Statistical Equations
Quantitative Methods (GEO 441)
Equation
Statistic
n
x
i
Mean
i =1
m=
n
n
2
Median Location
n
(x - x)
2
i
Sample Variance
s2 =
i =1
n -1
n
(x - m)
Sample Standard Deviation
2
i
s=
Z-Score
i =1
n
z=
(x - x)
s
n
(x - x)
3
i
Skewness

Quantitative Methods (GEO 441)
In-Class Test 2 Study Guide
Dr. Paul Marr
Please know the following for the second in-class test:
Know all of the components of correlation and regression.
o Know the definitions of the terms used (e.g. residual, coefficient

Quantitative Methods (GEO 441)
Hypothesis Testing
From Zar, 1984
Statistical procedures for addressing research questions involves formulating a concise statement of the hypothesis
to be tested. The hypothesis to be tested is referred to as the null hypot

Quantitative Methods (Geo 441)
Dr. Paul Marr
The formats below are to be used when completing the exercises and tests for this course.
Format for Hand Calculation:
Research Question: One hundred and fifty students in Geo 441 were surveyed
to determine if

Quantitative Methods (GEO 441)
In-Class Test 1 Study Guide
Dr. Paul Marr
Please know the following for the first in-class test:
What each of the statistical test/methods we covered does
The difference between laws, hypotheses, theories, etc
The major s

DAgostino Normality Test
SPSS Macro Usage Instructions
Quantitative Methods (Geo 441)
Dr. Paul Marr
Macro written by: Lawrence DeCarlo, Columbia University
Instructions:
1. Save the DAgostino Macro file (dagostino.sps) to your working folder.
2. Save the

Plus / Minus Grading System
Dr. Paul Marr
Note that grades are determined to 1 decimal place and rounded up to the next whole number.
Grade
A
AB+
B
BC+
C
D
F
Point Range (out of 100)
94 - 100
90 - 93
87 - 89
84 - 86
80 - 83
77 - 79
70 - 76
60 - 69
59 or l

Parametric Test
One-Sample t-Test for Means
(where is known and is unknown)
Equations taken from Zar, 1984
X
sX
t=
where
sX =
s
n
Example: Suppose that we know that the average income per household in Chester
county is $46,527. We want to know if average

Non-Parametric Test
Mann-Whitney U Test
Equations taken from Zar, 1984
U = n1 n 2 +
n1 ( n1 + 1)
R1
2
where R1 is the sum of the ranks for group 1
U ' = n1 n 2 U
Number of College Graduates
Example: We are interested in determining if the
number of colle

Non-Parametric Test
Kruskal-Wallis Analysis of Variance
Equations taken from Zar, 1984
k
R i2
12
3( N + 1)
H =
N ( N + 1)
i =1 n i
(t
C =1
3
i
ti )
H Corrected =
3
N N
H
C
(use correction factor when many ranks are tied)
Percent Education
Example:

Spatial Statistic
Mean and Weighted Mean Centers
Equations taken from Burt and Barber, 1996
Mean Center
n
n
X
i =1
X Coord =
Y
i
n
Y Coord =
i
i =1
n
n = 13
Example: The mean center is the average X and Y
coordinate for a series of points on a map. The me

Non-Parametric
2 Contingency Analysis (r x c Tables)
Equations taken from Zar, 1984
2 =
and
( f ij f ij ) 2
f
where
f ij =
( Row i )(Column j )
ij
n
df = ( Rows 1)(Columns 1)
Example: The map on the right shows
employment levels for two groups of
villages

Parametric Test
Directional Mean and Rayleighs Z
Equations taken from Zar, 1984
Mean Angle Calculation
n
X
cos
i 1
n
Rayleighs Z
n
a
Y
sin
i 1
a
n
r X 2 Y 2
cos a
X
r
sin a
Y
r
sin a
a arctan
cos a
z nr 2
Example: The direction (azimuth) of wind

Parametric Test
Grubbs G Outlier Test
Equation taken from Verma and Quiroz-Ruiz, 2006
Gmax
xn x
s
or Gmin
x xn
s
where xn is the suspected outlier, x is the mean, and s is the standard deviation. Gmax is used when the
suspect observation is greater than

Normality Test
Jarque-Bera Normality Test (JB)
Equations derived from Jarque and Bera, 1987
n
k3
( xi x )3
i 1
ns 3
k 3 2 k 4 2
JB n
6 24
n
k4
(x x)
i 1
i
ns 4
4
3
For this test the Jarques-Bera value is determined based on the skewness and kurtos

Non-Parametric Test
Cramrs
Equations taken from Zar, 1984.
=
f11 f 22 f12 f 21
C1C2 R1 R2
Presence
f11
f21
C1
Presence
Absence
Absence
f12
f22
C2
R1
R2
where f are the cell values, C are the column totals,
and R are the row totals.
Example: The Cramr (pr

Parametric Test
Dixon Q Outlier Test
Equation taken from Verma and Quiroz-Ruiz, 2006
Q=
x n x n 1
x n x1
where xn is the suspected outlier, xn-1 is the next ranked observation, and x1 is the last
ranked observation.
Example: The Dixon outlier test was ori

Spatial Statistic
Standard Distance
Equations taken from Burt and Barber, 1996
n
n
( X i X )2
(Y
i
Y )2
+ i =1
n
n
where X i and Y i are coordinate s,
SD =
i =1
and X and Y define the mean center.
Example: We are interested in finding the standard dista

Non-Parametric Test
Watsons U2 Test
Equations taken from Zar, 1984
U2 =
( d )2
n1n2
d k2 k
N2
N
where
N = n1 + n2
Example: The tornado tracks for Starke
and Marshall counties, Indiana were
compiled to determine if the directions
(azimuths) of the sa

Parametric Test
Analysis of Variance
Equations taken from Zar, 1984
ni
k
TSS =
i =1
ni
BSS =
i =1
2
X ij
C
( X )
where C =
N
j =1
2
X ij
j =1
C
ni
ni
2
ij
BSS
F =
WSS
X is the observation in row i, column j.
N is the pooled sample size.
ni is the observ

Nonparametric Test
Wilcoxon Signed Ranks T Test
Equations taken from Zar, 1984
T
n(n 1)
T
2
T
and
n(n 1)
T
2
or
T Positive Ranked Differences
and
T Negative Ranked Differences
Example: The data table shows stream flow
measurements for the Rio Isluga i

Non-Parametric Test
z Test for Correlated Proportions
Equations taken loosely from Kanji, 1995
z=
pd
where
pd =
bc
n
and =
(b c) 2
(b + c)
n
n(n 1)
Environmental Regulation Poll
2011 and 2012
Example: At the start of 2012 a new regulation which was des

Parametric Test
Simple Linear Regression
Spatial Application: Regression Residual Mapping
Example: The table to the right has data
concerning the homicide rate per 100,000
(Homicide) and percent of the population living
in poverty (PCTPov) for the lower 4

Non-Parametric Test
Turning Point Test for Randomness
Equations taken loosely from Kanji, 1995
Burd Run Dissolved Oxygen Data
z=
tp x
s
where
t p = peaks + troughs
x = 2 3 (n 2)
(16n) 29
s=
90
Example: The turning point test for
randomness is used to dete

Non-Parametric Test
Runs Test for Serial Randomness of Nominal Data
Equations taken from Zar, 1984
Example: the runs test is used to determine for serial randomness: whether or not observations occur in a sequence
in time or over space. In geographic stud

Parametric Test
Two-Sample t-Test for Means
Equations taken from Zar, 1984
X1 X 2
t=
1 1
+
n1 n2
sp
where s p =
2
(n1 1) s12 + (n2 1) s 2
n1 + n2 2
Asthma Rate per 10,000
Example: We are concerned that asthma is increasing in Denver, Colorado.
Our data se

Parametric Test
Simple Linear Regression
Equations taken from Zar, 1984
Linear Regression Equation
y i = a + bx i
xy
b=
x
2
xy = X Y
where
i i
( X )( Y )
i
i
n
x =
2
and
X i2
( X )
2
i
n
a = Y bX
Coefficient of Determination (r2) and Significance Testin

Parametric Test
Pearsons Parametric Correlation
Equations taken from Zar, 1984
xy
x y
r
2
where
2
x2
X2
xy XY
t
r
sr
X
2
n
y2
Y2
Y
2
n
X Y
n
1 (r ) 2
n2
where s r
Example: Temperature and dissolved oxygen readings were taken at Burd Run over

Parametric Test
Paired Sample t Test
Equations taken from Zar, 1984
t
d
sed
where sed
sd
n
where d is the mean of the difference between pairs, and
s d is the standard deviation of the difference, and se is
d
the standard error of the mean difference.
Ex

Non-Parametric Test
Spearmans Ranked Correlation
Equations taken from Zar, 1984
n
rs = 1
6 d i2
i =1
3
n n
where d is the difference between X and Y ranks.
t = rs n 1
% Days of Sun and Suicide Rate
Example: We are interested in determining whether the
am