This session covers the type of variables, level of measurement with an example, central tendency, and dispersions with applicability. Methods are illustrated with published examples.

1. A Story of Numbers:
Variables, Central Tendency, and
methods of Dispersion
Dr. Rajeev Kumar
MSW (TISS, Mumbai), M.Phil., (CIP, Ranchi), UGC-JRF, Ph.D. (IIT Kharagpur)
Lecture-3 : Research Methodology

2. Content of this session
In this session, we will discuss the following topics
• Types of variables
• Level of measurement with examples
• Central tendency with example (mean, mode, and median)
• Dispersion (standard deviation, variance, IQR, and range)
• Pairing of central tendency and dispersion
• Relation between level of measurement and central tendency
• Examples in published papers
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3. What is variable?
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4. Further classification of variables
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5. Types of variables
Categorical
Nominal
Ordinal
Continuous
Interval
Ratio
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6. Examples of Variables
Examples of nominal
Gender : Male, Female, TG
Response: Yes, No, can’t say
Weather: Hot, Cold, Rainy
Examples of Ordinal
Education: Primary, Middle, Matric,
Intermediate, Graduation
Socio-economic status: Low, middle,
high
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7. Continuous variables
Categorical
Nominal
Ordinal
Continuous
Interval
Ratio
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8. Converting continuous variables into categorical
Education in
years (Continuous
variables)
Education in
categories
(Categorical
/discrete
variables)
0 Illiterate
5 Primary
8 Middle
10 Matriculation
Marks
(Continuous
variables)
Converted into
divisions
(categorical/dis
crete variables)
38% III
48% II
65% I
80% I
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10. Two types of data presentation
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12. Lets understand the central tendency with an example
Mrs. Banerjee is a teacher, she conducted an unit test of three classes
for 5 marks. Principal of school asked her to present performance of each
class in aggregated number. Mrs. Banerjee calculated mean of each class.
But do these aggregation/ central tendency (mean) really represent
?marks of each class?
Class-B Marks
Sonia 0
James 1
Shahrukh 3
Pinchu 4
Mona 5
Aggregated
number
(Mean)
2.6
Class-A Marks
Suraj 4
Mamta 4.25
Salman 4.50
Nandu 4.75
Pappu 5
Aggregated
number
(Mean)
(4.5)
Class-C Marks
Bablu 0
Golu 0
Soni 0
Laxmi 5
Satish 0
Aggregated
number
(Mean)
1
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13. Whether these central tendency represented each class?
Students of Class-A were happy. Because (4.5) represented marks of
every student.
But Principal was not happy with marks of Class-B and Class-C. Because
(2.6) in class-B overrepresented the marks of Sonia and James, and
under-represented marks of Pinchu and Mona. In class-C (mean=1) did
not represent marks of any one.
Class-B Marks
Sonia 0
James 1
Shahrukh 3
Pinchu 4
Mona 5
Aggregated
number
(Mean)
2.6
Class-A Marks
Suraj 4
Mamta 4.25
Salman 4.50
Nandu 4.75
Pappu 5
Aggregated
number
(Mean)
(4.5)
Class-C Marks
Bablu 0
Golu 0
Soni 0
Laxmi 5
Satish 0
Aggregated
number
(Mean)
1
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14. Whether Class- B and Class-C need a different central tendency?
We need to calculate the median
of marks of class-B.
Step-1: arrange all number in
ascending order.
N=5, this is odd number
Here n= 5 so (n+1)/2= (5+1)/2
6/2= 3. Median will be the 3rd
number, which is 3.
It means, there are two types of
students in class: who get marks
above 3 and below 3
Class-B Marks
Sonia 0
James 1
Shahrukh 3
Pinchu 4
Mona 5
Aggregated
number (Mean)
2.6
Median 3
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15. Method of calculating median
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16. What about class-C ?
We need to calculate the mode. Here most occurring number is (0). It
means, we can say, most of the students got zero (0) marks
But still marks of Laxmi are represented by mode (central tendency).
Class-C Marks
Bablu 0
Golu 0
Soni 0
Laxmi 5
Satish 0
Aggregated
number
(Mean)
1
Mode 0
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17. What else we need to have the full picture of data?
1. What we need to describe the position of Pappu and Suraj in Class-
A?
2. How to define position of Sonia, James, Pinchu, and Mona in Class-B?
3. And position of Satish in Class-C?
A calculation of dispersion can answer all these three questions.
Class-B Marks
Sonia 0
James 1
Shahrukh 3
Pinchu 4
Mona 5
Aggregated
number
Mean=2.6
Median=3
Class-A Marks
Suraj 4
Mamta 4.25
Salman 4.50
Nandu 4.75
Pappu 5
Aggregated
number
Mean=(4.5)
Median=4.5
Class-C Marks
Bablu 0
Golu 0
Soni 0
Laxmi 5
Satish 0
Aggregated
number
Mean=1
Mode=0
Median=0
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18. Mode Range
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19. How to calculate the range? Range= Maximum-minimum
Class-B Marks
Sonia 0
James 1
Shahrukh 3
Pinchu 4
Mona 5
Aggregated
number
Mean=2.6
Median=3
Maximum=5
Minimum=0
Range=5-0=5
Range= (5,0)=5
Class-A Marks
Suraj 4
Mamta 4.25
Salman 4.50
Nandu 4.75
Pappu 5
Aggregated
number
Mean=(4.5)
Median=4.5
Maximum=5
Minimum=4
Range= 5-4
Range=1
(5, 4)=1
Class-C Marks
Bablu 0
Golu 0
Soni 0
Laxmi 5
Satish 0
Aggregated
number
Mean=1
Mode=0
Median=0
Maximum=5
Minimum=0
Range=5-0=5
Range=(0,5)=5
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20. How to calculate (SD) standard deviation?
Class-A
N= 5
Marks =X D= x- x̄ D D² Formula of SD
Suraj 4 4-4.5 -.50 .25
Mamta 4.25 4.25-4.50 -.25 .0625
Salman 4.50 4.50-4.50 0 0
Nandu 4.75 4.75-4.50 .25 .0625
Pappu 5 5-4.50 .50 .25
Aggregated
number
Mean=4.5
x̄ = 4.5
Mean=(4.5)
Median=4.5
Maximum=5
Minimum=4
Range= 5-4
Range=1
(5, 4)=1
=.625
ΣD²= .625
N=5
N-1= 5-1=4
SD= √.625/4
SD= √.15625
SD= .39
Mean (±SD)
4.5 (±.39)
Variance= SD²
Variance=(.39)²
Variance=
.39x.39
Variance= .15625
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22. The relation between percentile and quartile
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23. Example
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25. Examples of calculating inter-quartile range
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26. Understanding box-whisker plot and IQR
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28. The Finale!
Class-B Marks
Sonia 0
James 1
Shahrukh 3
Pinchu 4
Mona 5
Aggregated
number
(central
tendency and
dispersion)
SD=2.07
IQR= 4
Mean=2.6
Median=3
Maximum=5
Minimum=0
Range=5-0=5
Range=
(5,0)=5
Class-A Marks
Suraj 4
Mamta 4.25
Salman 4.50
Nandu 4.75
Pappu 5
Aggregated
number
(central
tendency
and
dispersion)
SD =.40
IQR=.75
Mean=(4.5)
Median=4.5
Maximum=
5
Minimum=4
Range= 5-4
Range=1
(5, 4)=1
Class-C Marks
Bablu 0
Golu 0
Soni 0
Laxmi 5
Satish 0
Aggregated
number
(central
tendency and
dispersion)
SD= 2.23
IQR=2.5
Mean=1
Mode=0
Median=0
Maximum=5
Minimum=0
Range=5-0=5
Range=(0,5)=5
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29. Interpretation of result: What does these numbers
say?
In the class-A (SD=.40; IQR=.75, Range=1) there was the least variability. The
performance of each student was very close to each other.
However the highest (SD=2.23) was noted in the class-C, where four students got zero
and only one students got 5, therefore, in this class there is unmatched performance.
The highest IQR=4 was found in class-B, which shows the highest difference between
performance of two groups.
The SD of class-B (2.07) is the five times higher than class-A.
It means, there is five more variability in the class-B.
We can also say that considering academic performance, Class-A is the most
homogenous.
While heterogeneity were noticed in class-C and Class-B.
The worst performance is seen in the class-C.
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30. Relation between central tendency and level
of measurement
Central Tendency Appropriate
dispersion
Mean Standard Deviation
and Range
Median Interquartile Range
and Range
Mode Range (if variable is
nominal)
IQR and Range (if
variable is Ordinal)
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31. Example of Median and IQR from published research (Kumar et al.,
2020)
Kumar, R., Suar, D., Singh, S. K., & Bhattacharya, S. D. (2020). Why do AIDS Sufferers on Antiretroviral
Therapy Die Early?—Evidence from Jharkhand in India. In Race, Ethnicity, Gender and Other Social
Characteristics as Factors in Health and Health Care Disparities. Emerald Publishing Limited. pp. 199-
216. https://doi.org/10.1108/S0275-495920200000038015
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32. Example of median and IQR from (Kumar et al., 2020)
Kumar, R., Suar, D., Singh, S. K., & Bhattacharya, S. D. (2020). Why do AIDS Sufferers on Antiretroviral Therapy Die Early?—Evidence from Jharkhand in India. In Race, Ethnicity, Gender and Other Social Characteristics as Factors in Health and Health Care
Disparities. Emerald Publishing Limited. pp. 199-216. https://doi.org/10.1108/S0275-495920200000038015
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33. An eminent professor and scientist, who taught me the central
tendency with example of class-A, class-B, class-c, and 5 children.
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Thanks for your patience and inquisitiveness
in the next session, we will cover the normality
curve and skewness