CHAPTER # 04
MEASURES CENTRAL TENDENCY OR AVERAGES
If the average tends to lie at the centre of the distribution then it is called
measures of central tendency or measures of location.
•
Arithmetic Mean / Mean ( X )
Sum of the values divided by their numbers is known as arithmetic mean. It is denoted by X
.i.e.
Methods of Arithmetic Mean
1. Direct Method
2. Short Cut Method /Indirect Method
3. Coding Method or Step-Deviation Method
1. Direct Method
For Ungrouped Data
A.M =
X=
∑X
Where
n = number of observation
n
For Grouped Data
A.M =
X=
∑ fx
∑f
Where
f = frequency
2. Short Cut Method
For Ungrouped Data
X = A+
∑D
Where A=Constant,
n
D= X - A
For Grouped Data
X = A+
∑ fD
∑f
3. Coding Method or Step-Deviation Method
For Ungrouped Data
X = A+
∑u
n
×h
Where
u=
X −A
D
or
h
h
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For Grouped Data
X = A+
•
∑ fu
∑f
×h
Weighted Arithmetic Mean ( X W )
If X 1. X 2 . X 3 .......... X n be n values with weights W1.W2 .W3 ..........Wn respectively then weighted
average ( X W ) is defined as:
XW =
X 1W1 + X 2W2 + X 3W3 + .......... + X nWn
W1 + W2 + W3 + .......... + Wn
Or
XW =
•
Where
∑W
W= weight
Combined Mean / Grand Mean ( X )
X=
•
∑WX
∑ nX
i
∑n
Geometric Mean (G.M)
The positive nth root of the product of n positive values is called geometric mean. It is denoted
by G.M. i.e.
For Ungrouped Data
G.M = n X 1. X 2 . X 3 .......... X n
Or
G.M = Anti log
∑ log X
n
For Grouped Data
G.M = Anti log
∑ f log x
∑f
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•
Weighted Geometric Mean ( G.M W )
G.M W = Anti log
•
∑W log x
∑W
Harmonic Mean (H.M)
Harmonic mean is defined as the reciprocal of the arithmetic mean of the reciprocal of values in
a data. It is denoted by H.M. i.e.
For Ungrouped Data
H .M =
n
1
∑x
For Grouped Data
H .M =
•
∑f
1
∑f x
Median ( X% )
The central value of an array is called median. It is denoted by X%
.
For Ungrouped Data
Median =
 n + 1
th terms
X% = Value of 
 2 
For Grouped Data
Median = X% = l +

h ∑ f

− C.F 

f  2


Where
l = lower class boundary
C.F = Cumulative Frequency
h = Class interval
f = Frequency
•
Other Positional Measures
1. Quartile
2. Deciles
3. Percentiles
1. Quartile
These are the values which divide a distribution into four equal parts.
Q1 , Q2 , Q3
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For Ungrouped Data
 n + 1
Q1 = Value of 
th terms
 4 
 2 ( n + 1) 
Median = Q2 = Value of 
 th terms
 4 
Lower Quartile =
Upper Quartile =
 3 ( n + 1) 
Q3 = Value of 
 th terms
 4 
For Grouped Data

h ∑ f

− C.F 

f  4



h  2∑ f
− C.F 
Median = Q2 = l + 

f  4



h  3∑ f
− C.F 
Upper Quartile = Q3 = l + 

f  4


Lower Quartile =
Q1 = l +
Where
l = lower class boundary
C.F = Cumulative Frequency
h = Class interval
f = Frequency
2. Deciles
These are the values which divide a distribution into ten equal parts.
D1 , D2 , D3 ………… D9
For Ungrouped Data
 n + 1
D1 = Value of 
th terms
 10 
 2 ( n + 1) 
D2 = Value of 
 th terms
 10 
 3 ( n + 1) 
D3 = Value of 
 th terms
 10 
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
 8 ( n + 1) 
D8 = Value of 
 th terms
 10 
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 9 ( n + 1) 
D9 = Value of 
 th terms
 10 
For Grouped Data

h ∑ f

− C.F 

f  10



h  2∑ f
D2 = l + 
− C.F 

f  10



h  3∑ f
D3 = l + 
− C.F 

f  10


D1 = l +
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .

h  8∑ f

− C.F 

f  10



h  9∑ f
D9 = l + 
− C.F 

f  10


D8 = l +
Where
l = lower class boundary
C.F = Cumulative Frequency
h = Class interval
f = Frequency
3. Percentiles
These are the values which divide a distribution into hundred equal parts.
P1 , P2 , P3 ………… P99
For Ungrouped Data
 n + 1
P1 = Value of 
th terms
 100 
 2 ( n + 1) 
P2 = Value of 
 th terms
 100 
 3 ( n + 1) 
P3 = Value of 
 th terms
 100 
. . . .
. . . .
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. . . .
. . . .
. . . .
. . . .
 98 ( n + 1) 
P98 = Value of 
 th terms
 100 
 99 ( n + 1) 
P99 = Value of 
 th terms
 100 
For Grouped Data

h ∑ f

− C .F 

f  100



h  2∑ f
P2 = l + 
− C.F 

f  100



h  3∑ f
P3 = l + 
− C.F 

f  100


P1 = l +
. . . .
. . . .
P50 = l +
. . . .
. . . .

h  50∑ f

− C.F 

f  100


. . . .
. . . .
. . . .
. . . .

h  98∑ f

− C.F 

f  100



h  99∑ f
P99 = l + 
− C.F 

f  100


P98 = l +
Where
l = lower class boundary
C.F = Cumulative Frequency
h = Class interval
f = Frequency
•
Mode ( X̂ )
“The mode which is defined as the value which occurs the greatest number of times in the data”
some time, there are more than one mode in the data, and sometimes there are no mode in the
data because each value occurs the same number of times. It is denoted by X̂ .
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For Ungrouped Data
Mode =
X̂ = Greatest value in data
For Grouped Data
Mode =
f1 − f 2
X̂ = l + ( f − f ) + ( f − f ) × h
1
2
1
3
Or
Mode =
X̂ =
f1 − f 2
×h
( 2 f1 − f 2 − f3 )
Where
l = lower class boundary
f1 = Frequency of the model class or Maximum frequency
f 2 = Frequency preceding f1
f3 = Frequency acceding f1
h = Class interval of the modal class
•
Relation Between A.M , G.M and H.M
A.M ≥ G.M ≥ H .M
•
Empirical relation b/w Mean , Median and Mode
Mode = 3 Median – 2Mean
Or
Xˆ = 3 X% − 2 X
________________________________________________________________________________________________
Hashim (0345-4755472)
E-mail: hashim_farooqi@hotmail.com
Url: http://www.pakchoicez.com
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