dispersion in Y. Second, it provides measurement in the units of the 

coordinate system. Thus, if spherical coordinates are being used, then the units will be decimal 

degrees. On the other hand, if projected coordinates are being used, then units will be in feet or 

meters or some other metric. 

A measure which overcomes these problems is the standard distance deviation (or 

standard distance, for short). This is the standard deviation of the distance of each point from 

the mean center and is expressed in measurement units (feet, meters, miles). It is the twodimensional equivalent of a standard deviation. 

The formula for it is: 

(4.8)

where diMC is the distance between each point, i, and the mean center and N is the total number 

of points. Note that 2 is subtracted from the number of points to produce an unbiased estimate of 

standard distance since there are two constants from which this distance is measured (mean of X, 

mean of Y).2 

The standard distance can be represented as a single vector rather than two vectors as 

with the standard deviation of the X and Y coordinates. Figure 4.12 shows the mean center and 

standard distance deviation of both robberies and burglaries for 1996 in Baltimore County 

With a weight for an observation, wi, the squared distance is weighted and the formula becomes: 

where diMC is the distance from the point to the mean center. Both summations are over all points, N. 

4 . 1 6

2 


Miles 0 2 4

Figure 4.11: 1996 Metropolitan Baltimore Auto Theft

Mean Center and Standard Deviations of X and Y Coordinates
Figure 4.12: 1996 Baltimore County Burglaries and Rob

Comparison of Mean Centers and Standard Distance Deviations 

0 

Miles 

2 

SDD of Burglaries 

SDD of Robberies 

MC Burglaries  

MC Robberies 

4 
 

 

 

 ቑ

 

 

represented as circles. It is clear that the spatial distributions of these two types of crime vary 

with robberies being slightly more concentrated. 

Standard Deviational Ellipse 

The standard distance deviation is a good single measure of the dispersion of the 

incidents around the mean center. However, with two dimensions, distributions are frequently 

skewed in one direction or another (a condition called anisotropy). 

Instead, there is another statistic that gives dispersion in two dimensions, the standard 

deviational ellipse (or ellipse, for short; Ebdon, 1988; Cromley, 1992). The standard deviational 

ellipse is derived from the bivariate distribution (Furfey, 1927; Neft, 1962; Bachhi, 1957) and is 

defined by: 

(4.9) 

The two standard deviations, in the X and Y directions, are orthogonal to each other and 

define an ellipse. Ebdon (1988) rotates the X and Y axis so that the sum of squares of distances 

between points and axes are minimized. By convention, it is shown as an ellipse. 

Aside from the mean X and mean Y, the formulas for these statistics are as follows (the 

observation subscript, ݅, has been dropped from the summation sign): 

 (4.10)

where all summations are for i=1 to N (Ebdon, 1988). 

2. Two standard deviations are calculated, one along the transposed X-axis and one

along the transposed Y-axis:

(4.11)

4 . 1 9

1. The Y-axis is rotated clockwise through an angle, θ, where
 

 

 

 (4.12)

 (4.13) 

 (4.14) 

 

 

 

 (4.15) 

4. The area of the ellipse is:

where ܺത and ܻത are the means of X and Y respectively, ߠ is the angle (in radians), and N is the 

number of points. Note, again, that 2 is subtracted from the number of points in both 

denominators to produce an unbiased estimate of the standard deviational ellipse since there are 

two constants from which the distance along each axis is measured (ܺത, 

ത; see endnote ݅݅). 

3. The X-axis and Y-axis of the ellipse are defined by:

Figure 4.13 shows the output of the ellipse routine and figure 4.14 maps the standard 

deviational ellipse of auto thefts in Baltimore City and Baltimore County for 1996. 

Geometric Mean 

The mean center routine (Mcsd) includes two additional means. First, there is the 

geometric mean, which is a mean associated with the mean of the logarithms. It is defined as: 

(4.16) 

4 . 2 0

(4.17)

where Π is the product term of each point value, i (i.e., the values of X or Y are multiplied times 

each other), Wi is the weight used (default=1), and N is the sample size (Everitt, 2011). The 

weights must be defined on the Primary File page, either in the Weights field or in the Intensity 

field (but not both together). 
Figure 4.13: 

Standard Deviational Ellipse Output 
Figure 4.14: 1996 Metropolitan Baltimore Auto Theft

Mean Center and Standard Deviational Ellipse

Miles

0 2 4
 

 

 ݊݅ܿ݋݉ݎܽܪ

  ݊݅ܿ݋݉ݎܽܪ

 

 

 

 

 

 

 

The equation can be evaluated by logarithms: 

(4.18) 

 (4.19)

(4.20) 

 (4.21) 

The geometric mean is the anti-log of the mean of the logarithms. If weights are used, 

then the logarithm of each X or Y value is weighted and the sum of the weighted logarithms are 

divided by the sum of the weights. If weights are not used, then the default weight is 1 and the 

sum of the weights will equal the sample size. The geometric mean is output as part of the Mcsd 

routine and has a >Gm= prefix before the user defined name. 

Uses 

The geometric mean is used when units are multiplied by each other (e.g., robberies 

increase by 5% one year, 3% the next, and 4% the next; Wikipedia, 2007a). One cannot take the 

simple mean because there is a cumulative change in the units. In most cases, this is not relevant 

to point (incident) locations since the coordinates of each incident are independent and are not 

multiplied by each other. However, the geometric mean can be useful because it converts all X 

and Y coordinates into logarithms and, thus, has the effect of discounting extreme values. 

Harmonic Mean

The harmonic mean is a lso a mean which discounts extreme va lues, but is calculated 

differently. It is defined as (Wikipedia, 2007b): 

(4.22)

(4.23)

4 . 2 3
 

where Wi is the weight used (default=1), Xi and Yi are the X and Y values, and N is the sample 

size. The weights have to be defined on the Primary File page, either in the Weights field or in 

the Intensity field (but not both together). 

The harmonic mean of X and Y is the inverse of the mean of the inverse of X and Y 

respectively (i.e., take the inverse; take the mean of the inverse; and invert the mean of the 

inverse). If weights are used, then each X or Y value is weighted by its inverse while the 

numerator is the sum of the weights. If weights are not used, then the default weight is 1 and the 

sum of weights will equal the sample size. The harmonic mean is output as part of the Mcsd 

routine and has a >Hm= prefix before the user-defined name. 

Uses 

Typically, harmonic means are used in calculating the average of rates, or quantities 

whose values are changing over time (Wikipedia, 2007b). For example, in calculating the 

average speed over multiple segments of equal length (see chapter 30 on Network Assignment), 

the harmonic mean should be used, not the arithmetic mean. If there are two adjacent road 

segments, each one mile in length and if a car travels over the first segment 20 miles per hour 

(mph) but over the second segment at 40 mph, the average speed is not 30 mph (the arithmetic 

mean), but 26.7 mph (the harmonic mean). The car takes 3 minutes to travel the first segment 

(60 minutes per hour times 1 mile divided by 20 mph ) and 1.5 minutes to travel the second 

segment (60 minutes per hour times 1 mile divided by 40 mph). Thus, the total time to travel the 

two miles is 4.5 minutes and the average speed is 26.7 mph. 

Again, for point (incident) locations, the harmonic mean would normally not be relevant 

since the coordinates of each of the incidents are independent. However, since the harmonic 

mean is weighted more heavily by the smaller values, it can be useful to discount cases which 

have outlying coordinates. 

In other words, the harmonic mean of X and Y respectively is the inverse of the mean of 

the inverse of X and Y respectively (i.e., take the inverse; take the mean of the inverse; and 

invert the mean of the inverse). If weights are used, then each X or Y value is weighted and the 

numerator is the sum of the weights. If weights are not used, then the sum of the weights will 

equal the sample size. The harmonic mean is output as part of the Mcsd routine and has a >Hm=

prefix before the user defined name. 

The geometric and harmonic means are discounted means that >hug= the center of the 

distribution. They differ from the mean center when there is a very skewed distribution. 

4 . 2 4
Figure 4.15:
 

To contrast the different means, figure 4.15 below shows five different means for 

Baltimore County motor vehicle thefts: 

1. Mean center;

2. Center of minimum distance;

3. Geometric mean;

4. Harmonic mean; and

5. Triangulated mean (discussed below)

In the example, the mean center, geometric mean, and harmonic mean fall very close to 

each other; however, they will not always be so. The center of minimum distance approximates 

the geographical center of the distribution. The triangulated mean is defined by the angularity 

and distance from the lower-left and upper-right corners of the data set (see below). 

Centrographic descriptors can be very powerful tools for examining spatial patterns. 

They are a first step in any spatial analysis, but an important one. The above example illustrates 

how they can be a basis for decision-making, even with small samples. A couple of other 

examples can be illustrated. 

Average Density 

The average density is the number of incidents divided by the area. It is a measure of 

the average number of events per unit of area; it is sometimes called the intensity. If the area is 

defined on the measurement parameters page, the routine uses that value for area; otherwise, it 

takes the rectangular area defined by the minimum and maximum X and Y values (the bounding 

rectangle). 

Output Files 

Calculating the Statistics 

Once the statistics have been selected, the user clicks on Compute to run the routine. The 

results are shown in a results table. 

4 . 2 6
 

Tabular Output 

For each of these statistics, CrimeStat produces tabular output. In CrimeStat, all tables 

are labeled by symbols, for example Mcsd for the mean center and standard distance deviation or 

Mcmd for the center of minimum distance. All tables present the sample size. 

Graphical Objects 

The six centrographic statistics can be output as graphical objects. The mean center and 

center of minimum distance are output as single points. The standard deviation of the X and Y 

coordinates is output as a rectangle. The standard distance deviation is output as a circle and the 

standard deviational ellipse is output as an ellipse. 

CrimeStat currently supports graphical outputs to ArcGIS >shp=, MapInfo >mif=, Google 

Earth ‘kml, or various Ascii files. Before running the calculation, the user should select the 

desired output files and specify a root name (e.g., Precinct1Burglaries). Figure 4.16 shows a 

dialog box for outputting a shape file to ArcGIS. For MapInfo output only, the user has to also 

indicate the name of the projection, the projection number and the datum number. These can be 

found in the MapInfo users guide. By default, CrimeStat will use the standard parameters for a 

spherical coordinate system (Earth projection, projection number 1, and datum number 33). If a 

user requires a different coordinate system, the appropriate values should be typed into the space. 

Figure 4.17 shows the selection of the MapInfo coordinate parameters. 

If requested, the output files are saved in the specified directory under the specified (root) 

name. For each statistic, CrimeStat will add prefix letters to the root name. 

MC<root> for the mean center 

MdnCntr<root> for the median center 

Mcmd<root> for center of minimum distance 

XYD<root> for the standard deviation of the X and Y coordinates 

SDD<root> for the standard distance deviation 

SDE<root> for the standard deviational ellipse. 

The >.shp= files can be read directly into ArcGIS as themes. The ‘kml files can be read 

directly into Google Earth. The >.mif= files have to be imported into MapInfo.

3 

In MapInfo, the comm`and is Table Import <MapInfo Interchange file> though it is a lot easier to use the 

MapInfo Universal Translator. 

4 . 2 7

3 
Figure 4.16: 

Outputting a Shape File to ArcView/ArcGIS
Figure 4.17: 

MapInfo Output Options 
 

Statistical Testing 

While the current version of CrimeStat does not conduct statistical tests that compare two 

distributions, it is possible to conduct such tests. Appendix A presents a discussion of the 

statistical tests that can be used. Instead, the discussion here will focus on using the outputs of 

the routines without formal testing. 

Decision-making Without Formal Tests 

Formal significance testing has the advantage of providing a consistent inference about 

whether the difference in two distributions is likely or unlikely to be due to chance. Almost all 

formal tests compare the distribution of a statistic with that of a random distribution. However, 

police departments frequently have to make decisions based on small samples, in which case the 

formal tests are less useful than they would with larger samples. Still, the centrographic statistics 

calculated in CrimeStat can be useful and can help a police department make decision even in the 

absence of formal tests. 

Examples of Centrographic Statistics 

Example 1: June and July Auto Thefts in Precinct 11 

We want to illustrate the use of these statistics to make decisions with two examples. 

The first is a comparison of crimes in small geographical areas. In most metropolitan areas, most 

analysts will concentrate on particular sub-areas of the jurisdiction, rather than on the jurisdiction 

itself. In Baltimore County, for instance, analysis is done both for the jurisdiction as a whole as 

well as by individual precincts. 

Below in Figure 4.18 are the standard deviational ellipses for 1996 auto thefts for June 

and July in Precinct 11 of Baltimore County. As can be seen, there was a spatial shift that 

occurred between June and July of that year, the result most probably of increased vacation 

travel to the Chesapeake Bay. While the comparison is very simple, involving looking at the 

graphical object created by CrimeStat, such a month to month comparison can be useful for 

police departments because it points to a shift in incident patterns, allowing the police 

department to reorient their patrol units. 

4 . 3 0
Figure 4.18: 
Example 2: Serial Burglaries in Baltimore City and Baltimore County 

The second example illustrates a rash of burglaries that occurred on both sides of the 

border of Baltimore City and Baltimore County. On one hand there were ten residential 

burglaries that occurred on the western edge of the City/County border within a short time period 

of each other and, on the other hand, there were 13 commercial burglaries that occurred in the 

central part of the metropolitan areas. Both police departments suspected that these two sets 

were the work of a serial burglar (or group of burglars). What they were not sure about was 

whether the two sets of burglaries were done by the same individuals or by different individuals. 

The number of incidents involved are too small for significance testing; only one of the 

parameters tested was significant and that could easily be due to chance. However, the police do 

have to make a guess about the possible perpetrator even with limited information. Let=s use 

CrimeStat to try and make a decision about the distributions. 

Figure 4.19 illustrates these distributions. The thirteen commercial burglaries are shown 

as squares while the ten residential burglaries are shown as triangles. Figure 4.20 plots the mean 

centers of the two distributions. They are close to each other, but not identical. An initial hunch 

would suggest that the robberies are committed by two perpetrators (or groups of perpetrators), 

but the mean centers are not different enough to truly confirm this expectation. 

Similarly, Figure 4.21 plots the center of minimum distance. Again, there is a difference 

in the distribution, but it is not great enough to truly rule out the single perpetrator theory. Figure 

4.22 plots the raw standard deviations, expressed as a rectangle by CrimeStat. The dispersion of 

incidents overlaps to a sizeable extent and the area defined by the rectangle is approximately the 

same. In other words, the search area of the perpetrator or perpetrators is approximately the 

same. This might argue for a single perpetrator, rather than two. Figure 4.23 shows the 

standard distance deviation of the two sets of incidents. Again, there is sizeable overlap and the 

search radiuses are approximately the same. 

Only with the standard deviational ellipse, however, is there a fundamental difference 

between the two distributions (figure 4.24). The pattern of commercial robberies is falling along 

a northeast-southwest orientation while that for residential robberies along a northwest-southeast 

axis. In other words, when the orientation of the incidents is examined, as defined by the 

standard deviational ellipse, there are two completely opposite patterns. Unless this difference 

can be explained by an obvious factor (e.g., the distribution of commercial establishments), it is 

probable that the two sets of robberies were committed by two different perpetrators (or groups 

of perpetrators). 

4 . 3 2
Figure 4.19: 
Figure 4.20: 
Figure 4.21: 
Figure 4.22: 
Figure 4.23: 
Figure 4.24: 
Directional Mean and Variance 

Centrographic statistics utilize the coordinates of a point, defined as an X and Y value on 

either a spherical or projected/Cartesian coordinate system. There is another type of metric that 

can be used for identifying incident locations, namely a polar coordinate system. A vector is a 

line with direction and length. In this system, there is a reference vector (usually 00 due North) 

and all locations are defined by angular deviations from this reference vector. By convention, 

angles are defined as deviations from 00, clockwise through 3600. Note the measurement scale is 

a circle which returns back on itself (i.e. 00 is also 3600). Point locations can be represented as 

vectors on a polar coordinate system. 

With such a system, ordinary statistics cannot be used. For example, if there are five 

points which on the northern side of the polar coordinate system and are defined by their angular 

deviations as 00, 100, 150, 3450, and 3500 from the reference vector (moving clockwise from due 

North), the statistical mean will produce an erroneous estimate of 1440. This vector would be 

southeast and will lie in an opposite direction from the distribution of points. 

Instead, statistics have to be calculated by trigonometric functions. The input for such a 

system is a set of vectors, defined as angular deviations from the reference vector and a distance 

vector. Both the angle and the distance vector are defined with respect to an origin. The routine 

can calculate angles directly or can convert all X and Y coordinates into angles with a bearing 

from an origin. For reading angles directly, the input is a set of vectors, defined as angular 

deviations from the reference vector. CrimeStat calculates the mean direction and the circular 

variance of a series of points defined by their angles. On the primary file screen, the user must 

select Direction (angles) as the coordinate system. 

If the angles are to be calculated from X/Y coordinates, the user must define an origin 

location. On the reference file page, the user can select among three origin points: 

1. The lower-left corner of the data set (the minimum X and Y values). This is the

default setting.

2. The upper-right corner of the data set (the maximum X and Y values); and

3. A user-defined point.

Users should be careful about choosing a particular location for an origin, either lowerleft, upper-right or user-defined. If there is a point at that origin, CrimeStat will drop that case 

since any calculations for a point with zero distance are indeterminate. Users should check that 

4 . 3 9
 

 

 

 

there is no point at the desired origin. If there is, then the origin should be adjusted slightly so 

that no point falls at that location (e.g., taking slightly smaller X and Y values for the lower-left 

corner or slightly larger X and Y values for the upper right corner). 

The routine converts all X and Y points into an angular deviation from true North relative 

to the specified origin and a distance from the origin. The bearing is calculated with different 

formulae depending on the quadrant that the point falls within. 

First Quadrant 

With the lower-left corner as the origin, all angles are in the first quadrant. The 

clockwise angle, θi, is calculated by: 

(4.24)

where Xi is the X-value of the point, Yi is the Y-value of the point, XO is the X-value of the 

origin, and YO is the Y-value of the origin. 

(4.25)

Third Quadrant 

With the upper-right corner as the origin, all angles are in the third quadrant. The 

clockwise angle, θi, is calculated by: 

(4.26)

where the angle, θi, is again in radians. Since there are 2π radians in a circle, π radians is 1800. 

Again, the angle in radians can be converted into degrees with formula 4.25 above. 

Second and Fourth Quadrants 

When the origin is user-defined, each point must be evaluated as to which quadrant it is 

in. The second and fourth quadrants define the clockwise angle, θi, differently: 

4 . 4 0

The angle, θi, is in radians and can be converted to polar coordinate degrees using:
 

Second quadrant 

Fourth quadrant 

 

 

 (4.29)

(4.27)

(4.28)

Once all X/Y coordinates are converted into angles, the mean angle is calculated. 

Mean Angle 

With either angular input or conversion from X/Y coordinates, the Mean Angle is the 

resultant of all individual vectors (i.e., points defined by their angles from the reference vector). 

It is an angle that summarizes the mean direction. Graphically, a resultant is the sum of all 

vectors and can be shown by laying each vector end to end. Statistically, it is defined as 

where the summation of sines and cosines is over the total number of points, i, defined by their 

angles, θi. Each angle, θi, can be weighted by the length of the vector, di. In an unweighted 

angle, di is assumed to be of equal length, 1. The absolute value of the ratio of the sum of the 

weighted sines to the sum of the weighted cosines is taken. All angles are in radians. In 

determining the mean angle, the quadrant of the resultant must be identified: 

Formulas 4.26, 4.27, 4.28 and 4.29 above are then used to convert the directional mean 

back to an X/Y coordinate, depending on which quadrant it falls within. 

4 . 4 1
 

 ݎ݈ܽݑܿݎ݅ܥ

 

where N is the number points (sample size). 

 

 

 

 (4.32)

 

Circular Variance 

The dispersion (or variance) of the angles are also defined by trigonometric functions. 

The unstandardized variance, R, is sometimes called the sample resultant length since it is the 

resultant of all vectors (angles): 

(4.30) 

Because R increases with sample size, it is standardized by dividing by N to produce a 

mean resultant length: 

(4.31) 

Finally, the average distance from the origin, D, is calculated and the circular variance is 

calculated by: 

This is the standardized variance which varies from 0 (no variability) to 1 (maximum 

variability). The details of the derivations can be found in Burt and Barber (1996) and Gaile and 

Barber (1980). 

Mean Distance 

The mean distance, 

,

ഥ is calculated directly from the X and Y coordinates. It is identified 

in relation to the defined origin. 

Directional Mean 

The directional mean is calculated as the intersection of the mean angle and the mean 

distance. It is not a unique position since distance and angularity are independent dimensions. 

Thus, the directional mean calculated using the minimum X and minimum Y location as the 

reference origin (the >lower left corner=) will yield a different location from the directional mean 

4 . 4 2

where di is the length of vector, i, with an angle (bearing) for the vector of θi. For the 

unweighted sample resultant, di is 1. 
calculated using the maximum X and maximum Y location as the origin (the >upper right 

corner=). There is a weighted and unweighted directional mean. Though CrimeStat calculates 

the location, users should be aware of the non-uniqueness of the location. The unweighted 

directional mean can be output with a >Dm= prefix. The weighted directional mean is not output. 

Triangulated Mean 

The triangulated mean is defined as the intersection of the two vectors, one from the 

lower-left corner of the study area (the minimum X and Y values) and the other from the upperright corner of the study area (the maximum X and Y values). It is calculated by estimating 

mean angles from each origin (lower left and upper right corners), translating these into 

equations, and finding the point at which these equations intersect (by setting the two functions 

equal to each other). 

Directional Mean Output 

The directional mean routine outputs nine statistics: 

1. The sample size;

2. The unweighted mean angle;

3. The weighted mean angle;

4. The unweighted circular variance;

5. The weighted circular variance;

6. The mean distance;

7. The intersection of the mean angle and the mean distance;

8. The X and Y coordinates for the triangulated mean; and

9. The X and Y coordinates for the weighted triangulated mean.

The directional mean and triangulated mean can be saved as ArcGIS >shp=, MapInfo >mif=, 

Google Earth ‘kml, or various Ascii files. The unweighted directional mean - the intersection of 

the mean angle and the mean distance, is output with the prefix >Dm= while the unweighted 

triangulated mean location is output with a >Tm= prefix. The weighted triangulated mean is 

output with a >TmWt= prefix. See the example below. 

Figure 4.25 shows the unweighted triangular mean for 1996 Baltimore County robberies 

and compares it to the two directional means calculated using the lower-left corner (Dmean1) 

and the upper-right corner (Dmean2) respectively as origins. As can be seen, the two directional 

means fall at different locations. Lines have been drawn from each origin point to their 

4 . 4 3

respective directional means and are extended until they intersect. As seen, the triangulated 

mean falls at the location where the two vectors (i.e., mean angles) intersect. 

Because the triangulated mean is calculated with vector geometry, it will not necessarily 

capture the central tendency of a distribution. Asymmetrical distributions can cause it to be 

placed in peripheral locations. On the other hand, if the distribution is relatively balanced in 

each direction, it can capture the center of orientation perhaps better than other means, as figure 

4.25 shows. Appendix A includes a discussion of how to formally test the mean direction 

between two different distributions. 

Convex Hull 

The convex hull is a boundary drawn around the distribution of points. It is a relatively 

simple concept, at least on the surface. Intuitively, it represents a polygon that circumscribes all 

the points in the distribution such that no point lies outside of the polygon. 

The complexity comes because there are different ways to define a convex hull. The 

most basic algorithm is the Graham scan (Graham, 1972). Starting with one point known to be 

on the convex hull, typically the point with the lowest X coordinate, the algorithm sorts the 

remaining points in angular order around this in a counterclockwise manner. If the angle formed 

by the next point and the last edge is less than 180 degrees, then that point is added to the hull. If 

the angle is greater than 180 degrees, then the chain of nodes starting from the last edge must be 

deleted. The routine proceeds until the hull closes back on itself (de Berg, van Kreveld, 

Overmans, and Schwarzkopf, 2000). 

Many alternative algorithms have been proposed. Among these are the >gift wrap= (Chand 

and Kapur, 1970; Skiena, 1997), the Quick Hull, the ADivide and conquer@ (Preparata and Hong, 

1977), and the incremental (Kallay, 1984) algorithms. Even more complexity has been 

introduced by the mathematics of fractals where an almost infinite number of borders could be 

defined (Lam and De Cola, 1993). In most implementations, though, a simplified algorithm is 

used to produce the convex hull. 

CrimeStat implements the >gift wrap= algorithm. Starting with the point with the lowest Y 

coordinate, A, it searches for another point, B, such that all other points lie to the left of the line 

AB. It then finds another point, C, such that all remaining points lie to the left of the line BC. It 

continues in this way until it reaches the original point A again. It is like >wrapping a gift= around 

the outside of the points. 

4 . 4 5
 

The routine outputs three statistics: 

1. The sample size;

2. The number of points in the convex hull

3. The X and Y coordinates for each of the points in the convex hull

The convex hull can be saved as ArcGIS >shp=, MapInfo >mif=, Google Earth ‘kml, or 

various Ascii files with a 'Chull' prefix. 

Figure 4.26 shows the convex hull of Baltimore County robberies for 1996. As seen, the 

hull occupies a relatively smaller part of Baltimore County. Figure 4.27, on the other hand 

shows the convex hull of 1996 Baltimore County burglaries. As seen, the convex hull of the 

burglaries cover a much larger area than for the robberies. 

Uses and Limitations of the Convex Hull 

A convex hull can be useful for displaying the geographical extent of a distribution. 

Simple comparisons, such as in Figures 4.26 and 4.27, can show whether one distribution has a 

greater extent than another. Further, as we shall see, a convex hull can be useful for describing 

the geographical spread of a crime hot spot, essentially indicating where the crimes are 

distributed. 

On the other hand, a convex hull is vulnerable to extreme values. If one incident is 

isolated, the hull will of necessity be large. The mean center, too, is influenced by extreme 

values but not to the same extent since it averages the location of all points. The convex hull, on 

the other hand, is defined by the most extreme points. A comparison of different crime types or 

the same crime type for different years using the convex hull may only show the variability of 

the extreme values, rather than any central property of the distribution. Therefore, caution must 

be used in interpreting the meaning of a hull. 

4 . 4 6
Figure 4.26: 
Figure 4.27: 
 

References 

Bachi, R. (1957). Statistical Analysis of Geographical Series. Central Bureau of Statistics, 

Kaplan School, Hebrew University: Jerusalem. 

Burt, J. E. & Barber, G. M. (1996). Elementary Statistics for Geographers (second edition). The 

Guilford Press: New York. 

Chand, D & Kapur, S. (1970). An algorithm for convex polytopes. J. ACM, 17, 78-86. 

Cromley, R. G. (1992). Digital Cartography. Prentice Hall: Englewood Cliffs, NJ. 

de Berg, M.; van Kreveld, M.; Overmans, M.; & Schwarzkopf, O. (2000). Convex hulls: mixing 

things. In Computational Geometry: Algorithms and Applications, 2nd rev. ed. Springer-Verlag: 

Berlin, 235-250. 

Ebdon, D. (1988). Statistics in Geography (second edition with corrections). Blackwell: Oxford. 

Everitt, B. S. (2011). Cluster Analysis (5th edition). J. Wiley: London. 

Furfey, P. H. (1927). A note on Lefever's 'Standard deviational ellipse'. American Journal of 

Sociology. XXIII, 94-98. 

Gaile, G. L. & Burt, J. E. (1980). Directional Statistics. Concepts and Techniques in Modern 

Geography No. 25. Institute of British Geographers, Norwich, England: Geo Books. 

Graham, R (1972). An efficient algorithm for determining the convex hull of a finite planar point 

set. Info. Proc. Letters, 1, 132-133. 

Hammond, R., & McCullagh, P. (1978). Quantitative Techniques in Geography: An 

Introduction. Second Edition. Clarendon Press: Oxford, England. 

Hultquist, J., Brown, L. & Holmes, J. (1971). Centro: a program for centrographic measures. 

Discussion paper no. 21, Department of Geography, Ohio State University: Columbus, OH. 

Kallay, M. (1984). The complexity of incremental convex hull algorithms in Rd, Info. Proc. 

Letters 19, 197. 

4 . 4 9
 

References (continued)

Kuhn, H. W. & Kuenne, R. E. (1962). An efficient algorithm for the numerical solution of the 

generalized Weber problem in spatial economics, Journal of Regional Science 4, 21-33. 

Lam, N. Siu-ngan & De Cola, L. (1993). Fractals in Geography. The Blackburn Press: 

Caldwell, NJ. 

Langworthy, R. H. & Jefferis, E. (1998). The utility of standard deviational ellipses for project 

evaluation. Discussion paper, National Institute of Justice: Washington, DC. 

Lefever, D. (1926). Measuring geographic concentration by means of the standard deviational 

ellipse. American Journal of Sociology, 32(1): 88-94. 

Neft, D. S. (1962). Statistical Analysis for Areal Distributions. Ph.D. dissertation, Columbia 

University: New York. 

Preparata, F. & Hong, S. J. (1977). Convex hulls of finite sets of points in two and three 

dimensions, Comm. ACM, 20, 87-93. 

Skiena, S. S. (1997). Convex hull. §8.6.2 in The Algorithm Design Manual. Springer-Verlag: 

New York, 351-354. 

Stephenson, L. (1980). Centrographic analysis of crime. In D. George-Abeyie & K. Harries 

(eds), Crime, A Spatial Perspective, Columbia University Press: New York. 

4 . 5 0
 

 

 

 

 

Endnotes 

i. CrimeStat=s implementation of the Kuhn and Kuenne algorithm is as follows (from Burt

and Barber, 1996, 112-113):

1. Let t be the number of the iteration. For the first iteration only (i.e., t=1) the

weighted mean center is taken as the initial estimate of the median location, Xt 

and Yt.

2. Calculate the distance from each point, i, to the current estimate of the median

location, dict, where i is a single point and ct is the current estimate of the median

location during iteration t.

a. If the coordinates are spherical, then Great Circle distances are used.

b. If the coordinates are projected, then Euclidean distances are used.

3. Weight each case by a weight, Wi, and calculate:

where e is the base of the natural logarithm(2.7183..). 

a. If no weights are defined in the primary file, Wi is assumed to be 1.

b. If weights are defined in the primary file, Wi takes their values.

Note that as the distance, dict, approaches 0, then ݁ିௗ೔೎೟ becomes 1. 

4. Calculate a new estimate of the center of minimum distance from:

where Xi and Yi are the coordinates of point i (either lat/lon for spherical or feet 

or meters for projected). 

4 . 5 1
 

 

 

 

 

 

 

 

Endnotes (continued)

5. Check to see how much change has occurred since the last iteration

a. If either the X or Y coordinates have changed by greater than 0.000001

between iterations, substitute Xt+1 for Xt

 and Yt+1 for Yt and repeat steps

B through D.

b. If both the change in X and the change in Y is less than or equal to

0.000001, then the estimated Xt and Yt coordinates are taken as the center

of median distance.

ii. Formulas for the new axes provided by Ebdon (1988) and Cromley (1992) yield standard

deviational ellipses that are too small, for two different reasons. First, they produce

transformed axes that are too small. If the distribution of points is random and even in all

directions, ideally the standard deviational ellipse should be equal to the standard

distance deviation, since Sx = Sy. The formula used here has this property. Since the

formula for the standard distance deviation is (equation 4.8):

4 . 5 2

Similarly, the formulas for the transformed axes are (4.9, 4.10): 
 

 

 

 

Endnotes (continued)

However, if Sx = Sy, then θ = 0, cos θ = 1, Sin θ = 0 and, therefore: 

which is the same as for the standard distance deviation (SDD) under the same 

conditions. The formulas used by Ebdon (1988) and Cromley (1992) produce axes which 

are √2 times too small. 

The second problem with the Ebdon and Cromley formulas is that they do not correct for 

degrees of freedom and, hence, produce too small a standard deviational ellipse. Since 

there are two constants in each equation, ܺത and ܻത, then there are only N-2 degrees of 

freedom. The cumulative effect of using transformed axes that are too small and not 

correcting for degrees of freedom yields a much smaller ellipse than that used here. 

4 . 5 3
Attachments 

4 . 5 4
Using Spatial Measures of Central Tendency with Network Analyst 

to Identify Routes Used by Motor Vehicle Thieves 

Philip R. Canter 

Baltimore County Police Department 

Towson, Maryland 

Motor vehicle thefts have been steadily declining countywide over the last 5 

years, but one police precinct in southwest Baltimore County was experiencing 

significant increases over several months. Cases were concentrated in several 

communities, but directed deployment and saturated patrols had minimal impact. In 

addition to increasing patrols in target communities, the precinct commander was 

interested in deploying police on roads possibly used by motor vehicle thieves. Police 

analysts had addresses for theft and recovery locations; it was a matter of using the 

existing highway network to connect the two locations. 

To avoid analyzing dozens of paired locations, analysts decided to set up a 

database using one location representing the origin of motor vehicle thefts for a 

particular community. The origin was computed using CrimeStat’s median center for 

motor vehicle theft locations reported for a particular community. The median 

center is the position of minimum average travel and is less affected by extreme 

locations compared to the arithmetic mean center. The database consisted of the 

median center paired with a recovery location. Using Network Analyst, a least-effort 

route was computed for cases reported by community. A count was assigned to each 

link along a roadway identified by Network Analyst. Analysts used the count to 

thematically weight links in ArcView. The precinct commander deployed resources 

along these routes with orders to stop suspicious vehicles. This operation resulted in 

27 arrests, and a reduction in motor vehicle thefts. 
Centrographic Analysis 

Man With A Gun Calls For Service 

Charlotte, N.C., 1989 

James L. LeBeau 

Administration of Justice 

Southern Illinois University – Carbondale 

Hurricane Hugo arrived on Friday, September 22, 1989 in Charlotte, North 

Carolina. That weekend experienced the highest counts of Man With A Gun calls for 

service for the year. The locations of the calls during the Hugo Weekend are 

compared with the following New Year’s Eve weekend. 

CrimeStat was used to compare the two weekends. Compared to the New

Year’s Eve weekend: 1) Hugo’s mean and median centers are more easterly; 2) 

Hugo’s ellipse is larger and more circular; and 3) Hugo‘s ellipse shifts more to the 

east and southeast. The abrupt spatial change of Man With A Gun calls during a

natural disaster might indicate more instances of defensive gun use for protection of 

property.