statistics

1. Chapter 1

2. - (singular sense) statistics is a science
that deals with techniques for
collecting, presenting, analyzing, and
drawing conclusions from data.
- (plural sense) numerical descriptions
by which we enhance the
understanding of data.

3. Philippines in Figures
Population
(August 2007)
88.57M
Projected Population (2010) 94.01M
Inflation Rate (October 2011) 5.2%
Balance of Trade (August 2011) $-804M
Exports (August 2011) $4.053B
Imports (August 2011) $4.926B
Unemployment (July 2011) 7.1%
Underemployment (July 2011) 19.1%
s Office
Simple Literacy (2000) 92.3%
Functional Literacy (2008) 86.4%
Average Family Income (2009) P206,000
GNP (Q4 2010) P2,760.1B
GDP (Q4 2010) P2,421.9B
Source: National Statistic

4. Statistics plays a vital role in almost all fields
 Economics
 Agriculture
 Market Research
 Engineering
 Education
 Computer Science
 Biology
 Chemistry
 Physics
 Veterinary Medicine
 Psychology

5.  Variables
 Any characteristic that can take on different
values for different individuals as far as the
group is concerned.
Example:
age, sex, height, weight, degree program
 Constant
 Any characteristic that is of the same for every
member of the group as far as the group is
concerned

6.  Quantitative (Numerical) Variable
 A variable that quantifies an element of a
population.
 Example:
height, weight, age, distance
 Qualitative (Categorical) Variables
 A variable that categorizes or describes an
element of a population.
 Example:
Gender, Degree Program, Civil Status

7.  Continuous Variable
 can take on an infinite number of values; A
variable that can assume an uncountable
number of values; can assume any value along
a line interval, including every possible value
between any two values;associated with
measurement
Ex: height, weight, length of time to finish an exam,
temperature

8.  Discrete Variable
 A variable that can assume a countable number
of values; that can have only a finite number of
values between any two values or can take on
only designated values; typically restricted to
whole countable numbers; associated with
counting
Ex: no. of students in a STAT 21 class, no. of
dormitories in a university, brand of milk

9. 1. Nominal Scale
 classifies different objects into categories based upon
some defined characteristics.
Data categories are mutually exclusive (an object can
belong to only one category).
Data categories have no logical order.
Example: Sex, Degree Program, Religion
2. Ordinal Scale
 Classifies different objects into order (rank)
Data categories are mutually exclusive.
Data categories have some logical order.
Data categories are scaled according to the amount
of the particular characteristic they possess.
Example: Job Position, Military Rank, Grading System

10. 3. Interval Scale (Equal Unit Scale)
Data categories are mutually exclusive.
Data categories have a logical order.
Data categories are scaled according to the amount of
the characteristic they possess.
Equal differences in the characteristic are represented
by equal differences in the numbers assigned to the
categories.
The point zero is just another point on the scale.
Example: Temperature, IQ, GPA, Achievement Test Score

11. 4. Ratio
Data categories are mutually exclusive.
Data categories have a logical order.
Data categories are scaled according to the amount of
the characteristic they possess.
Equal differences in the characteristic are represented
by equal differences in the numbers assigned to the
categories.
The point zero reflects an absence of the
characteristic.
Example: Height, Weight, Area, Length

12. 1. Classify the following variables asqualitative or
quantitative
a. Choice of diet (vegetarian, nonvegetarian)
b. Time spent in browsing the web last week
c. Ownership of laptop (yes, no)
d. Educational attainment (elem grad, HS grad,
college grad)
e. Distance from boarding house to university gym
f. Opinion on the verdict on CJ Corona (approve,
undecided, disapprove)

13. 2. Classify the following variables according to
scale of measurement
a. Choice of diet (vegetarian, nonvegetarian)
b. Time spent in browsing the web last week
c. Ownership of laptop (yes, no)
d. Educational attainment (elem grad, HS grad,
college, grad)
e. Educational attainment (defined as the number
of years in schooling)
f. Distance from boarding house to university gym
g. Attitude towards RH Bill (favor, neutral, oppose)

14. Population
 includes all members of some defined group
Example: All the residents of Baybay
All the VSU students for SY 2013-2014
Sample
 a subset of a population
Parameter: Descriptive measure of population
Ex: mean (μ), standard deviation (σ)
Statistic: Descriptive measure of sample
Ex: mean ( ), standard deviation (s)x

15. Sampling is the process of selecting a small number
of elements from a larger defined target group of
elements such that the information gathered from
the small group will allow judgments to be made
about the larger groups; -the process of selecting
a number of individuals for a study in such a way
that the individuals represent the larger group
from which they were selected.
Why sample?
it is not always feasible to gather information on all
members of a population.

16. Population
Sample
Sampling Inference
(generalization)

17. Probability sampling
all elements in the population has a known
chance of being included in the sample
Ex: Simple random sampling, systematic
sampling, cluster sampling, stratified
sampling
Non-probability sampling
elements in the sample are selected on the
basis of their availability or based on the
researcher’s judgment
Ex: convenience sampling, quota sampling

18. Simple Random Sampling
 Each element in the population has a known
and equal probability of selection.
Each possible sample of a given size (n) has a
known and equal probability of being the
sample actually selected.
This implies that every element is selected
independently of every other element.
Maybe done with replacement or without
replacement.

19. Systematic Sampling
 The sample is chosen by selecting a random
starting point and then picking every k element in
succession from the sampling frame.
 The sampling interval, k, is determined by
dividing the population size N by the sample size
n and rounding to the nearest integer.
 When the ordering of the elements is related to
the characteristic of interest, systematic sampling
increases the representativeness of the sample.
If the ordering of the elements produces a cyclical
pattern, systematic sampling may decrease the
representativeness of the sample.

20. Cluster Sampling
The target population is first divided into mutually
exclusive and collectively exhaustive subpopulations,
or clusters.
Then a random sample of clusters is selected, based
on a probability sampling technique such as SRS.
For each selected cluster, either all the elements are
included in the sample (one-stage) or a sample of
elements is drawn probabilistically (two-stage).
Elements within a cluster should be as heterogeneous
as possible, but clusters themselves should be as
homogeneous as possible. Ideally, each cluster should
be a small-scale representation of the population.

21. Stratified Sampling
The elements within a stratum should be as
homogeneous as possible, but the elements in
different strata should be as heterogeneous as
possible.
The stratification variables should also be closely
related to the characteristic of interest.
Finally, the variables should decrease the cost of the
stratification process by being easy to measure and
apply.
In proportionate stratified sampling, the size of the
sample drawn from each stratum is proportionate to
the relative size of that stratum in the total population.

22. Descriptive Statistics
Use of numerical information to summarize, simplify, and
present data.
Organize and summarize data for clear presentation and
easy interpretation
Computation of measures of location and variation
Construction of tables and graphs
Inferential statistics
 techniques that use sample data to make general
statements about a population making decisions and
drawing conclusions about populations
allows meaningful generalizations only if the subjects in
the sample are representative of the population
Estimation and hypotheses testing

23. Classify the following statements either descriptive or
inferential in nature.
1. As of July 2011 the unemployment rate of the Philippines
is at 7.1%.
2. Students with good mathematical background are
expected to perform well in STAT 21.
3. A survey was conducted using a random sample of 2000
respondents. One of the questions in the survey is about
the attitude of the respondents on the RH bill.
a. Of the 2000 respondents, 78% are opposed and 20% are in
favor. The rest are undecided.
b. Majority of the Filipinos are in favor of the RH bill.

24. 

N
i
Ni XXXX
1
21 ...
where: i is the index of the summation
Xi is the summand
1 is the lower limit of the summation
N is the upper limit of the summation.
-The Greek letter sigma, or Σ, is used to stand for summation
-The expression ΣX means "add all the scores for variable X." -
- Formally, if there are N observations on X represented by X1,
X2, ..., XN

25. )1(
)( 22


nn
XXn ii
Consider the scores on the first quiz of a small class.
6, 7, 7, 7, 8, 8, 8, 9, 10
Compute:
1)
2)
3)

n
i
iX
3
n
X
n
i
i1

26. Rules on summation
1. When there are two variables X and Y, ΣX
indicates the sum of the X’s, and ΣY refers to
the sum of the Y’s.
i X Y
1 2 5
2 6 6
3 4 -3
4 10 11
5 12 10
6 3 -9
ΣX=37 ΣY=20

27. 2. When two variables (X and Y) are multiplied
together, the product is represented by the
expression XY. The expression ΣXY means
"sum the products of X and Y."
i X Y XY
1
2
3
2
3
4
4
1
3
8
3
12
X = 9 Y = 8 XY = 23

28. 3. The squared value of a score is represented
by the symbol X2. The expression ΣX2 means
"sum of the squared scores."
i X X2
1
2
3
4
3
1
4
2
9
1
16
4
X = 10 X2 = 30

29. 4. When two variables (X and Y) are added
together, the sum is represented by the
expression X + Y. The expression Σ(X + Y)
means "sum the sums of X and Y." This is
equivalent to the expression ΣX + ΣY.
i X Y X + Y
1
2
3
2
3
4
4
1
3
6
4
7
X = 9 Y = 8 (X+Y) = 17

30. 5. When a constant value, C, is added to every
score, it is necessary to use parentheses to
represent the sum of these new scores, Σ(X +
C).
C=4
i X X + 4
1
2
3
1
4
6
5
8
10
(X+4) = 23

31. 6. If a constant value, C, is multiplied to every
score, the sum is represented by the
expression ΣCX.
C=2
I X 2X
1
2
3
1
3
5
2
6
10
X = 9 2X = 18

32. 7. If a constant value, C, is to be added n
times, the expression is and this is just equal
to NC.
Example: (C = 4, N = 10)
40)10(44
44444444444
10
1
10
1






i
i

33. 8. If a and b are constants, then
Example: (a = 5, b = 10)
     

N
i
i
N
i
N
i
iii YbXabYaX
11 1
i X Y 5X 10Y 5X + 10Y
1
2
3
2
3
4
4
1
3
10
15
20
40
10
30
50
25
50
X = 9 Y = 8 (5X+10Y) = 125
