1.-Lecture-Notes-in-Statistics-POWERPOINT.pptx

Statistics -
science that deals with the collection,
organization, presentation, analysis and
interpretation of data to come up with
meaningful information.

Statistics -
•-It refers to numerical observations of
almost any kind. Statistical data took the
forms of figures on important information’s
on tax return, populations, birth, deaths,
trade and others that are considered
important information to the state

Two Major Areas of Statistics
•Descriptive Statistics comprises those
methods concerned with collecting and
describing a set of data as to yield
meaningful information.

Two Major Areas of Statistics
•Inferential Statistics comprises those methods
concerned with analysis of subset of data
leading to predictions or inferences about the
entire set of data.

Example: (Distinction between Descriptive and
Inferential Statistics)
Academic records of the graduating classes during the
past 5 years in a certain university show that 72% of the
entering freshman eventually graduated. The numerical value,
72% is a descriptive statistics. If you are a member of the
present freshman class and conclude from this study that your
chances of graduating are better than 70%, you have made a
statistical inference that is subject to uncertainty

Population
• - consists of the total collection of observations.
It is a groups or aggregates of people, animals,
objects, materials, happenings or things of any
form.
•consist of the totality of observations with
which we are concerned.

Sample-
• is a subject of measurement taken from the
population. It is a few members of the
population to represent their
characteristics.
•a subset of the population

Example:
•A candidate for political office hires a polling
firms to asses his chances in the upcoming
election. The population consists of all
voters in the candidate’s district. The sample
consists of all voters in the candidate’s
district interviewed by the polling firm.

• Parameter- any numerical value describing a
characteristic of a population.
- use to represent a certain population
• Statistics - any numerical value describing a
characteristic of a sample.
- use to give information about unknown
values in the corresponding population

• Variable- refers to the characteristics or property
whereby members of the group or set vary from
another.
-A characteristic or information of interest that is
observable or measurable from every individual or
object under consideration.
Example: Gender, Age, Occupation

Types of Variables
Qualitative or Categorical
• Measures a quality or characteristics
• variables have values that can only be placed into
categories, such as “yes” and “no.”
Quantitative or Numerical
•Measures a numerical quantity or amount
•Answers question “ how much” or “how many”

Data
Categorical Numerical
Discrete Continuous

Examples:
 MaritalStatus
 PoliticalParty
 EyeColor
(Definedcategories) Examples:
 NumberofChildren
 Defectsperhour
(Counteditems)
Examples:
 Weight
 Voltage
(Measuredcharacteristics)

Types of Quantitative variable
•Discrete
-Assumes only a finite or countable
number of values
•Continuous
-Assumes infinitely many values
corresponding to the points on a line interval

Dependent variable is sometimes
called criterion variables while the
Independent variable is sometimes
called predictor variables or a
variable that can be controlled or
manipulated.

Example
Suppose the investigator is interested in the relationships
between two variables: the effect of information about the
gender of a job applicant on hiring decisions made by
personnel managers. The hiring decision is the dependent
variable because it is thought to depend on the information
about the gender of the applicant, while the gender of the
applicant is independent because it is assumed to influence
the dependent variable and does not “depend” on the other
variable.

Example
“Study on the effect of Psychological
stress on blood pressure” the
Independent variable is the amount
Psychological stress an individual feeling
and the dependent variable is the
individuals blood pressure.

Measurement Scales
In order to determine what statistical tool
used in the analysis of gathered data, it is
important to know the different type of
measuring scale use in the data gathering or
collection. A measuring scale can have one or
more of the following mathematical attributes:
magnitude, an equal interval between adjacent
units, and an absolute zero point.

Measurement Scales
• Nominal Scale used with variables that are qualitative
in nature. The data collected are simply labels,
categories or nameless without any implicit or explicit
ordering of the categories or explicit ordering of the
labels. The observations or subjects belong to the
same category. It is the lowest level of measurement.
Nominal scale does not posses any of the attributes of
magnitude, equal interval or absolute zero point

Measurement Scales
Sex/ Gender
The complexion of students
Hair Color of students

Measurement Scales
•Ordinal scale has a relative low level of property
of magnitude, but it does not have the property
of equal intervals between the adjacent units.
This is concerned with the ranking or order of
the objects measured. The level of measurement
is higher than nominal.

Measurement Scales
Example
Winners of a contest
Faculty Rank
Military Rank

Measurement Scales
Interval scale has its property of magnitude
and equal interval between two adjacent
units, but it does have an absolute zero
point. The data collected can be ordered or
rank. The unit measurement is constant. The
level of measurement is higher than the
ordinal.

Measurement Scales
•Ratio scale is the highest level of measurement
scale. It has all the properties of an interval
scale, that is, it has magnitude and equal
intervals plus the absolute zero point. There is a
constant size interval between each successive
unit on the measurement scale. Furthermore,
there is a physical significance to this zero.

Measurement Scales
• Determine the following measurement scales
1. Temperature in Celsius scale
2. IQ
3. The reaction time to a particular drug
4. The number of visits to a Doctor,
5. The weight loss of on diet individual,
6. The average score of CAT score of CAS students

Data Collection and Presentation
Any statistical investigations it must necessarily be
based on accurate data. To ensure the accuracy of data the
researcher or investigator must know the right source and
methods of collecting them.
There are two types of data the primary and
secondary data. A primary data refers to the information
gathered directly from an original source or based on the
direct or first hand experience while secondary data refers
to the information, which are previously gathered by
previous individual.

Methods of Collecting Data
• Direct of Interview Method- it is person-to-person
exchange between the interviewee and the
interviewer. It provides consistent and more precise
information.
Disadvantages- Time consuming, Expensive and has
limited coverage.

Indirect or Questionnaire Method-
written responses are given to prepared
questions. Questionnaire is a list of
question to illicit answers to the
problems of the study.

• Registration Method- a method of gathering
information or data is enforced by the law.
• Observation Method- the investigator observes the
behavior of the subject and their outcomes. The
subject usually cannot talk or write and it requires a
proper recording of the behavior at the appropriate
time and situation.

Experiment method- it is a method of gathering
data when the objective is to determine the cause
and effect relationship of a phenomena under
controlled conditions.

Organization and Presentation of Data
After the collection of data, one of the most important
parts of research work is the analysis and interpretation of the
gathered data. Accuracy will greatly enhance in an orderly and
concise method of presentation of the data gathered. This
benefits not only the researcher and the reader, but most is
the statistician who will make the analysis of the data. Data
analysis are easily determined if the data are properly
determined.

Methods of Data Presentation
•Textual
•Tabular
•Graphical

Textual
Presents data in narrative form to describe the data collected
Example:
“Business confidence in Metro Manila continued to deteriorate due to
political uncertainties and the slowdown in the economies of the country’s
major trading partners. The survey showed that out of 177 respondents, only
23.3 percent said that they were willing to expands this year” (Philippine
Daily Inquirer, September 24, 2001)

• Tabular
Presents data in condensed form by arranging them
systematically in rows and columns
A statistical table that can be constructed to present a data
collected is the Frequency Distribution Table

• Frequency Distribution Table
tabular arrangement of data by grouping the
values into mutually exclusive classes and showing the
number of observations falling in each class.

• Example
Table 1. Frequency Distribution Table for Blood Types of 25 Patients
Afflicted with certain Disease
Blood Type Number of Patients
O 7
A 6
B 8
AB 5

• Example
• Table 2. Frequency Distribution Table for Number of Cars Owned by
A Sample of 30 Families in a Certain Residential Area
Number of Cars Owned Number of Families
0 15
1 10
2 3
3 2

• Example
Table 3. Frequency Distribution Table of Distance Traveled (in
Km) from Warehouse by 25 Trucks in a certain Company
Classes Frequency
50-54 3
55-59 7
60-64 9
65-69 5

• Grouped Frequency distribution of Interval Data
• Step 1. Compute the range R. The range is equal to the difference of the highest
score and the lowest score.
R = Highest observed value-lowest observed value
• Step 2 Decide on an adequate number of classes of intervals, k. There should be
not too many and not too few classes. Usually between 5 to 20 classes will do. If
the number of classes can not be decided upon, the formula K= √N can be used,
where n is the number of observations in the data set
or k =1 + 3.3 Log N

• Step 3. Compute the ratio of Rand K.
C=R/K
• Step 4. Determine the class width or size c by rounding top the
nearest value whose precisions is the same as those of the raw data.
• Step 5 Organize the class interval. See to it that the lowest interval
begins with a number that is a multiple of the interval size.

Grouped Frequency distribution of Interval Data
• Step 6. Tally each score to the category of class interval.
• Step 7. Count the tally column and summarize it under
column f.

• Step 8.
Build additional columns to obtain other information about the
distributional characteristics of the data. These are :
Class Boundaries- Numbers that are halfway between the upper
limit of a class and the lower limit of the next class.
Class Mark or Midpoint- average of the lower and upper limits of
a given class interval

• Step 8.
• Relative Frequency- obtained by dividing the frequency of that
class by the total number of observations.
• Relative Percentage- obtained by multiplying the relative
frequency by 100%
• Cumulative frequency, CF- the accumulated frequency of a class.
Less than CF or Grater than CF

•Example: Suppose we are given the weights
of 40 graduate school male students in a
certain university. ( the numbers below are
called raw data)

120 133 180 138
140 150 170 153
161 149 124 168
148 139 161 142
130 143 137 147
165 138 147 167
156 148 128 118
146 150 149 129
142 158 152 150
175 151 142 157

•Step 1. Range = highest observation - lowest
observation
R = 118-120 = 62
•Step 2. K = √40 = 6.32 = 6
•Step 3 C = 62/6 = 10.33 = 10
•Step 4, 5 and 6

Class limits Class Boundaries Tally f X RF% <CF >CF
118 - 127 117.5 - 127.5 122.5
128 - 137 127.5 - 137.5 132.5
138 - 147 137.5 - 147.5 142.5
148 - 157 147.5 - 157.5 152.5
158 - 167 157.5 - 167.5 162.5
168 - 177 167.5 - 177.5 172.5
178 - 187 177.5 - 187.5 182.5

•Graphical
•Presents the data in pictorial form
•Designed and constructed to attract and hold
attention of readers
•Common Graphs constructed are line, bar and
pie charts.

•Types of Graph
Line Graph
•Useful for showing trends over a period of time

Figure1 Beef and Chicken Consumption of Adults from 1994 to 2000
Beef and Chicken Consumption
0
10
20
30
40
50
60
70
80
90
1 2 3 4 5 6 7
Ye ar
Annual
pounds
per
person
Beef
Chicken

•Bar Graph
•Consist of series of rectangular bars where
the length of the bar represents the
magnitude to be demonstrated.

Figure 2. Students Classifications in a Statistics Class
Classification of Students
0
100
200
300
400
500
600
Fresman Sophomore Junior Senior
Classification
Frequency

Pie graph-
provides easy measurement and fast
presentation of nominal data divided
into a few categories.

• Figure 3. Political Views of Household Heads in a Certain Municipality
Political View s
Conserva
tive, 565
Moderate
, 450
Liberal,
425
Radical,
380 Conservative
Moderate
Liberal
Radical

•Pictogram
Pictures or symbols are used to represent certain
quantity or volume
. Preferred Canteen of a Sample of 100 College Students
Canteen X 
Canteen Y 
Canteen Z 

•Histogram
•A bar graph of a frequency distribution table.
•Class Boundaries are represented by the width
of the bars and the frequencies that fall within
the classes are represented by the height of the
bars.

44.5-54.5 54.5-64.5 64.5-74.5 74.5-84.5 84.5-94.5
Frequency
40
30
20
10

Sampling Techniques
•Probability Sampling
•Non Probability Sampling

Sampling Techniques (Probability Sampling)
•Random Sampling
Lottery Sampling
Use of table of random numbers
• Systematic Sampling-methods will be develop
which we call systematic and draw the sample
similar to random sampling by drawing item say
multiple of from the sample.

•Types of Systematic sampling
•Stratified Sampling- population must be first
divided into homogeneity to avoid the
possibility of drawing samples whose member
come only from one stratum. It is often called
stratified proportional sampling.

• example
Strata Population
Distribution of
population
% share n*(%share)
Number of
Sample units
High income 500 500/10000 0.05 1000*.05 50
Middle Income 2500 2500/10000 0.25 1000*.25 250
Low Income 7000 7000/10000 0.7 1000*.7 700
N 10000 1000
n 1000

• Cluster Sampling sometimes referred to as an area
sampling because it is frequently applied on geographical
basis. On this basis, districts or blocks of a municipality or
city is selected. These districts or blocks constitutes the
cluster.
For instance, if a community has a lower, middle and
upper income residents living side by side, we may use this
community as a source of a sample to study the different
socio-economic status. By concentrating on this particular
area, we can save time, effort and money.
This is reverse of the stratified proportional sampling.

•Multi-Stage sampling – this technique uses
several stages or phases in getting the
sample.

Sampling Techniques (Non-Probability Sampling)
not all members are given equal chances to be chosen.
Types of Non Random Sampling
Purposive Sampling- this is based on the criteria laid down
by the researcher. People who satisfy the criteria are
interviewed.
For instance- a researcher may want to find out the
central bank circular, instead of interviewing the executives
of all banks, he purposely can choose to interview the key
executives of only five biggest banks in the country.

•Quota Sampling- this is relatively quick and
inexpensive method to operate. Each interviewer is
given definite instruction about the section of the
public he is to question, but the final choice of the
actual person is left to his own convenience or
preference.
• For instance –a researcher want to determine
the favorite teams of the PBA. His quota is say 100
basketball fans. The researcher may reached his
quota by going to the ARANETA coliseum to enjoy
watching the game at the same time interview
thrilled viewers during the game.

Convenience Sampling-
a researcher may get his sample at his own
convenience

Measures of Central Tendency
•Frequency distributions provide useful behavior
of the data. However, they do not provide with
measures, which could quantitatively
summarize the characteristics of the
population. Hence, we further need to come up
with other measurable characteristics of the
data to describe the population.

•Quantities that describe statistical data are
numerical descriptive measures. They are quantities
computed from a given set of observations and are
used to derive information from data collected by
the researcher. There are several descriptive
measures. The most commonly used are the
measures of location, dispersion, skewness, and
kurtosis.

•A measure of central tendency describes the
"center" of a given set of data. It is a single value
about which the observations tend to cluster.
The common measures of central tendency are
the arithmetic mean or simply mean, median
and mode.

•Arithmetic Mean (or simply, mean)
•Ungroup
• Group
n
fx
X



n
x
X




• Median the middle value of an array
• Lme- Lower limit of the Median Class
• N- number of samples
• Cfb- Cumulative frequency below the median Class
• Fme- frequency within the median class
• Cme- class interval size
me
me
b
me c
f
cf
N
L














 2
X
~

Mode – is the observation which occurs most
frequently in the data set
mo
mo c
d
d
d
L 











2
1
1
X

Mode –
Lmo- Lower limit of the modal class
•d1- difference between the frequency of modal
class and the frequency of the next lower class
•d2-ifference between the frequency of modal
class and the frequency of the next lower class
•Cmo- Class interval size

Measures of Variability
Measure of central tendency means to describe
the given set of data. These measures indicated the
point where the items are centrally located. However,
they do not show whether the terms in the distribution
are far from or close to each other.

• For instance take five sets of observations:
Set A: 15, 15, 17, 18, 20
Set B: 15, 16, 16, 18, 20
Set C: 14, 15, 16, 19, 21
Set D: 11, 13, 18, 18, 25
Set E: 14, 15, 18, 19, 19

• For instance take five sets of observations:
Set A: 15, 15, 17, 18, 20
Set B: 15, 16, 16, 18, 20
Set C: 14, 15, 16, 19, 21
Set D: 11, 13, 18, 18, 25
Set E: 14, 15, 18, 19, 19
The five sets of observations have the same mean of 17, but do not
totally describe each of the five sets.

• The measures of central position are of little value
unless degree of spread or variability which occurs
about them are given. Hence, the description of the
set of data becomes more meaningful if the degree of
clustering a central point is measured. Information on
how apart the observations are from each other in
every set will be very useful. Set D is the most
variable. In set A and B we cannot right away see the
spread of the values of the items.

All these can be answered
through the use of the
measures of spread or
variability.

•Range
• Is the simplest of the measures of spread or variability?
It is the difference between the highest and the lowest score.
From the above illustration Set A and B the range is 5, while
C is 7, D is 14 and E is 5.
• Although range is the easiest to compute and easiest to
understand, it is also least satisfactory since its values is
dependent only upon the two extremes and does not
consider the scatter of the values in between these two
extremes.

• Range
For instance, consider the following test scores of two students:
Liza 17 18 7 15 14 13
Ana 18 10 17 11 18 10
If we compare the test scores we see that liza’s scores have higher range
than Ana. These ranges tell us that liza’s score are apparently more scattered than
Ana. If we look closely at liza’s score, except for 7, her scores are more consistent or
more clustered than Ana. Can we say that liza’s score are more scattered or variable
than Ana?
The range is not considered a stable measure of variability because its
values can fluctuate greatly with the change in just a single score-either the highest
or the lowest.

• Standard Deviation
• Is a special form of average deviations from the mean,
it is therefore also affected by all the individual values
of the items in the distribution.
-For instance, if the standard deviation of IQ
scores of a class of 50 students is numerically big, we
can say that there is heterogeneity in their intelligence,
while it is small, we can say there is homogeneity in
their intelligence.

•Standard Deviation
•Ungrouped data
•Grouped data
 
1
2



N
X
X
 
1
2



N
X
X
f

• Example:
A simple interest designed to investigate the effect of drug on a
cognitive task such as coding. An experimental group of subjects, who
receive a drug, and a control group, who do not receive a drug are
used. Each group contains 10 subjects. Let as assume the scores on the
coding task for the groups are as follows:
• Experimental Group(EG): 5 7 17 31 45 47 68 85 96 99
• Control Group(CG): 29 36 37 42 49 58 62 63 69 70
• Mean EG = 50.0 Mean CG = 51.5

•The investigator might be led to conclude from
inspecting the means that the drug had a little or
no effect on the performance of the subject.
sd EG = 35.63
sd CG = 14.86

• The experimental group being much more variable in the
performance than the control group. Quite clearly the
treatment appears to be exerting a substantial influence on
the variation in performance, although it’s on the level of
performance is negligible.
• Variability of different data sets can be compared.
However the range, Mean Average deviation and Standard
deviation are not appropriate measures to use especially
when they have different means or different units. This led
to another measure of dispersion.

Coefficient of variation - is the ratio of the standard
deviation to its mean, expressed in percent. It is relative
measure of dispersion useful when comparing dispersion of
two or more data sets with different units.
%
100
x
CV
Mean
deviation
Standard


• Example: A random sample of 7 students were thought in
Mathematics in standard classroom situation. Another sample of 9
students taught themselves using programmed text and consulted
the teacher only when they had questions. At the end of the
semester, both students were given standardized exams.
Standard Experimental
Mean 91.9 142.3
Standard Deviation 20.9 20.8
Coefficient of Variation 22.7% 14.6%
The experimental group is less dispersed than the standard
group because of smaller Coefficient of Variation.

Standard Scores
A value taken from the distribution
whose mean X and whose standard
deviation sd has a standardized value
defined by
sd
X
X
Z



Standard Scores
• A standard score have a mean of 0 and a standard deviation
of 1 or a Z score measures the distance from X from the mean
per 1 standard deviation.
• Standard scores are frequently use to obtained
comparability of observations obtained by different procedures.
• Illustrations: Consider examinations in English and Mathematics
applied to the same group of individuals, and assume the means
and standard deviation to be as follows:

Standard Scores
In his performance in English is one standard
deviation unit below average, while in Mathematics his
performance 0.50 standard deviation above the
average. Quite clearly, this individual did much more
poorly in English than in Mathematics relative to the
performance of the group of individuals taking the
examinations.

0
.
1
8
65
57



English
5
.
0
12
52
58


s
Mathematic

Standard Scores
• A score of 65 in English is equivalent to a score of 52 in mathematics. A
score of one standard deviation above the mean on English examination is
65 + 8 or 73 while in Mathematics examination is 52 + 12 or 64. If an
individual makes a score of 57 in English examination and 58 in
Mathematics examination we may compare their relative performance on
the two subjects.
Examination Mean sd
English 65 8
Mathematics 52 12

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