This document discusses various measures of central tendency including arithmetic mean, median, and mode. It provides definitions and formulas for calculating each measure along with examples using raw data and grouped data. The three main points covered are: 1. Measures of central tendency (mean, median, mode) calculate the central or typical value in a data set and are used to describe data distributions. 2. The arithmetic mean is the sum of all values divided by the number of values, the median is the middle number when values are arranged in order, and the mode is the most frequent value. 3. Formulas and methods are provided for calculating each measure using both raw ungrouped data as well as grouped frequency distribution data

1. Suresh Babu G
Measures of Central Tendency
Suresh Babu G
Assistant Professor
CTE CPAS Paippad, Kottayam

2. Suresh Babu G
Measures of Central Tendency
• A measure of central tendency is a summary
statistic that represents the centre point or typical
value of a dataset.
• In statistics, the three most common measures of
central tendency are
1. Arithmetic Mean
2. Median
3. Mode
There are two more types of average ie, Geometric
Mean and Harmonic Mean
• Each of these measures calculates the location of
the central point using a different method.

3. Suresh Babu G
Arithmetic Mean
It is defined as the sum of the values of all
observations divided by the number of
observations and is usually denoted by X̅ .
If there are N observations as X1, X2, X3,…..Xn
N
Xn....X3X2X1
X


N
X
X
 Where ΣX = Sum of all observation
N = Total number of observation

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Arithmetic Mean for Ungrouped Data
Individual Series
Direct Method
Example:
Calculate Arithmetic Mean from the data showing
marks of students in a class in an psychology
test : 40, 50, 55, 78, 58.
N
X
X

2.565
5878555040
 
X
N
X
X


5. Suresh Babu G
Arithmetic Mean for Ungrouped Data
Discrete Series
Direct Method
N
fX
X


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Example
Calculate the AM from the following:
Mark No: of
students
fX
22 5 110
25 10 250
30 15 450
37 7 259
45 3 135
50 10 500
N = 50 Σfx = 1704
Marks 22 25 30 37 45 50
No: of students 5 10 15 7 3 10
N
fX
X


08.34
50
1704 X
08.34
50
1704

X

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Continuous Series
Direct Method
N
fm
or
N
fX
X 

Where M = Mid value

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Example:
From the following data calculate AM
Marks 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50
No: of Students 5 3 7 25 20
Class Frequenc
y (f)
Mid value
(x)
fx
0 – 10 5 5 25
10 – 20 3 15 45
20 – 30 7 25 175
30 – 40 25 35 875
40 - 50 20 45 900
N = 60 Σfx = 2020
N
fX
X


67.33
60
2020


X
X
33.67

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Short Cut Method
N
fd
AX


Where
A = Assumed Mean
d = deveation of mid values from the
assumed mean d = m-A
N = Number of observations
N
fd
AX



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Example:
From the following data calculate AM
Marks 0 - 10 10 - 20 20 - 30 30 - 40 40 - 50
No: of Students 5 3 7 25 20
Class Freque
ncy (f)
Mid
value
(x)
d= x-A fd
0 – 10 5 5 -20 -100
10 – 20 3 15 -10 -30
20 – 30 7 25 0 0
30 – 40 25 35 10 250
40 - 50 20 45 20 400
N = 60 Σfd=520
25
A = 25
N
fd
AX


66.33
66.825
60
52025



X
X

11. Suresh Babu G
Median
Median is the middle element when the data set is
arranged in order of the magnitude.
Median of Ungrouped Data
Individual Series
Median = Size of item
Where N is number of observations
2
)1( N
th

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Example : The following data provides marks of
seven students. Calculate median
110, 115, 140, 117, 109, 113, 120
Arrange the data in ascending order
109, 110, 113, 115, 117, 120, 140
Median = Size of item
Median = Size of item = Size of item
Median = Size of 4 item
Median = 115
2
)1( N th
2
)17(  th
2
8
th
th

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Example 2 : Calculate the median
38, 24, 45, 50, 85, 60, 95, 40, 56, 63
Ascending order
24, 38, 40, 45, 50, 56, 60, 63, 85, 95
Median = Size of item
Median = Size of item = Size of item
Median = Size of 5.5 item
Median = 50 + 0.5(56-50) = 50 + 0.5 X 6 = 53
Median = 53
2
)1( N
2
)110( 
2
11
th th
th
th

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Median of Grouped Data
Discrete Series
Median = Size of item
Example:
Calculate Median
2
)1( N th
Marks No: of students
10 2
20 4
30 10
40 4
N = Total frequency

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Marks No: of students Cumulative
Frequency
10 2 2
20 4 6
30 10 16
40 4 20
Median = Size of item
Median = Size of item = Size of item
Median = Size of 10.5 item
10.5 is easily located at 16 of cf corresponding mark is 30
so Median =30
2
)1( N
th
2
)120( 
th
2
21
th
th

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Continues Series
Median =
Where L = Lower limit of the median class
cf = Cumulative frequency of the classes
preceding the median class
f = frequency of the median class
h = magnitude of the median class
interval
h
f
cfN
L 


)2/(

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Example: Find median
Marks No of Students
0 – 10 4
10 – 20 12
20 – 30 24
30 – 40 36
40 – 50 20
50 – 60 16
60 – 70 8
70 - 80 5

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Median =
Median = = = 30 + 6.25
Medan = 36.25
Marks No of
Students
Cumulative
Frequency
0 – 10 4 4
10 – 20 12 16
20 – 30 24 40
30 – 40 36 76
40 – 50 20 96
50 – 60 16 112
60 – 70 8 120
70 - 80 5 125
Size of N/2 th item
= Size of 125/2 th item
= Size of 62.5 th item
Median class = 30 – 40
L = 30
cf = 40
f = 36
h = 10
2
N
h
f
cfN
L 


)2/(
1030 36
402/125
 
1030 36
5.22


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Mode
Mode is the most frequently observed data value.
Ungrouped data – Individual Series
Example
Find mode
1, 2, 3, 4, 4, 5
Mode is 4 ( as 4 repeats 2 times )

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Mode of Grouped Data
Discrete Series
Example : Find Mode
By inspecting the data value, the maximum
frequency is 20 ie, 30 mark repeats 20 times so
the mode value is 30 mark
Marks No. of Students
10 2
20 8
30 20
40 10
50 5
30

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Continuous Series:
Inspection Method
Mode =
Where
L = lower limit of the modal class
D1 = Difference between the frequency of the model class
and the frequency of the class preceding the modal class
(ignoring signs)
D2 = Difference between the frequency of the model class
and the frequency of the class succeeding the modal
class (ignoring signs)
h = class interval if the distribution
h
DD
DL 


21
1

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Example
Find mode value
Marks No of Students
0 – 10 5
10 – 20 7
20 – 30 8
30 – 40 20
40 – 50 10
50 – 60 6
60 – 70 2
70 - 80 2

23. Suresh Babu G
The most frequently
occurring data
value is between
30-40 which
occurs 20 times.
The model class is
30 – 40.
Mode = D1 = 20 – 8 = 12
Mode = D2 = 20 – 10 = 10
Mode = 30 + 5.45 L = 30
Mode = 30.45 h = 10
Marks No of Students
0 – 10 5
10 – 20 7
20 – 30 8
30 – 40 20
40 – 50 10
50 – 60 6
60 – 70 2
70 - 80 2
h
DD
DL 


21
1
1030 1012
12
 

24. Suresh Babu G
Another Method to find mode value is
Mode = 3Median – 2 Mean

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