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Research Methodology
Dr. Unnikrishnan P.C.
Professor, EEE
Dr. Unnikrishnan P.C.



BTech. : EEE, NSS College of Engineering,
1981-85.
MTech: Control & Instrumentation, IIT
Bombay,1990-92.
PhD. : EEE, Karpagam University,
Coimbatore, 2010-2016.
Dr. Unnikrishnan P.C.



1986-1996 : Assistant Professor and
Associate Professor, Rajasthan Technical
University, Kota, India
1996-2016 : Assistant Professor, Academic
Coordinator, Registrar, Head of Section and
Head of the Department at Colleges of
Technology, Ministry of Manpower, Muscat,
Sultanate of Oman.
2016 : Professor, EEE, RSET
Module III
 Descriptive and Inferential Statistics
Research Process
Step VI. Analyze Data(Test Hypothesis if any)
Data analysis methods
• Qualitative data analysis- Analysis of content
of interview to identify the main themes that
emerge from responses given by
respondents. This is done in 4 steps
– Identify main themes.
– Assign codes to the main themes.
– Classify responses into the main themes.
– Integrate themes and responses into the text of
your report.
6
Family life of daily workers
HYD
HYD1
HYD2
HDD
1.
1
1
0
1
2.
0
0
0
0
3.
1
0
1
0
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Quantitative Data Analysis
• Used in well designed and well administered surveys using
properly constructed and worded questionnaire.
• Data can be analyzed manually or with a computer (SPSS
for Windows)
• Before analysis, it becomes necessary to make certain
transformations of data
–
–
–
–
Identifying and coding missing values
Computing totals and new variables
Reversing scale items
Re-coding and categorization
8
Missing Value Imputation Techniques
• Hot deck imputation (Values taken from
matching respondents)
• Predicted mean imputation (Values predicted
using certain statistical procedures)
• Last value carried forward (Based on
previously observed values)
• Group means (Values determined by
calculating variable’s group mean)
9
Computing Totals and New Variables
ExampleTotal income I = I1+I2+I3+I4
Expenditure vide Chapter VI= E
Taxable Income = TI= I-E
Tax=T= (TI-200000)*0.1 if TI<500001
Tax=T=(30000+(TI-500000)*0.2) if TI<1000001
Tax=T=(130000+(TI-10000000)*0.3) if TI>1000000
10
Reversing scale items
Responses of negative questions are to be reverse
scored before analysis- see example
Strongly
agree
She makes tasty food
*
I get very angry when
she snore in the night
*
Agree
Neither Disagree
agree or
disagree
Strongly
disagree
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Re-Coding Variables
• Turning a continuous variable into categorical
variable or collapsing some observations into
ranges
Example – age, income etc.
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Data Analysis Contd…….
Generally, analysis of data involves one or more
of the following tasks
1. Computation of descriptive statistics
2. Regression analysis
3. Correlation analysis
4. Testing hypotheses
5. Factor analysis
6. Discriminant analysis
7. Conjoint analysis
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Data analysis
• Descriptive statistics- It allow the researcher to
describe the data and examine relationship
between the variables. Used to summarize a
study sample prior to analyzing a study’s primary
hypotheses (frequency tables, histograms,
measures of central tendency, dispersion,
correlation, skewness)
• Inferential statistics- It allow the researcher to
examine causal relationships (t-test, ANOVA, chisquare and regression)
14
Measures of central tendency
Amongst the measures of central tendency,
the three most important ones are the
arithmetic average or mean, median and
mode.
Geometric mean and harmonic mean are
also sometimes used.
Mean
Mean, also known as arithmetic average, is the
most common measure of central tendency and may
be defined as the value which we get by dividing the
total of the values of various given items in a series
by the total number of items.
Median
Median is the value of the middle item of series
when it is arranged in ascending or descending
order of magnitude. It divides the series into
two halves; in one half all items are less than
median, whereas in the other half all items have
values higher than median.
Mode
Mode is the most commonly or frequently occurring
value in a series. The mode in a distribution is that item
around which there is maximum concentration. In
general, mode is the size of the item which has the
maximum frequency, but at items such an item may not
be mode on account of the effect of the frequencies of
the neighboring items. Like median, mode is a
positional average and is not affected by the values
of extreme items. it is, therefore, useful in all situations
where we want to eliminate the effect of extreme
variations.
Geometric mean
Geometric mean is also useful under certain
conditions. It is defined as the nth root of the
product of the values of n times in a given
series. Symbolically, we can put it thus:
MEASURES OF DISPERSION
An averages can represent a series only as best as a
single figure can, but it certainly cannot reveal the
entire story of any phenomenon under study. Specially
it fails to give any idea about the scatter of the values of
items of a variable in the series around the true value
of average. In order to measure this scatter, statistical
devices called measures of dispersion are calculated.
Important measures of dispersion are (a) range, (b)
mean deviation, and (c) standard deviation.
Range
Range is the simplest possible measure of
dispersion and is defined as the difference
between the values of the extreme items of a
series.
Mean deviation
Mean deviation is the average of
difference of the values of items from
some average of the series. Such a
difference is technically described as
deviation. In calculating mean deviation we
ignore the minus sign of deviations while
taking their total for obtaining the mean
deviation.
Mean deviation
Standard Deviation
Standard deviation is most widely used measure of
dispersion of a series and is commonly denoted by the
symbol ‘’ Standard deviation is defined as the
square-root of the average of squares of deviations.
Square of standard deviation is known as variance.
Example
The owner of a restaurant is interested in how much people
spend at the restaurant. He examines 10 randomly selected
receipts for parties of four and writes down the following data (In
Rupees)
440, 500, 380, 960, 420, 470, 400, 390, 460, 500
Calculate Mean, Median, Mode, Range, Variance and Sandard
Deviation
SAMPLING DISTRIBUTIONS
Some important sampling distributions,
which are commonly used
(1) sampling distribution of mean
(2) sampling distribution of proportion
(3) student’s ‘t’ distribution
(4) F distribution
(5) Chi-square distribution.
(1)Sampling distribution of mean
Sampling distribution of mean refers to the
probability distribution of all the possible
means of random samples of a given size
that we take from a population. If samples
are taken from a normal population,
, the
sampling distribution of mean would also be
normal with mean
and standard
deviation = s p n , where m is the mean of the
population, s p is the standard deviation of the
population and n means the number of items in
a sample.
Sampling distribution of mean
• The mean of the sampling distribution of the mean
is the mean of the population from which the
scores were sampled. Therefore, if a population has
a mean μ, then the mean of the sampling
distribution of the mean is also μ. The symbol μM is
used to refer to the mean of the sampling
distribution of the mean. Therefore, the formula for
the mean of the sampling distribution of the mean
can be written as:
μM = μ
VARIANCE
The variance of the sampling distribution of the
mean is computed as follows:
2
𝜎
𝜎2 𝑀 =
𝑁
i.e. the variance of the sampling distribution
of the mean is the population variance
divided by N, the sample size.Thus, the larger
the sample size, the smaller the variance of the
sampling distribution of the mean.
CENTRAL LIMIT THEOREM
The central limit theorem states that:
Given a population with a finite mean μ and a finite
non-zero variance σ2, the sampling distribution of
the mean approaches a normal distribution with a
mean of μ and a variance of σ2/N as N, the sample
size, increases.
What is remarkable is that regardless of the shape of
the parent population, the sampling distribution of
the mean approaches a normal distribution as N
increases.
Simulation of a sampling distribution. The parent population is
uniform.
See that the
distribution for N = 2 is
far from a normal
distribution.
For N = 10 the
distribution is close to
a normal distribution.
Notice that the means
of the two distributions
are the same(16), but
that the spread of the
distribution for N = 10
is smaller.
Sampling distribution of mean
When sampling is from a population which is
not normal (may be positively or negatively
skewed), even then, as per the central limit
theorem, the sampling distribution of
mean tends quite closer to the normal
distribution, provided the number of
sample items is large.
Simulation of a sampling distribution. The parent population is
very non-normal.
The sampling distribution of the
mean approximates a normal
distribution even when the
parent population is very nonnormal. If you look closely you
can see that the sampling
distributions do have a slight
positive skew. The larger the
sample size, the closer the
sampling distribution of the mean
would be to a normal
distribution.
(2) Sampling Distribution of Proportion
Population Proportion ( )
Population proportion is part of a population with a
particular attribute, expressed as a fraction, decimal
or percentage of the whole population.
For a finite population, the population proportion is
the number of members in the population with a
particular attribute divided by the number of
members in the population.
Sample Proportions …..
Two common statistics are the sample
proportion, 𝑷, and sample mean, 𝒙. Sample
statistics are random variables and therefore vary
from sample to sample.
For instance, consider taking two random samples,
each sample consisting of 5 students, from a class
and calculating the mean height of the students in
each sample. Would you expect both sample means
to be exactly the same?
Sample Proportions …..
As a result, sample statistics also have a
distribution called the sampling distribution.
These sampling distributions have a mean and
standard deviation. However, we refer to the
standard deviation of a sampling distribution as
the standard error. Thus, the standard error is
simply the standard deviation of a sampling
distribution.
Sampling Distributions for Sample
Proportion 𝐩
If numerous repetitions of samples are taken, the
distribution of p is said to approximate a normal
curve distribution.
We can estimate the true population proportion,
p, by p and the true standard deviation of p
by 𝑠. 𝑒. (p) =
𝑝(1−𝑝)
𝑁
where 𝑠. 𝑒. (p) is
interpreted as the standard error of p.
Example
Suppose the proportion of all college students who
have used cocaine in the past 6 months is p = 0.40. For
a class of size N = 200, representative of all college
students on use of cocaine, what is the chance that the
proportion of students who have used cocaine in the
past 6 months is 0.32 (or 32%)?
NOTE: This would imply that 32% of the sample
students said "yes" to having used cocaine, or 64 of the
200 said "yes". This means the sample proportion 𝑃 is
64/200 or 32%
Solution
The mean of the sample proportion p is p and the
standard error of p is 𝑠. 𝑒. (p) =
𝑝(1−𝑝)
𝑛
. For this
marijuana example, we are given that p = 0.4. We then
determine
𝑠. 𝑒. p =
𝑝 1−𝑝
𝑛
=
0.4(1−0.4)
200
= 0.0346
So, the sample proportion p is about normal with
mean p = 0.40 and 𝑠. 𝑒. p = 0.0346.
student’s ‘t’ distribution
Student’s ‘t’ distribution
When population standard deviation 𝜎 is not known and the
sample is of small size (i.e., 𝑛 ≤ 30), we use t distribution for the
sampling distribution of mean and find the t variable as:
Given N independent measurements 𝑥𝑖 (N-Sample size and N-1
is called the degrees of freedom)
(𝑥 − 𝜇)
𝑡= 𝜎
𝑁
Where
1
𝜎=
𝑁 −1
𝑛
𝑥𝑖 − 𝑥
𝑖=1
2
Example:
The CEO of light bulbs manufacturing company
claims that an average light bulb lasts 300 days.
A researcher randomly selects 15 bulbs for
testing. The sampled bulbs last an average of
290 days, with a standard deviation of 50 days.
If the CEO’s claim were true, what is the
probability that 15 randomly selected bulbs
would have an average life of no more than 290
days?
Solution
Sample Mean 𝑥 = 290
The standard deviation of the sample = 50
The population mean =300
Sample Size N=15
The t distribution is
𝑡=
(𝑥−𝜇)
𝜎
𝑁
=
290 −300
50
15
=
−10
12.9099
= 0.7746
• The degrees of freedom are equal to 15 – 1 = 14.
F distribution
• The F distribution is the probability
distribution associated with the f statistic.
The f Statistic
Steps required to compute an f statistic:
• Select a random sample of size n1 from a
normal population, having a standard
deviation equal to σ1.
• Select an independent random sample of
size n2 from a normal population, having a
standard deviation equal to σ2.
• The f statistic is the ratio
of s12/σ12 and s22/σ22.
The f Statistic ……
• The following equivalent equations are commonly
used to compute an f statistic:
f = [ s12/σ12 ] / [ s22/σ22 ]
f = [ s12 * σ22 ] / [ s22 * σ12 ]
f = [ Χ21 / v1 ] / [ Χ22 / v2 ]
f = [ Χ21 * v2 ] / [ Χ22 * v1 ]
where σ1 is the standard deviation of population
1, s1 is the standard deviation of the sample drawn
from population 1.
The f Statistic ……
σ2 is the standard deviation of population 2, s2 is the
standard deviation of the sample drawn from
population 2,
Χ21 & Χ22 is the chi-square statistic for the sample
drawn from population 1 & 2 respectively
v1 & v2 is the degrees of freedom for Χ21 & Χ22 .
Note that degrees of freedom v1 = n1 - 1, and
degrees of freedom v2 = n2 - 1 .
The F Distribution
• The distribution of all possible values of
the f statistic is called an F distribution,
with v1 = n1 - 1 andv2 = n2 - 1 degrees of
freedom.
Example
Suppose you randomly select 7 women
from a population of women, and 12 men
from a population of men. The table below
shows the standard deviation in each
sample and in each population.
Compute the f statistic.
Population
Women
Men
Population standard Sample standard
deviation
deviation
30
50
35
45
Solution
• The f statistic can be computed from the
population and sample standard deviations, using
the following equation:
• f = [ s12/σ12 ] / [ s22/σ22 ]
where σ1 is the standard deviation of population
1, s1 is the standard deviation of the sample drawn
from population 1, σ2 is the standard deviation of
population 2, and s1 is the standard deviation of the
sample drawn from population 2.
Solution ….
As you can see from the equation, there are actually
two ways to compute an f statistic from these data. If
the women's data appears in the numerator, we can
calculate an f statistic as follows:
f = ( 352 / 302 ) / ( 452 / 502 ) = (1225 / 900) / (2025 /
2500) = 1.361 / 0.81 = 1.68
For this calculation, the numerator degrees of
freedom v1 are 7 - 1 or 6; and the denominator
degrees of freedom v2 are 12 - 1 or 11.
Solution ….
On the other hand, if the men's data appears in the
numerator, we can calculate an f statistic as follows:
f = ( 452 / 502 ) / ( 352 / 302 ) = (2025 / 2500) / (1225 /
900) = 0.81 / 1.361 = 0.595
For this calculation, the numerator degrees of
freedom v1 are 12 - 1 or 11; and the denominator
degrees of freedom v2 are 7 - 1 or 6.
Chi- square distribution
Chi- square distribution is encountered when we deal
with collections of values that involve adding up
squares. Variances of samples require us to add a
collection of squared quantities and thus have
distributions that are related to chi-square
distribution. If we take each one of a collection of
sample variances, divide them by the known
population variance and multiply these quotients by
(n – 1), where n means the number of items in the
sample, we shall obtain a chi-square distribution.
Thus,
would have the same distribution as
chi-square distribution with (n – 1) degrees of
freedom.