Moments in statistics

MOMENTS
Subject: SMDM
Submitted to:
Prof. S.C.Singh
To be presented by
Sanjay Saw
Roll No-36
FMS-BHU

Sr. No content Slide
no
1 Moment system 4-5
2 Notations used 6
3 Moment about mean 7-8
4 Moment about arbitrary point 9-10
5 Relationship b/w central moments and moment about any
arbitrary point
11
6 Moment about zero or origin 12
7 Numerical problem 13-17
8 Reference 18

A quantity of data which by its mere bulk may
be incapable of entering the mind is to be
replaced by relatively few quantities which
shall adequately represent the whole or which
in other words, shall contain as much as
possible ,ideally the whole ,of the relevant
information contained in the original data.
R.A.Fisher

MOMENT SYSTEM
 Sub group of measurement of central
tendency i.e variability and skewness.
 Includes measurement like mean ,average
deviation ,std. deviation and so on.
 It is analogous to the term “moment”used in
physics.
 Size of class intervals represent the force
and
deviation of mid value of each class from the
observation represent the distance.

CONTD..
Moments basically represents a convenient
and unifying method for summarizing certain
descriptive statistical measures.

Notation of moments
 Greek letter (mu)
 if deviation are taken from the actual
mean.
 Greek letter (nu) or ’.
 if deviation are taken from some
assumed
mean (or arbitrary value other than zero).

MOMENTS ABOUT MEAN
let x1,x2,x3………,xn n observations with
mean x̅.
Then the rth moment about the actual mean of
a variable both for ungrouped and grouped
data is given by:
For ungrouped data:
r= 1/n*(x- x̅)^r r=1,2,3,4….

CONTD….
When r=1, 1=1/n* x ;A.M
r=2, 2= 1/n*(x- x̅
)^2 ;σ2(variance)
r=3, 3= 1/n*(x- x̅
)^3 ;indicates symmetry or asymmetry
r=4, 4= 1/n*(x- x̅
)^4 ;kurtosis(flatness) of the frequency
curve
For grouped data:
r= 1/N*f(x- x̅
)^r r=1,2,3,4; N= fi

MOMENTS ABOUT ARBITRARY POINTS
When from the data it is being feel that the
actual mean is bit difficult to find out or in
fractions.the moments are first calculated
about an assumed mean say A and then
converted about the actual mean
For grouped data
’r=1/ n* f(x-A)^r ;r=1,2,3,4

CONTD……
For ungrouped data
’r=1/ n* (x-A)^r ;r=1,2,3,4
For r=1,we have
’r=1/ n* (x-A) =1/n* (x)-A
= x̅ -A

RELATION BETWEEN CENTRAL MOMENTS AND
MOMENTS ABOUT ANY ARBITRARY POINTS
1=’1
2=’2-(’1)^2
3=’3-3’2’1+2(’1)^3
4=’4-4’3’1+6’2(’1)^2- 3(’1)4

MOMENT ABOUT ZERO OR ORIGIN
The moments about zero or origin are obtained
as follows:
Vr=1/n*fx^r ;r=1,2,3,4
The first moment about origin gives the mean
V1=1/n* fx ; A.M

EXAMPLE
Following is the data on early earning(in rs)of
employees in a company:
Earnin
g
50-70 70-90 90-110 110-130 130-150 150-170 170-190
No of
workers
4 8 12 20 6 7 3
Calculate the first four moments about the
point 120 .convert the results into moments
about the mean also find out the first moment
about origin.

SOLUTION
class Mid
value(
m=x)
Solution:-
Frequency(
f)
f*x d=(m-
A)/20
fd Fd^2 Fd^
3
Fd^4
50-70 60 4 240 -3 -12 36 -108 324
70-90 80 8 640 -2 -16 32 -64 128
90-110 100 12 1200 -1 -12 12 -12 12
110-130 120=A 20 2400 0 0 0 0 0
130-150 140 6 840 1 6 6 6 6
150-170 160 7 1120 2 14 28 56 112
170-190 180 3 540 3 9 27 81 243
60 6980 -11 141 -41 825

The moments about some arbitrary origin or point(A=120)is given by
’r=1/ n* f(x-A)^r ;for grouped data
=1/n*(fd^r)h^r ;d=(m-A)/h or (m-A)=hd here (x=mid value i.e=m)
For A=120 and x=m,we get
’1=1/n* fd*h=1/60*(-11)*20= -3.66
’2=1/n* fd^2*h^2=1/60*(141)*20^2=940
’3=1/n* fd^3*h^3=1/60*(-41)*20^3=-5466.66
’4=1/n* fd^4*h^4=1/60*(825)*20^4=2200000

CONTD..
The moments about actual mean (’2=940)is
given by
1=’1=-3.66
2=’2-(’1)^2=940-(-3.66)^2=926.55
3=’3-3’2’1+2(’1)^3=4774.83
4=’4-4’3’1+6’2(’1)^2- 3(’1)4
=2195107.20

CONTD…
Since 3 is positive ,therefore the given
distribution is positively skewed.
The moment about origin is
Vr=1/n*fx^r ; (x=mid value i.e=m)
When r=1; v1=1/60*6980=116.33334

REFERENCE
Business statistics 2nd edition by J.K.SHARMA

Moments in statistics

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