This document summarizes moments and moment systems. It defines moments as a method for summarizing descriptive statistical measures that represents variability and skewness analogous to moments in physics. It discusses notations used for moments, calculating moments about the mean and arbitrary points, and the relationships between central moments and moments about arbitrary points. The document also includes an example calculating first four moments about an arbitrary point of 120 for employee earnings data and converting them to moments about the mean. It determines the distribution is positively skewed based on the positive value of the third central moment.

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

2. 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

3. 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

4. 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

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

6. 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).

7. 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….

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

9. 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

10. 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

11. 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

12. 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

13. EXAMPLE
Following is the data on early earning(in rs)of
employees in a company:
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.
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

14. SOLUTION
Solution:-
class Mid
value(
m=x)
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

15. 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

16. 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

17. 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

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

19. THANK YOU ALL