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 x ̅.

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- x ̅)^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:

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

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

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

THANK YOU ALL