Appendix A: 

Some Notes on the 

Statistical Comparison of Two Samples 

By 

Ned Levine 

Ned Levine & Associates 

Houston, TX 

The following presents methods for testing the spatial differences between two 

distributions. At this point, CrimeStat does not include routines for testing the differences 

between two or more samples. The following is provided for the reader=s information. 

Chapter 4 discussed the calculation of these statistics as a single distribution. 

Differences in the Mean Center of Two Samples 

For differences between two samples in the mean center, it is necessary to test both 

differences in the X coordinate and differences in the Y coordinates. Since CrimeStat outputs 

the mean X, the mean Y, the standard deviation of X, and the standard deviation of Y, a simple 

t-test can be set up. The null hypothesis is that the mean centers are equal 

H0: μXA = μXB (A.1) 

μYA = μYB (A.2) 

and the alternative hypothesis is that the mean centers are not equal 

H1: μXA =/ μXB (A.3) 

μYA =/ μYB (A.4) 

Because the true standard deviations of sample A, σXA and σYA, and sample B, σXB and σYB, 

are not known, the sample standard deviations are taken, SXA, SYA, SXB and SYA. However, 

since there are two different variables being tested (mean of X and mean of Y for groups 1 and 

2), the alternative hypothesis has two fundamentally different interpretations: 

Comparison I: That EITHER μXA =/ μXB OR μYA =/ μYB is true (A.5) 

A.1 
 

 

 

Comparison II: That BOTH μXA =/ μXB AND μYA =/ μYB are true (A.6) 

In the first case, the mean centers will be considered not being equal if either the mean of 

X or the mean of Y are significantly different. In the second case, both the mean of and the 

mean of Y must be significantly different for the mean centers to be considered not equal. The 

first case is clearly easier to fulfill than the second. 

Significance levels 

By tradition, significance tests for comparisons between two means are made at the α#.05 

or α#.01 levels, though there is nothing absolute about those levels. The significance levels are 

selected to minimize Type 1 Errors, inadvertently declaring a difference in the means when, in 

reality, there is not a difference. Thus, a test establishes that the likelihood of falsely rejecting the 

null hypothesis be less than one-in-twenty (less strict) or one-in-one hundred (more strict). 

However, with multiple comparisons, the chances increase for finding >significance= due 

to the multiple tests. For example, with two tests - a difference in the means of the X 

coordinate and a difference in the means of the Y coordinate, the likelihood of rejecting the first 

null hypothesis (μXA =/ μXB) is one-in-twenty and the likelihood of rejecting the second null 

hypothesis (μYA =/ μYB) is also one-in-twenty, then the likelihood of rejecting either one null 

hypothesis or the other is actually one-in-ten. 

To handle this situation, comparison I - the >either/or= condition, a Bonferoni test is 

appropriate (Anselin, 1995; Systat, 1996). Because the likelihood of achieving a given 

significance level increases with multiple tests, a >penalty= must be assigned in finding either the 

differences in means for the X coordinate or differences in means for the Y coordinates 

significant. The Bonferoni criteria divides the critical probability level by the number of tests. 

Thus, if the α#.05 level is taken for rejecting the null hypothesis, the critical probability for each 

mean must be .025 (.05/2); that is, differences in either the mean of X or mean of Y between two 

groups must yield a significance level less than .025. 

For comparison II - the >both/and= condition, on the other hand, the test is more stringent 

since the differences between the means of X and the means of Y must both be significant. 

Following the logic of the Bonferoni criteria, the critical probability level is multiplied by the 

number of tests. Thus, if the α=.05 level is taken for rejecting the null hypothesis, then both 

tests must be significant at the α#.10 level (i.e., .05*2).1 

There are limits to the Bonferoni logic. For example, if there were 10 tests, having a threshold significance 

A.2 

1 
sample standard deviations) of both the X and Y coordinates:

(A.7)

(A.8)

 

 

 

Tests 

The statistics used are for the t-test of the difference between means (Kanji, 1993). 

a. First, test for equality of variances by taking the ratio of the variances (squared

with (NA - 1) and (NB - 1) degrees of freedom for groups A and B respectively. 

This test is usually done with the larger of the variances in the numerator. Since 

there are two variances being compared (for X and Y, respectively), the logic 

should follow either I or II above (i.e., if either are to be true, then the critical α

will be actually α/2 for each; if both must be true, then the critical α will be 

actually 2*α for each). 

b. Second, if the variances are considered equal, then a t-test for two group means

with unknown, but equal, variances can be used (Kanji, 1993; 28). Let

(A.9)

 (A.10) 

where the summations are for i=1 to N within each group separately. Then the 

test becomes 

level of .005 (.05 / 10) for the >either/or= conditions and a threshold significance level of .50 (.05 * 10) for 

the >both/and= would lead to an excessively difficult test in the first case and a much too easy test in the 

second. Thus, the Bonferoni logic should be applied to only a few tests (e.g., 5 or fewer). 

A.3
with degrees of freedom 

with degrees of freedom for each test. 

c. Third, if the variances are not equal, then a t-test for two group means with

unknown and unequal variances should be used (Kanji, 1993; 29).

(A.11) 

(A.12) 

(A.13) 

(A.14) 

(A.15) 

(A.16) 

(A.17) 

(A.18) 

(A.19) 

A.4
 (A.20)

for both the X and Y test. Even though this latter formula is cumbersome, in 

practice, if the sample size of each group is greater than 100, then the t-values for 

infinity can be taken as a reasonable approximation and the above degrees of 

freedom need not be tested: 

i. t=1.645 for α=.05; t=2.326 for α=.01 for a one-tail test

ii. t=1.645 for α=.10; t=1.960 for α=.05; t=2.327 for α=.02; t=2.576

for α=.01 for a two-tail test

d. The significance levels are those selected above. For comparison I - that either

differences in the means of X or differences in the means of Y are significant, the

critical probability level is α/2 (e.g., .05/2 = .025; .01/2 = .005). For comparison

II - that both differences in the means of X and differences in the means of Y are

significant, the critical probability level is α*2 (e.g., .05*2 = .10; .01*2 - .02).

e. Reject the null hypothesis if:

Comparison I: Either tested t-value (tx or ty) is greater than the Critical t 

for α/2 

Comparison II: Both tested t-values (tx and ty) are greater than the critical t 

for α*2 

Example 1: Burglaries and Robberies in Baltimore County 

To illustrate, compare the distribution of burglaries in Baltimore County with those of 

robberies, both for 1996. Figure A.1 shows the mean center of all robberies (blue square) and 

all residential burglaries (red triangle). As can be seen, the mean centers are located within 

Baltimore City, a property of the unusual shape of the county (which surrounds the city on three 

sides). Thus, these mean centers cannot be considered an unbiased estimate of the metropolitan 

area, but unbiased estimates for the County only. When the relative positions of the two mean 

centers are compared, the center of robberies is south and west of the center for burglaries. Is 

this difference significant or not? 

A.5
Figure A.1:

Mean center of burglaries 

Mean center of robberies 
 

 

 

 

 

 

To test this, the standard deviations of the two distributions are first compared and the 

F-test of the larger to the smaller variance is used (equations A.1 and A.2). CrimeStat provides

the standard deviation of both the X and Y coordinates; the variance is the square of the standard

deviation. In this case, the variance for burglaries is slightly larger than for robberies for both

the X and Y coordinates.

(A.21) 

(A.22) 

Because both samples are fairly large (1180 robberies and 6051 burglaries respectively), 

the degrees of freedom are also very large. The F-tables are a little indeterminate with large 

samples, but the variance ratio approaches 1.00 as the sample reaches infinity. An approximate 

critical F-ratio can be obtained by the next largest pair of values in the table (1.22 for p#.05 and 

1.32 for p#.01). Using this criterion, differences in the variances for the X coordinate are 

probably not significant while that for the Y coordinates definitely are significant. 

Consequently, the test for a difference in means with unequal variances is used (equations A.17 

and A.18). 

(A.23) 

(A.24) 

Therefore, whether we use the >either/or= test (critical α#.025) or the >both/and= test 

(critical α#.1), we find that the difference in the mean centers is highly significant. Burglaries 

have a different center of gravity than robberies in Baltimore County. 

Differences in the Standard Distance Deviation of Two Samples 

Since the standard distance deviation, SXY (equation 4.6 in Chapter 4) is a standard 

deviation, differences in the standard distances of two groups can be compared with an equality 

of variance test (Kanji, 1993, 37), 

A.7
 

 

 

 

 (A.25)

with (NA - 1) and (NB - 1) degrees of freedom for groups A and B, respectively. This test is 

usually done with the larger of the variances in the numerator. Since there is only one variance 

being compared, the critical α are as listed in the tables. 

Baltimore County Burglary Example (continued) 

From CrimeStat, we find that the standard distance deviation of burglaries is 8.44 miles 

while that for robberies is 7.42 miles. The F-test of the difference is calculated by 

 (A.26)

with 6050 and 1180 degrees of freedom respectively. Again, the F-tables are slightly 

indeterminate with respect to large samples, but the next largest F beyond infinity is 1.25 for 

p#.05 and 1.38 for p#.01. Thus, it appears that burglaries have a significantly greater 

dispersion than robberies, at least at the p#.05 level. 

Differences in the Standard Deviational Ellipse of Two Samples 

Figure A.2 shows the standard deviational ellipse of all robberies (light blue) and all 

residential burglaries (light red). As can be seen, the dispersion of incidents, as defined by the 

standard deviational ellipse, is greater for burglaries than for robberies. In a standard 

deviational ellipse, there are actually six variables being compared: 

1. Mean of X

2. Mean of Y

3. Angle of rotation

4. Standard deviation along the transformed X axis

5. Standard deviation along the transformed Y axis

6. Area of the ellipse

A.8
Figure A.2:

SDE of burglaries 

SDE of robberies 
Differences in the mean centers 

Comparisons between the two mean centers can be tested with equations A.9 through 

A.12 if the variance test of equations A.7 and A.8 show equality or equation A.13 through A.20

if the variances are unequal.

Differences in the angle of rotation 

Unfortunately, to our knowledge, there is not a formal test for the difference in the angle 

of rotation. Until this test is developed, we have to rely on subjective judgment. 

Differences in the standard deviations along the transformed axes 

The differences in the standard deviations along the transformed axes (X and Y) can be 

tested with an equality of variance test (Kanji, 1993, 37), 

 (A.27) 

 (A.28) 

with NA.1 and NB-1 degrees of freedom for groups A and B respectively. This test is usually 

conducted with the larger of the variances in the numerator. The example above for comparing 

the mean centers of Baltimore County burglaries and robberies illustrated the use of this test. 

Differences in the areas of the two ellipses 

Since an area is a variance, the differences in the areas of the two ellipses can be 

compared with an equality of variance test (Kanji, 1993, 37), 

 (A.29)

with NA.1 and NB-1 degrees of freedom for groups A and B respectively. This test is conducted 

with the larger of the variances in the numerator. 

A.10
Significance levels 

The testing of each of these parameters for the difference between two ellipses is even 

more complicated than the difference between two mean centers since there are up to six 

parameters which must be tested (differences in mean X, mean Y, angle of rotation, standard 

deviation along transformed X axis, standard deviation along transformed Y axis, and area of 

ellipse). However, as with differences in mean center of two groups, there are two different 

interpretations of differences. 

Comparison I: That the two ellipses differ on ANY of the parameters (A.30) 

Comparison II: That the two ellipses differ on ALL parameters (A.31) 

In the first case, the critical probability level, α, must be divided by the number of 

parameters being tested, α/p. In theory, this could involve up to six tests, though in practice 

some of these may not be tested (e.g., the angle of rotation). For example, if five of the 

parameters are being estimated, then the critical probability level at α#.05 is actually α# .01 

(.05/5). 

In the second case, the critical probability level, α, is multiplied by the number of 

parameters being tested, α*p, since all tests must be significant for the two ellipses to be 

considered as different. For example, if five of the parameters are being estimated, then the 

critical probability level, say, at α#.05 is actually α#.25 (.05*5). 

Differences in the Mean Direction Between Two Groups 

Statistical tests of different angular distributions can be made with the directional mean 

and variance statistics. To test the difference in the angle of rotation between two groups, a 

Watson-Williams test can be used (Kanji, 1993; 153-54). The steps in the test are as follows: 

(A.32) 

2. For each sample separately, A and B, the following measures are calculated

(A.33) 

A.11

1.
5. An F-test of the two angular means is calculated with

the resultant vector.

 

(A.34)

where θj and θk are the individual angles for the respective groups, A and A. 

3. Calculate the resultant lengths of each group

(A.35) 

(A.36) 

4. Resultant lengths for the combined sample are calculated as well as the length of

 (A.37) 

 (A.38) 

 (A.39) 

 (A.40) 

(A.41)

(A.42)

where 

 (A.43)

with k being identified from a maximum likelihood Von Mises distribution by referencing R* 

with 1 and N-2 degrees of freedom (Gaile & Burt, 1980; Mardia, 1972). Some of the reference 

k=s are given in Table A.1 below (from Kanji, 1993, table 38; Mardia, 1972). 

A.12
6. Reject the null hypothesis of no angular difference if the calculated F is greater 

than the critical value F1, N-2. 

Example 2: Angular comparisons between two groups 

A second example is that of sets of angular measurements from two different groups, A 

and B. Table A.2 provides the data for the two sets. The angular mean for Group A is 144.830 

with a directional variance of 0.35 while the angular mean for Group B is 258.950 with a 

directional variance of 0.47. The higher directional variance for Group B suggests that there is 

more angular variability than for Group A. 

Table A.1: 

Maximum Likelihood Estimates for Given R* in the Von Mises Case 

(Kanji, 1993, table 38; Mardia, 1972) 

R* k 

0.00 0.00000 

0.05 0.10013 

0.10 0.20101 

0.15 0.30344 

0.20 0.40828 

0.25 0.51649 

0.30 0.62922 

0.35 0.74783 

0.40 0.87408 

0.45 1.01022 

0.50 1.15932 

0.55 1.32570 

0.60 1.51574 

0.65 1.73945 

0.70 2.01363 

0.75 2.36930 

0.80 2.87129 

0.85 3.68041 

0.90 5.3047 

0.95 10.2716 

1.00 infinity 

A.13 
Group A Group B

Measured 

Incident Angle 

1 160

2 184 

3 240 

4 100 

5 95 

6 120 

7 190 

8 340 

 Measured 

Incident Angle

 1 196 

2 212 

3 297 

4 280 

5 235 

6 353 

 

 

 

 

 

 

 

 

 

 

Using the Watson-Wheeler test, we compare these two distributions. 

1. All angles are converted into radians (equation A.32). 

2. The cosines and sines of each angle are taken and are summed within groups 

(equations A.33 and A.34). 

CA = -3.1981 SA = 2.2533 

CB = -.8078 SB = -4.1381 

Table A.2: 

Comparison of Two Groups for Angular Measurements 

Angle of Deviation From Due North 

 

3. The resultants are calculated (equations A.35 and A.36). 

RA = 3.9121 

RB = 4.2162 

A.14 
 

4. Combined sample characteristics are defined (equations A.37 through A.41). 

C = -4.0059 

S = -1.8848 

R = 4.4271 

N = 14 

R*

 = 0.5806 

5. Once the parameter, k, is obtained (approximated from Table A.1 or obtained 

from Mardia, 1972 or Kanji, 1993), g is calculated, and an F-test is constructed 

(equations A.42 and A.43). 

k = 1.44 

g = 0.7396 

F = 5.59 

6. The critical F for 1 and 12 degrees of freedom is 4.75 for p#.05 and F=9.33 for 

p#.01. Since F=5.59 is between these two critical F values, the test is significant 

at the p#.05 level, but not at the p#.01 level. Nevertheless, we reject the null 

hypothesis of no angular differences between the two groups. Group A has a 

different angular distribution than Group B. 

A.15 
References 

Anselin, L. (1995). Local indicators of spatial association - LISA. Geographical Analysis. 

27, No. 2 (April), 93-115. 

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. 

Kanji, G. K. (1993). 100 Statistical Tests. Sage Publications: Thousand Oaks, CA. 

Mardia, K.V. (1972). Statistics of Directional Data. Academic Press: New York. 

Systat, Inc. (2008). Systat 13: Statistics I. SPSS, Inc.: Chicago. 

A.16