Topics under Research Methodology

1. Unit III:
 Data preparation: editing, coding, classification
and tabulation of data,
 Measures of central tendency,
 Probability concepts, Theoretical distributions:
Binomial distributions, Normal distribution, and
Poisson distribution.

2. Random Variable
A variable which assumes different numerical values as a
result of random experiment is known as random
variable.-
A random variable may be discrete or continuous.

3. In statistical, frequency distribution can broadly be
classified under two heads:- (i) Actual Frequency
distribution & (ii) Theoretical Frequency distribution
(i) Actual Frequency distribution
Actual frequency distribution is also known as ‘Observed
frequency distribution’. It is constructed on the basis of
data available in statistical experiments. For example,
distribution constructed on the basis of actual data of marks
obtained by students of a class.

4. (ii)Theoretical Frequency Distribution
Theoretical Frequency distribution is also called as ‘Expected or
Probability or Model Frequency Distribution’. We may define
theoretical frequency distribution as a distribution of frequencies
which is not based on actual experiments but is constructed
through expected frequencies obtained by mathematical
computation based on certain hypothesis. For example, two coins
are tossed four times and upcoming of head is taken as success,
then expected frequencies would be as follows:-
.
Elements of the
sample space
No. of Success
(Heads)
Probability Expected
Frequencies
T, T 0 ¼ 1
T, H 1 ¼
2
H, T 1 ¼
H, H 2 ¼ 1

5. (ii)Theoretical Frequency Distribution (Cont…)
Similarly, if four coins are tossed 80 times, the expected
frequencies of getting head according to binomial theorem
would be as follows:-
Expected or Probability or Model Frequency Distribution’.
No. of Success
(Heads)
Probability Expected
Frequencies
0 1/16 80 x 1/16 = 5
1 4/16 80 x 4/16 = 20
2 6/16 80 x 6/16 = 30
3 4/16 80 x 4/16 = 20
4 1/16 80 x 1/16 = 5

6. Utility or Importance of the Theoretical Frequency
Distribution
Theoretical frequency distribution is base of modern statistics.
Its importance may be placed under the following heads:-
1. Estimate of nature & trend of frequency distribution-
2. Basis of logical decision-
3. Forecasting-
4. Substitute of Actual data-
5. Test of Sampling
6. Solution of Various Problems of Daily Life- for example;
a ready-made garment manufacturer decides the quantities
of various sizes on the basis of normal distribution.

7. Types of Theoretical or Probability Distribution
In modern statistics different types of frequency distribution are
constructed as explained in the following chart:-
Among these, the following three distribution more popular:-
I. Binomial Distribution
II. Poisson Distribution
III. Normal Distribution
Discrete Frequency
Distribution
1. Binomial
Distribution
2. Poisson
Distribution
3. Rectangular
Distribution
4. Geometric
Distribution
Continuous Frequency
Distribution
1. Normal
Distribution
2. Student’s t-
distribution
3. Chi-square
distribution
4. F-distribution

8. I. Binomial Distribution
Binomial distribution is associated with the name of Swiss
mathematician James Bernoulli. Therefore, it is also known as Bernoulli
Distribution.
Definition of Binomial distribution:-
Binomial distribution is a discrete frequency distribution, which is based
on probability of success (desired event) and failure. This distribution, in
the form of probability density function, can be expressed as follows:-
𝑃 𝑥 = 𝑛𝐶𝑥𝑝𝑥𝑞𝑛 − 𝑥
𝑃 𝑟 = 𝑛𝐶𝑟𝑝𝑟𝑞𝑛 − 𝑟
Where;
p = Probability of success
q = Probability of failure or 1-p
n = number of trials,
x or r = number of successes in ‘n’ trial

9. Conditions for Application or Assumptions of Binomial
Distribution
Binomial distribution can be applied only under following
conditions:-
1. Finite number of trials-
2. Mutually exclusive outcomes-
3. Same probability in each trial-
4. All trial independent-
5. Discrete Variable-

10. Characteristics or Properties of Binomial Distribution
The main characteristics of Binomial Distribution are as
follows:-
1. Theoretical frequency distribution-
2. Discrete frequency distribution-
3. Presentation by line graph-
4. Shape of Binomial Distribution- The shape of binomial
distribution depends on the values of p and q. If p & q
both are equal, i.e., ½ , then binomial distribution be
perfectly symmetrical. If p & q both are not equal (p ≠ q),
the binomial distribution will be asymmetrical.
5. Main Parameters- p & q are two main parameters of
binomial distribution and the entire distribution can be
obtained with the help of these parameters.

11. Characteristics or Properties of Binomial Distribution (Cont…)
The main characteristics of Binomial Distribution are as follows:-
6. Mean, S.D. & Variance- Mean = 𝑛𝑝, 𝑆. 𝐷. =
𝑠𝑞𝑢𝑎𝑟𝑒 𝑟𝑜𝑜𝑡 𝑜𝑓 𝑛𝑝𝑞 & 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = 𝑛𝑝𝑞
7. Uses of Binomial Distribution-viz, tossing of coins, throwing of
dice, etc.
8. Sequence of p & q- The number of successes can be placed in
ascending or descending order
9. Expected Frequencies- The expected frequencies can be obtained by
multiplication of N in probabilities of binomial distribution.
10. Constant Values- In binomial distribution, the following formulae
are used for various constant values-
𝐹𝑖𝑟𝑠𝑡 𝑀𝑜𝑚𝑒𝑛𝑡 𝛍1 = 0
𝑆𝑒𝑐𝑜𝑛𝑑 𝑀𝑜𝑚𝑒𝑛𝑡 𝛍2 = 𝑛𝑝𝑞
𝑇ℎ𝑖𝑟𝑑 𝑀𝑜𝑚𝑒𝑛𝑡 𝛍3 = 𝑛𝑝𝑞 (𝑞 − 𝑝)
𝐹𝑜𝑢𝑟𝑡ℎ 𝑀𝑜𝑚𝑒𝑛𝑡 𝛍4 = 3𝑛2𝑝2𝑞2 + 𝑛𝑝𝑞 (1 − 6𝑝𝑞)
𝑀𝑒𝑎𝑛 = 𝑛𝑝
𝑆𝐷 = 𝑆𝑞𝑢𝑎𝑟𝑒 𝑟𝑜𝑜𝑡 𝑜𝑓 𝑛𝑝𝑞

12. General Rule of Binomial Distribution
While constructing binomial expansion, the following points
should be kept in mind:-
1. Number of Terms- The number of terms in a binomial
distribution is always 𝑛 + 1. For example, there will be three
terms in p + q 2 and 6 terms in p + q 5.
2. Sequence of Exponents- If binomial distribution is expressed
as p + q n, then the exponents of p will be marked in
decreasing order while those of q in an increasing order.
3. Sum of Exponents- The sum of exponents of p and q, for any
single term will always be equal to n.
4. Symmetrical Order of Coefficients- The coefficients in
binomial distribution are always symmetrical.
5. Sum of Coefficients- Sum of coefficients of all terms in a
binomial distribution is equal to 2n..
6. Sequence of p and q- The general form of binomial
distribution is the expansion of p + q n, in which the number
of successes is written in a descending order.

13. Illustrations of Binomial Distribution
1. Give full expansion of Binomial Distribution whose
arithmetic mean is 3 and variance is 2.
Solution;
Given Arithmetic mean = np 3
Variance = npq = 2
q = npq/np = 2/3, p = 1-q = 1- 2/3 = 1/3
n = np/p 3/1/3 = 9
Binomial Dist. = p + q n= (1/3 + 2/3)n
=nC0pnq0 + nC1pn-1q1 + nC2pn-2q2……

14. II. Poisson Distribution
Poisson distribution is a discrete probability distribution
originated by French Mathematician Simon Denis Poisson.
It is used in such cases where the value of p is very small,
i.e., p approaches zero (p→0) and the value of n is very large
since in these cases binomial distribution does not give
appropriate theoretical frequencies, Poisson distribution is
found very appropriate. It is worth mentioning that poisson
distribution is a limiting form of binomial distribution as n
moves towards infinity and p moves towards zero but np or
mean remains constant and finite.
Poisson distribution is used to describe the behaviour of rare
events such as number of germs in one drop of pure water,
number of printing errors per page, etc.

15. Uses of Poisson Distribution
1) In insurance problems to count the number of causalities,
2) In determining number of deaths due to suicides or rare
disease,
3) In counting the number of defects per item in statistical
quality control,
4) Number of accidents taking place per day on a busy road,
5) In biology to count the number of bacteria.

16. Properties/Characteristics of Poisson Distribution-
The main characteristics of Poisson distribution are as follows:-
1) Nature of Distribution- Poisson distribution is a discrete
probability distribution in which the number of successes are
given in whole numbers such as 0, 1, 2, 3, …. Etc.
2) Uses of Poisson Distribution- Poisson distribution is used in
those conditions where the value of p is very small, i.e., p
approaches zero (p→0) and the value of q is almost equal to 1
(q→1) and the value of n is very large.
3) Main Parameter- The main parameter of the poisson
distribution is mean (m = np). If the value of mean is known,
the entire distribution can be constructed.
4) Shape of Distribution- The Poisson distribution is always
positively skewed. However, with the increase in the value of
mean, the distribution shifts to right & skewness is reduced.

17. Properties/Characteristics of Poisson Distribution (Cont…)-
5) Constant Values-
i) Arithmetic mean = m = np,
ii) SD = square root of np,
iii) Variance = np,
iv) 𝛍1 = 0,
v) 𝛍2 = 𝑚,
vi) 𝛍3 = 𝑚,
vii) 𝛍4 = 𝑚 + 3𝑚2,
viii) 𝛽1 =
𝛍3
2
𝛍2
3 =
𝑚2
𝑚3 =
1
𝑚
ix) 𝛽2 =
𝛍4
𝛍2
2 =
𝑚+3𝑚2
𝑚2 = 3 +
1
𝑚
6. Assumption of Distribution- Poisson distribution is based on the following
assumptions:-
i) The occurrence or non-occurrence of an event does not influence the
other events,
ii) The probability of happening of more than one event in a very small
interval is negligible,
The value of e-m is very important in calculation of poisson distribution

18. III. Normal Distribution
Meaning of Normal Distribution-
Normal Distribution is a continuous probability distribution in
which the relative frequencies of a continuous variable are
distributed according to normal probability law.
In simple words, it is symmetrical distribution in which the
frequencies are distributed about the mean of distribution.
It has been observed that “The perfect smooth and
symmetrical curve, resulting from the expansion of the
binomial p + q n, when n approaches infinity is known as
the normal curve. We can say that the normal curve represents
a continuous & infinite binomial distribution”.

19. Assumptions of Normal Distribution-
The normal distribution is based on certain assumptions:-
1) Independent Causes- The forces or causes affecting
events are independent of one another.
2) Multiple Causation- The causal forces are numerous &
all causes are of approximately equal importance.
3) Symmetrical- The deviations from mean on either side is
equal in number and size.

20. Properties/Characteristics of Normal Distribution-
The main characteristics of Poisson distribution are as follows:-
1) Bell Shaped-
2) Continuous Distribution-
3) Equality of Central Values-
4) Uni-apex (Uni-modal)-
5) Equal Distance of Quartiles from Median-
6) Asymptotic to the Base Line-
7) Parameters of Distribution-
8) Relationship between QD & SD-
9) Relationship between MD & SD
10) Area Relationship-
i) 𝑋 ± 1𝜎 covers 68.27%
ii) 𝑋 ± 2𝜎 covers 95.45%
iii) 𝑋 ± 1𝜎covers 99.73%