CHAPTER # 05
MEASURES OF DISPERSION, MOMENTS AND
SKEWNESS
A quantity that measures how the data are dispersed about the average is called
measures of dispersion.
•
Range (R)
The range is a simplest measure of dispersion. “It is defined as the difference b/w the largest
and smallest observation in a set of data.” It is denoted by “R”. This is an absolute measure of
dispersion.
For Ungrouped Data
Range = R =
Where
Xm − Xo
X m = the largest value.
X o = the smallest value.
For Grouped Data
Range = R = Upper class boundary of the highest class – lower class boundary of the
lowest class
Or
Range = R = Class Marks (X) of the highest class – Class Marks of the lowest class
•
Semi Inter Quartile Range or Quartile Deviation
The semi inter-quartile range or quartile deviation is defined as half of the difference b/w the
third and the first quartiles. Symbolically it is given by the
S.I.Q.R = Q.D =
Where
Q3 − Q1
2
Q1 = First, Lower quartile
Q3 = Third, Upper quartile
This is an absolute measure of dispersion.
•
Mean Deviation or Average Deviation
The mean deviation is defined as the average of the deviation of the values from an average
(Mean, Median), the deviation are taken without considering algebraic signs.
1. Mean Deviation From Mean
For Ungrouped Data
M.D =
∑ X −X
n
Or
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M.D =
∑ X − Mean
n
For Grouped Data
M.D =
∑f X−X
∑f
Or
M.D =
∑f
X − Mean
∑f
2. Mean Deviation From Median
For Ungrouped Data
M.D =
∑ X − X%
n
Or
M.D =
∑ X − Median
n
For Grouped Data
M.D =
∑ f X − X%
∑f
Or
M.D =
•
∑f
X − Median
∑f
Standard Deviation (S)
The standard deviation is defined as the positive square root of the mean of the squared
deviation of the values from their mean. Thus the standard deviation of a set of n values
X 1. X 2 . X 3 .......... X n .it is denoted by ‘S’. This is an absolute measure of dispersion.
Methods of Standard Deviation
I. Direct Method
II. Short Cut Method
III. Coding Method or Step-Deviation Method
1. Direct Method
For Ungrouped Data
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S.D = S =
∑X
∑X
− 
 n
S.D = S =
∑( X − X )
2
n



2
2
n
For Grouped Data
S.D = S =
∑ fX
∑f
S.D = S =
∑ f ( X −X)
∑f
2
 ∑ fX
−
 ∑f




2
2
2. Short Cut Method
For Ungrouped Data
∑D
S.D = S =
∑D
− 

 n 
2
n
2
Where D= X – A
For Grouped Data
∑ fD
∑f
S.D = S =
2
 ∑ fD 
−
 ∑ f 


2
3. Coding Method or Step-Deviation Method
For Ungrouped Data
S.D = S =
h×
∑u
2
n
 ∑u 
− 

 n 
2
Where
u=
X−A
D
or
h
h
For Grouped Data
S.D = S =
•
h×
∑ fu
∑f
2
 ∑ fu 
−
 ∑ f 


2
Combined Standard Deviation ( Sc )
For two set of values
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Sc =
2
n1S12 + n2 S 22
n1n2
+
X1 − X 2 )
2 (
n1 + n2
( n1 + n2 )
For three or more sets of data
Sc =
∑ n  S
i
2
i
( X −X)
i
∑n
2


i
•
Variance ( S 2 )
The variance is defined as the mean of the squared deviation from mean. It is denoted by ‘ S 2 ’
Or
The square of the standard c=deviation is called variance. It is denoted by ‘ S 2 ’
Methods of Standard Deviation
1. Direct Method
2. Short Cut Method
3. Coding Method or Step-Deviation Method
1. Direct Method
For Ungrouped Data
Var(X) =
Var(X) =
2
S =
2
S =
∑X
2
n
∑X
− 
 n
∑( X − X )



2
2
n
For Grouped Data
Var(X) =
Var(X) =
2
S =
2
S =
∑ fX
∑f
2
 ∑ fX
−
 ∑f

∑ f ( X −X)
∑f



2
2
2. Short Cut Method
For Ungrouped Data
Var(X) =
2
S =
∑D
n
2
∑D
− 

 n 
2
Where D= X – A
For Grouped Data
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Var(X) =
2
S =
∑ fD
∑f
2
 ∑ fD 
−
 ∑ f 


2
3. Coding Method or Step-Deviation Method
For Ungrouped Data
 u 2  u 2 
∑ − ∑  
Var(X) = S 2 = h × 
 n  
 n

 

2
Where
u=
X−A
D
or
h
h
For Grouped Data
2
 fu 2 
 
fu
∑
∑

−
Var(X) = S 2 = h × 
 ∑ f  
 ∑f

 

2
Combined Variance ( Sc )
2
•
For two set of values
Sc
2
2
n1S12 + n2 S22
n1n2
+
X1 − X 2 )
=
2 (
n1 + n2
( n1 + n2 )
For three or more sets of data
2
Sc =
∑ n  S
i
2
i
( X −X)
i
∑n
2


i
•
Relative Measure of Dispersion
1. Coefficient Of Range
Coefficient of Range =
Xm − Xo
Xm + Xo
2. Coefficient Of Quartile Deviation
Coefficient of Q.D =
Where
Q3 − Q1
Q3 + Q1
Q1 = First, Lower quartile
Q3 = Third, Upper quartile
3. Coefficient Of Mean Deviation From Mean
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Coefficient of M.D from Mean =
Mean Deviation From Mean
Mean
Or
Coefficient of M.D from Mean =
M .D From X
X
4. Coefficient Of Mean Deviation From Median
Coefficient of M.D from Median =
Mean Deviation From Median
Median
Or
Coefficient of M.D from Mean =
M .D From X%
X%
5. Coefficient Of Standard Deviation
Coefficient of S.D =
S .D
X
6. Coefficient Of Variation (C.V)
“The coefficient of variation expresses the standard deviation as a percentage in terms of
arithmetic mean”. It is used as a criterion of consistent performance, the smaller
coefficient of variation, and the more consistent in the performance.
Or
“Coefficient of variation is used to compare the variability of two or more than two
series”.
Coefficient of Variation = C.V =
•
S .D
×100
X
Relationship Between Measures of Dispersion
1. For Normal Distribution
I. Mean Deviation = M.D = 0.7979 S.D
II. Quartile Deviation = Q.D = 0.6745 S.D
2. For Moderately Skewed Distribution
I. Mean Deviation = M.D =
3
S.D
4
II. Quartile Deviation = Q.D =
2
S.D
3
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III. Quartile Deviation = Q.D =
•
5
M.D
6
Moments
A moment designates the power to which deviation are raised before averaging them.
Methods of Standard Deviation
1. Moments about Mean or Central Moments
2. Moments about Origin or Zero
3. Moments about Provisional Mean or Arbitrary Value (Non Central Moment)
1. Moments about Mean or Central Moments
For Ungrouped Data
µ1 = m1 = ∑
( x− x)
n
=0
2
µ2 = m2
∑( x − x )
=
3
µ3 = m3
∑( x − x )
=
4
µ 4 = m4
∑( x − x )
=
= Variance
n
n
n
For Grouped Data
µ1 = m1 = ∑
f ( x−x)
∑f
=0
2
µ 2 = m2
∑ f ( x− x)
=
∑f
3
µ3 = m3
∑ f ( x− x)
=
∑f
4
µ4 = m4
∑ f ( x− x)
=
∑f
= Variance
2. Moments about Origin or Zero
For Ungrouped Data
µ '1 = m'1 = ∑
n
x
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µ 2 = m2
'
'
∑x
=
2
n
x3
µ '3 = m ' 3 = ∑
n
µ 4 = m4
'
'
∑x
=
4
n
For Grouped Data
µ '1 = m'1 = ∑
fx
∑f
µ 2 = m2
'
'
∑ fx
=
∑f
µ '3 = m ' 3 = ∑
2
fx3
∑f
µ 4 = m4
'
'
∑ fx
=
∑f
4
3. Moments about Provisional Mean or Arbitrary Value (Non Central Moment)
4.
Methods of Standard Deviation
i. Direct Method
ii. Short Cut Method
iii. Coding Method or Step-Deviation Method
i. Direct Method
For Ungrouped Data
µ '1 = m'1 = ∑
( x − A)
Where A is constant
n
2
µ 2 = m2
∑ ( x − A)
=
3
µ 3 = m3
∑ ( x − A)
=
4
µ 4 = m4
∑ ( x − A)
=
'
'
'
'
'
'
n
n
n
For Grouped Data
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µ '1 = m'1 = ∑
f ( x − A)
Where A is constant
∑f
2
µ 2 = m2
∑ f ( x − A)
=
∑f
3
µ 3 = m3
∑ f ( x − A)
=
∑f
4
µ 4 = m4
∑ f ( x − A)
=
∑f
'
'
'
'
'
'
ii. Short Cut Method
For Ungrouped Data
D
µ '1 = m'1 = ∑
n
Where D= X - A
µ 2 = m2
∑D
=
µ 3 = m3
∑D
=
'
'
'
'
2
n
3
n
D4
µ ' 4 = m' 4 = ∑
n
For Grouped Data
µ '1 = m'1 = ∑
fD
µ ' 2 = m' 2 = ∑
fD 2
Where D= X - A
∑f
∑f
µ 3 = m3
'
'
∑ fD
=
∑f
µ ' 4 = m' 4 = ∑
3
fD 4
∑f
iii. Coding Method or Step Deviation Method
For Ungrouped Data
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u
µ '1 = m'1 = ∑ × h
n
Where
u=
X −A
D
or
h
h
u2
µ ' 2 = m' 2 = ∑ × h 2
n
µ 3 = m3
'
'
∑u
=
3
× h3
n
u4
µ ' 4 = m' 4 = ∑ × h 4
n
For Grouped Data
µ '1 = m'1 = ∑
fu
µ ' 2 = m'2 = ∑
fu 2
∑f
×h
∑f
µ 3 = m3
∑ fu
=
∑f
µ 4 = m4
∑ fu
=
∑f
'
'
•
'
'
Where
u=
X −A
D
or
h
h
× h2
3
× h3
4
× h4
Relation Between Central moments in Terms of Non Central Moments
µ1 = m1 = µ '1 − µ1' = 0
µ2 = m2 = µ '2 − ( µ1' ) = Varaince
2
µ3 = m3 = µ '3 − 3µ1' µ 2' + 2 ( µ1' )
3
µ4 = m4 = µ '4 − 4 µ1' µ3' + 6 ( µ1' ) µ2 ' − 3 ( µ1' )
2
•
4
Moments – Ration
µ3 2
µ 23
µ
β 2 = b2 = 42
µ2
β1 = b1 =
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•
Sheppard’s Correction for Moments of Group Data
µ 2 (corrected ) = µ 2 (uncorrected ) −
h2
12
µ3 (corrected ) = µ3
µ 4 (corrected ) = µ4 (uncorrected ) −
•
Charliers Check
i.
∑ f ( u + 1) = ∑ fu + ∑ f
ii.
∑ f ( u + 1)
2
= ∑ fu 2 + 2∑ fu + ∑ f
iii.
∑ f ( u + 1)
3
= ∑ fu 3 + 3∑ fu 2 + 3∑ f u + ∑ f
iv.
•
h2
7 4
µ2 (uncorrected ) +
h
2
240
∑ f ( u + 1)
4
= ∑ fu 4 + 4∑ fu 3 + 6∑ f u 2 + 4∑ f u + ∑ f
Symmetry
In a symmetrical distribution a deviation below the mean exactly equals the corresponding
deviation above the mean. It is called symmetry.
For symmetrical distribution the following relations hold.
Mean = Median = Mode
Q3 - Median = Median - Q1
u3 = m3 = 0
β1 = b1 = 0
•
Skewness
Skewness is the lack of symmetry in a distribution around some central value i.e. means Median
or Mode. It is the degree of asymmetry.
Mean ≠ Median ≠ Mode
Q3 - Median ≠ Median - Q1
u3 = m3 ≠ 0
β1 = b1 ≠ 0
There are two types of Skewness.
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1. Positive Skewness
If the frequency curve has a longer tail to right, the distribution is said to be positively
skewed.
2. Negative Skewness
If the frequency curve has a longer tail to left, the distribution is said to negatively skewed.
•
Coefficient of Skewness (SK)
Karl Pearson’s Coefficient of Skewness
SK =
Mean − Mode
S .D
SK =
3 ( Mean − Median )
S .D
Bowly’s Quartile Coefficient of Skewness
SK =
Q3 + Q1 − 2 Median
Q3 − Q1
Moment Coefficient of Skewness
SK =
•
Kurtosis
Moment coefficient
β1 ( β 2 + 3)
2 ( 5β 2 − 6β1 − 9 )
β 2 is an important measure of kurtosis. These measures define as:
β 2 = b2 =
The moment coefficient
µ4
µ2 2
β 2 is a pure numbers and independent of the origin and unit of
measurement.
If
If
If
β 2 > 3 distribution is Leptokurtic
β 2 = 3 distribution is Normal or Mesokurtic
β 2 < 3 distribution is Platy Kurtic
Or
K=
Q.D
For Normal distribution, K = 0.263
P90 − P10
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