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GEOG 090 – Quantitative Methods in Geography

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- The Scientific Method
- Exploratory methods (descriptive statistics)
- Confirmatory methods (inferential statistics)

- Mathematical Notation
- Summation notation
- Pi notation
- Factorial notation
- Combinations

organize

surprise

validate

- 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

formalize

Theory

Laws

Model

- 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 quantitativemethods in the context of the scientific method in at least two distinct fashions:

organize

surprise

Concepts

Description

Hypothesis

formalize

validate

Theory

Laws

Model

- Exploratory methods of analysis focus on generating and suggesting hypotheses
- Confirmatory methods are applied in order to test the utility and validity of hypotheses

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

- 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:

The expression is equivalent to:

refers to where the

sum of terms ends

indicates what we

are summing up

indicates we are

taking a sum

refers to where the

sum of terms begins

Summation Notation: Simplification

- 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

Example I:All observations are included in the sum:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Example II:Only observations 3 through 5 are included in the sum:

- Rule I:Summing a constantn times yields a result of na:

- Here we are simply using the summation notation to carry out a multiplication, e.g.:

- Rule II:Constants may be taken outside of the summation sign

- Rule II:Constants may be taken outside of the summation sign

- 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

Summation Notation: Rules

- Rule III:The order in which addition operations are carried out is unimportant

+

- Rule III:The order in which addition operations are carried out is unimportant

- 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

Summation Notation: Rules

- Rule IV:Exponents are handled differently depending on whether they are applied to the observation term or the whole sum

Example: Now let the values of a set (n = 3) of x values be:

x1 = 4, x2 = 5, x3 = 6

- Rule IV:Exponents are handled differently depending on whether they are applied to the observation term or the whole sum

Summation Notation: Rules

- Rule V:Products are handled much like exponents

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

- Rule V: Products are handled much like exponents

Summation Notation: Compound Sums

- 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:

Columns

Rows

- 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

- 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

- There is also a convention that 0! = 1
- Factorials are not defined for negative integers or nonintegers

- 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 nitems is denoted by:

or

- 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:

- 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)

- 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))

Mailboxes in Grad Workroom (315)

. . .

Jingfeng Xiao

. . .

. . .

. . .

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