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GEOG 090 – Quantitative Methods in Geography
• The Scientific Method
– Exploratory methods (descriptive statistics)
– Confirmatory methods (inferential statistics)
• Mathematical Notation
– Summation notation
– Pi notation
– Factorial notation
– Combinations
The Scientific Method
• Both physical scientists and social scientists (in our
context, physical and human geographers) often
make use of the scientific method in their attempts
to learn about the world
Concepts Description Hypothesis
Theory Laws Model
organize surprise
validateformalize
The Scientific Method
• The scientific method gives us a means by which
to approach the problems we wish to solve
• The core of this method is the forming and testing of
hypotheses
• A very loose definition of hypotheses is potential
answers to questions
• Geographers use quantitative methods in the
context of the scientific method in at least two distinct
fashions:
Two Sorts of Approaches
• Exploratory methods of analysis focus on
generating and suggesting hypotheses
• Confirmatory methods are applied in order to test
the utility and validity of hypotheses
Concepts Description Hypothesis
Theory Laws Model
organize surprise
validateformalize
Two Sorts of Statistics
• Descriptive statistics
• To describe and summarize the characteristics of
the sample
• Fall within the class of exploratory techniques
• Inferential statistics
• To infer something about the population from the
sample
• Lie within the class of confirmatory methods
Mathematical Notation
• The mathematical notation used most often in
this course is the summation notation
• The Greek letter is used as a shorthand way of
indicating that a sum is to be taken:
ni
iix
1
nxxx 21
The expression is equivalent to:
Summation Notation: Components
ni
iix
1indicates we are taking a sum
refers to where the sum of terms begins
refers to where the sum of terms ends
indicates what we are summing up
• A summation will often be written leaving out the upper and/or lower limits of the summation, assuming that all of the terms available are to be summed
Summation Notation: Simplification
ni
ii
n
iii xxx
1 1
Summation Notation: Examples
10
110987654321
ii xxxxxxxxxxx
10987654321
5
3iix
1, 2, 3, 4, 5, 6, 7, 8, 9, 10
Example I: All observations are included in the sum:
Example II: Only observations 3 through 5 are included in the sum:
12543543 xxx
Summation Notation: Rules
• Rule I: Summing a constant n times yields a result
of na:
n
i
naaaaa1
5
1
4i
• Here we are simply using the summation notation
to carry out a multiplication, e.g.:
205444444
Summation Notation: Rules
• Rule II: Constants may be taken outside of the summation sign
n
i
n
iii xaax
1 1
n
ini axaxaxax
121
n
iin xaxxxa
121 )(
• Example: Now let a = 3, and let the values of a set (n = 3) of x and y values be:
x1 = 4, x2 = 5, x3 = 6
y1 = 7, y2 = 8, y3 = 9
3
2
3
2
33)65(3i i
ii xaax
• Rule II: Constants may be taken outside of the summation sign
45)654(3ii xaax
Summation Notation: Rules
• Rule III: The order in which addition operations are carried out is unimportant
n
i
n
i
n
iiiii yxyx
1 1 1
)(
)( 1321 nn xxxxx
)( 1321 nn yyyyy
+
• Example: Now let a = 3, and let the values of a set (n = 3) of x and y values be:
x1 = 4, x2 = 5, x3 = 6
y1 = 7, y2 = 8, y3 = 9
2
1
2
1
2
1
24)87()54()(i i i
iiii yxyx
• Rule III: The order in which addition operations are carried out is unimportant
2
1
24)85()74()(i
ii yx
• Rule IV: Exponents are handled differently depending on whether they are applied to the observation term or the whole sum
kn
n
i
kkki xxxx
121
Summation Notation: Rules
kn
kn
ii xxxx )( 21
1
• Example: Now let the values of a set (n = 3) of x values be:
x1 = 4, x2 = 5, x3 = 6
77654 2222ix
• Rule IV: Exponents are handled differently depending on whether they are applied to the observation term or the whole sum
225)654()( 22 ix
• Rule V: Products are handled much like exponents
n
innii yxyxyxyx
12211 )(
n
i
n
i
n
iiiii yxyx
1 1 1
)()( 211 1
21 nn
n
i
n
iii yyyxxxyx
Summation Notation: Rules
• Example: Now let the values of a set (n = 3) of x and y values be:
x1 = 4, x2 = 5, x3 = 6
y1 = 7, y2 = 8, y3 = 9
122968574ii yx
• Rule V: Products are handled much like exponents
360)987()654(ii yx
• We frequently use tabular data (or data drawn from matrices), with which we can construct sums of both the rows and the columns (compound sums), using subscript i to denote the row index and the subscript j to denote the column index:
Summation Notation: Compound Sums
11x 12x 13x
22x 23x21x
2
1
3
1i jijx
)( 232221131211 xxxxxx
Rows
Columns
Pi Notation
n
n
ii xxxx
21
1
• Whereas the summation notation refers to the addition of terms, the product notation applies to the multiplication of terms
• It is denoted by the following capital Green letter
(pi), and is used in the same way as the summation notation
n
innii yxyxyxyx
12211 )())(()(
• There is also a convention that 0! = 1
• Factorials are not defined for negative integers or nonintegers
Factorial• The factorial of a positive integer, n, is equal to
the product of the first n integers
• Factorials can be denoted by an exclamation point
n
i
in1
!
5
1
12012345!5i
i
Combinations
• Combinations refer to the number of possible outcomes that particular probability experiments may have
• Specifically, the number of ways that r items may be chosen from a group of n items is denoted by:
)!(!
!
rnr
n
r
n
)!(!
!),(
rnr
nrnC
or
Combinations
• Example – Suppose the landscape can be characterized by five land cover types: forest (F), grassland (G), shrubland (S), agriculture (A), and water (W). A region has only two land cover types, the number of possible combinations is:
10)123()12(
12345
)!25(!2
!5)2,5(
C
)!(!
!),(
rnr
nrnC
Combinations
• Ten possible combinations:
F – G, F – S, F – A, F – W
G – S, G – A, G – W
S – A, S – W
A – W
F (forest), G (grassland), S (shrubland),
A (Agriculture), W (Water)
Assignment I
• Textbook, p39-40, #3 - #5
• #3 is about summation notation
• #4 is about factorial
• #5 is about combinations
• Due: January 26th (Thursday) (preferably at the
beginning of class, or put in my mailbox before
5pm – (Rm 315))