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))

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. . .

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Jingfeng Xiao

Mailboxes in Grad Workroom (315)

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