1. 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

2. 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
organize
surprise
Concepts
Description
Hypothesis
formalize
validate
Theory
Laws
Model

3. 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:

4. 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
validate
formalize

5. 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

6. 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:
The expression is equivalent to:

7. Summation Notation: Components
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

8. 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

9. Summation Notation: Examples
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:

10. Summation Notation: Rules
Rule I: Summing a constant n times yields a result of na:
Here we are simply using the summation notation to carry out a multiplication, e.g.:

11. Summation Notation: Rules
Rule II: Constants may be taken outside of the summation sign

12. 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

13. Summation Notation: Rules
Rule III: The order in which addition operations are carried out is unimportant +

14. 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

15. Summation Notation: Rules
Rule IV: Exponents are handled differently depending on whether they are applied to the observation term or the whole sum

16. 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

17. Summation Notation: Rules
Rule V: Products are handled much like exponents

18. 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

19. 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

20. Pi Notation
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

21. 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
There is also a convention that 0! = 1
Factorials are not defined for negative integers or nonintegers

22. 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: or

23. 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:

24. 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)

25. 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)) 

26. Mailboxes in Grad Workroom (315)
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Jingfeng Xiao
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