Unit 4 rm

General Classification of variables
1. Qualitative Variables (Categorical variables)
2. Quantitative Variables
(i) Discrete Variable
(ii) Continuous Variable
Prime considerations
(i) Observability
(ii) Count ability
(iii) Measurability

Variable
A Variable is a phenomenon that change from time to time,
place to place, and individual to individual etc.
Discrete variable
It can assume value from a limit set of numbers
Example :
(i) Number of flowers in a plant
(ii) Number of students in a class
(iii) Number of Non-defective screws in a box containing screws
(iv) Number of retail outlets
[Observable, countable]

Continuous variable [Observable,
Measurable]
Variables that can take any value within a range
Example
i) Leaf length of a particular plant
i) Weight of an apple
i) Distance travelled by a car
Experimental variable
The Variable whose effect is going to be known is called
experimental variable
Controlled variable
The effectiveness of an experimental variable is examined by
comparing with other variable, is called controlled variable

Structure of Variables in scientific
Investigations
i) Department Variable(s)
i) Independent Variable(s)
Dependent variable
A Variable is said to be dependent if it changes as a
result of change in the independent variable(s)
Independent Variable
Any variable that can be manipulated by the researcher is
known as independent variable
Example
(i) Leaf weight = f[Leaf length]
(ii) Profit of a company = f[sales]
(iii) Agricultural production = f[Rainfall, Soilfertility, Technology
Adoption]

Illustrations for variables
Name of Instrument Variable of study
1. Spectrophotometer (Biomedical
Engineering)
concentration of a given solution
2. Barometer (Electrical Engineering) Atmospheric Pressure
3. Electricity Meter / Energy Meter (Electrical
and Communication Engineering)
Energy dissipated from the circuit
resistance
4. Tensile tester (Mechanical Engineering) Tension testing
5. Hydrometer (Civil Engineering) Density of liquids
6. Packet Broker (Computer Engineering) Bandwidth analysis
NOTE: Clearly spell out the nature of the variable (Discrete/Continuous) with unit)

Nominal scale variables
i) These are unordered categorical variables
Nominal variables are often binary:
1-Presence, 0-Absence
Example: Sex of the respondent (Male, Female)
Hair color (Black, White, Grey)
Presence of absence of depression (1. Presence, 0-absence)
Ordinal Scale Variables
They are ranked data where there is an ordering of categories

STATISTICAL DISTRIBUTIONS
Discrete distributions: Based on Discrete Variable
1. Binomial Distribution
2. Poisson Distribution
3. Geometric Distribution
Continuous Distributions: Based on Continuous Variable
1. Normal Distribution
2. Exponential Distribution
3. Weibull Distribution
Points to be considered
1. Definition
2. Example
3. Applications

BINOMIAL DISTRIBUTION
James Bernoulli
Details Success Failure
Product Manufacturing Confirms to quality standards Not confirming to equality
standards
Plant experiment Seed germinated Seed not germinated
Medical Experiment Medicine cured Medicine Not cured

Experiment
A plant biologist wanted to test the quality of the seeds. He has conducted the
experiment is 100 pots. In each pot has kept 5 seeds. The results are
presented in the following table
Number of (x) Seeds Germinated 0 1 2 3 4 5 Total
Number of Pots (f) 15 25 30 20 6 4 100=N

Definition
The probability distribution of the random variable ‘x’, the number
of success is ‘n’ Binomial trails, is called binomial distribution and
is given by the formula
Where =0 otherwise
n⇒Number of trails
p⇒Probability of success
q⇒Probability of failure
x⇒Number of successes in ‘n’ trails
n,p ⇒ parameters of the distribution
p+q=1

Poisson Distribution
Prof. S.D. Poisson
Occurrence of Rare Event
1. Occurrence of flood in the last century is a country
2. Identification of printing mistakes occured in a dictionary
with large number of pages
3. Occurrence of deaths due to a rare disease

DATA
The following data is related to the occurrence of number of
floods in the last century in India
Number of floods occurred (x) 0 1 2 3 4≥ Total
Number of years (f) 85 9 3 2 1 N=100

Definition
The probability distribution of the random variable ‘x’, takes
the form
Is called prisson distribution, where

Geometric Distribution
A random variable ‘x’ is said to have geometric distribution if it
assumes non-negative values and it is given by:
Applications
1. Finding inefficiency of a telephone exchange system
during busy periods of time
2. To help managers to reduce the system trails occurring
prior to success to reduce costs.

Normal Distribution
Laplace
A continuous random variable ‘x’ is said to be normally
distributed and its probability density function is given by:

Properties of Normal Distribution

Weibull Distribution
Applications
1) Reliability and software
2) Probability that the drill bit with fail before 10 hours of
usage
3) Determination of Hazard Rate in order to set a service of
wear and strength of a component
A Continuous random variable ‘x’ has a Weibull distribution with parameter α and β
and its probability density function is given by

SAMPLING METHODS
Population
1) homogeneous population (SIMILAR UNITS)
2) heterogeneous population (DISSIMILAR UNITS)
SAMPLE
1) Finite sample:
2) Infinite sample

SAMPLING METHODS AND TYPE OF
POPULATION
Method of Sampling Type of Population
1. Simple Random sampling Homogenous population
2. Stratified sampling Heterogeneous population
3. Systematic sampling Homogenous population
4. Cluster sampling Mixed Type!

SIMPLE RANDOM SAMPLING (SRS)
SRSWOR : Simple Random sampling Without Replacement
SRSWR : Simple random Sampling With Replacement

STRATIFIED SAMPLING : How to Draw
Samples ?
Strata : DIVISIONS (PLURAL)
Stratum : DIVISION (SINGULAR)

SYSTEMATIC SAMPLING
Population SIZE =N
Sample SIZE=n
Systematic sample Number=R
In systematic sampling N= k . r (k is an integer)

CLUSTER SAMPLINGS
Clusters
(i) Clusters
(ii)Equal size
Unequal size
•Each cluster will consist of Homogenous units Mostly in practical
situations, we find clusters of unequal size
•We will select the required number of clusters RANDOMLY from the
CLUSTERS
•We survey all the units in the selected clusters

EXAMPLE
We observe that there are 7 clusters
The clusters consists of unequal number of units
Selected 3 clusters RANDOMLY out of 7 clusters
Let the selected clusters are c2
, c5
and c7
We collect the required from all the units in clusters 2,5, and 7
This method is called cluster sampling
NOTE : The units from the selected clusters are HOMOGENOUS / Nearly
Homogenous
c1

LARGE SAMPLE Sample size ‘n’ ≥ 30
SMALL SAMPLE Sample size ‘n’ <30
Tests based on size of samples
Student’s t-test (n<30)
Chi-square test (n>30)
F-test (n<30)
TESTS BASED ON
SAMPLES

BASIC TABLE FOR UNDERSTANDING
STATIC Based on Sample
PARAMETER Based on
Population

Applications of student’s t-distribution

To test for the difference between two sample means

To test the Independence of attributes

Simple Correlation / Bivariate
Correlation

Karl Pearson’s coefficient of
correlation

Computational Layout for computing
coefficient of correlation

Partial Correlation Coefficient

Multiple correlation coefficient

Regression Analysis
Classification of Regression Analysis
•Simple Regression Analysis
•Multiple Regression Analysis

FACTOR ANALYSIS
Meaning
It is a multivariate statistical technique to identify the
factors underlying the variables by means of clubbing
related variables in the same factor. Variables are
clubbed into different factors on the basis of their
interrelationship.
The number of data set should be at least five per variable
Number of variables = 15
Size of the sample = 75

Example
Objective of the study
A market researcher wants to determine the underlying
benefits consumers seek from the purchase of a car
Variables under study
X1: I like a car that has stylish Interior
X2: I like a car that looks great
X3: I prefer a car that gives high mileage
X4: I prefer a car with low maintenance
X5: I prefer a car that provides a good value for money
Rating Scale
0 Strongly Disagreeing
1 Disagreeing
2 Agreeing
3 Agreeing to a great extent
4 Strongly Agreeing

DATA STRUCTURE
Perform Factor Analysis and Identify the Factors
Example
Respondent X1 X2 X3 X4 X5
1 0 2 3 1 0
2 2 0 0 4 1
. . . . . .
. . . . . .
75 0 2 4 3 0
Cost Factor X3, X4, X5
Style Factor X1, X2

Discriminant Analysis
Meaning
Discriminant Analysis is a multivariate statistical technique
used for classifying a set of observations into predefined
groups. The purpose is to determine the predicator
variables on the basis of groups determined. The form of
the discriminate function is given by
Where c => a constant
bi => Discreminant Coefficient
Xi => Predictor Variables

Example
To study the successfulness or not for a new improved digital
camera
Characteristics under the study
X1: Durability of the camera
X2: performance of the camera
X3: Style
Category : Buyer of the Digital camera
Non-Buyer of the Digital camera
Rating Scale
0 Poor
1 Fair
2 Good
3 Better
4 Excellent

k
Perform discriminant Analysis and find linear combination
of variables which discriminates between the two groups
Example

Cluster Analysis
Meaning
It is a multivariate statistical technique for grouping cases
of data based on SIMILARITY of responses to several
variables / objects. The purpose of cluster analysis is to place
subjects / objects into groups, or clusters, suggested by the
data, such that objects in a cluster are homogeneous in some
sense, and objects in different clusters are dissimilar to a
great extent

Example
A canteen manager wishes to study the clusters of students
preference of 5 brands of carbonated Soft Drinks
Brands of Carbonated Soft Drinks
X1
: Coke
X2
: Pepsi
X3
: Thumps up
X4
: Sprite
X5
: Dew
Rating Scale
0: Very rarely
1: Normally
3: Often
4: Quite often

k
Perform cluster Analysis and find the Homogeneous clusters of
the Carbonated Soft Drinks
Example:
Cluster 1: Coke, Pepsi, Thumps Up
Cluster 2: Sprite, Dew

ANALYSIS OF VARIANCE / DESIGNS
OF EXPERIMENTS
Meaning
To study the variation is the data set in a systematic manner
using F-test

DESIGN
Objective : To study the variation is relief time among the
patients suffering from a particular ailment
The above data structure is called Completely Randomized
Design

RANDOMIZED BLOCK DESIGN
Objective
To study the variation in relief time among the patients
suffering from a particular ailment. Variation in Relief Time (2
Factors)
Effect of Drug (Between Drugs)
Effect of Age (Between Age group)
Data Structure
Drugs Administered : A,B,C,D and E
Age group of Patients : < 14, 14 - 35, 35 - 60, 60≥

LATIN SQUARE DESIGN (LSD)
Meaning
i) A LSD is an arrangement of n2
observations (objects)
i) An Observation (object) can occur only once in the
row/column

LATIN SQUARE DESIGN
Objective :
To study the variation in relief time among the patients suffering
from a particular ailment
Variation in Relief Time (3 Factors)
(1) Effect of drug (Between Drugs) [D1
,D2
,D3
,D4
]
(2) Effect of Age (Between Age Groups)
[ <14, 14-35, 35-60, 60≥ ]
(3) Effect of Administering time [Between Administering
Time]
A: 4 AM
B: 8 AM
C: 2 PM
D: 6 PM

SAMPLE PROBLEM-RANDOMIZED BLOCK
DESIGN
Based on the data given below carry out the analysis and
comment on your results
[Given F0.05
=4.76 for (3.6) d.f. and F0.05
=5.14 for (2,6) d.f.]

SAMPLE PROBLEM – Latin Square Design
An agricultural experiment was conducted and the
results are given below. The design adopted is a LSD.
Analyze the data and comment on your results.
[Given F0.05
=4.76 for (3.6) d.f]

Time Series Analysis
MEANING : A Phenomenon relating to time is called a time series
set up
COMPONENTS OF TIME SERIES
1. Trend
2. Seasonal variation
3. Cyclical variation
4. Irregular Variation
Time series Model
Yt
=T+S+C+I (Additive Model)
or
Yt
=T.S.C.I (Multiplicative Model)

IRREGULAR VARIATION
Data relating to Irregular pattern (figure only)
∙ BOOMS
* DEPRESSIONS
− Static

Unit 4 rm

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Unit 4 rm