Lecture 9 Moments

8/16/2019 Lecture 9 Moments

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Lecture 7

Dr. SHAHID QAISAR

MTH 262: Statistics andProbablity Teory

!"MSATS Institute o# In#or$ationTecnolo%y& Sai'al


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Review of Previous Lecture

In last lecture we discussed:

Measures of Dispersion

Variance and Standard Deviation

Coefficient of Variation

Properties of variance and standard deviation

2


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Objectives of Current Lecture

In the current lecture:

Moments

Moments about Mean (Central Moments)

Moments about any arbitrary ri!in

Moments about "ero

#elated $%cel Demos

3


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Objectives of Current Lecture

In the current lecture:

#elation b&w central moments and moments about ori!in

Moment #atios

S'ewness

urtosis

4


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Moments

moment is a *uantitative measure of the shape of a set of points+

,he first moment is called the mean which describes the center of the

distribution+

,he second moment is the variance which describes the spread of the

observations around the center+

ther moments describe other aspects of a distribution such as how the

distribution is s'ewed from its mean or pea'ed+

moment desi!nates the power to which deviations are raised before

avera!in! them+

5


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Central (or Mean) Moments

In mean moments- the deviations are ta'en from the mean+

For Ungroupe Data!

In .eneral-

6

( )r Population Moment about Mean/

r

ith

r

x

N

µ µ

−=

∑

( )r Sample Moment about Mean/

r

ith

r

x xm

n

−=

∑


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Formula for "roupe Data!

7

( )

( )

r Population Moment about Mean/

r Sample Moment about Mean/

r

ith

r

r

ith

r

f x

f

f x xm f

µ µ

−=

−=

∑∑

∑∑


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#$ample! Calculate first four moments about the mean for the followin! set

of e%amination mar's:

%olution! 0or solution- move to MS1$%cel+

8

X

45

3237

46

39

36

41

4836


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()a$*le: Calculate: frst our moments about mean or theollowing reuenc! "istribution:

Solution: #or solution$ mo%e to &'()*cel+

9

&eig'ts (grams) Freuenc (f)

23145 6

431785 78

7831795 7

7931755 78

7531725 3

7231745 5

7431985 3

,otal 28


10/29

Moments about (arbitrar) Origin

1,


11/29

Moments about *ero

11


12/29

Moments about *ero

#$ample! Calculate first four moments about ;ero (ori!in) for the followin!

set of e%amination mar's:

%olution! 0or solution- move to MS1$%cel+

12

X

45

3237

46

39

36

41

4836


13/29

Moments about *ero

()a$*le: Calculate: frst our moments about -ero .origin/ orthe ollowing reuenc! "istribution:

Solution: #or solution$ mo%e to &'()*cel+

13

&eig'ts (grams) Freuenc (f)

23145 6

431785 78

7831795 7

7931755 78

7531725 3

7231745 5

7431985 3

,otal 28


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Conversion from Moments about

Mean to Moments about Origion

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Moment Ratios

15

9

< 5

7 9< 9

9 9

- µ µ

β β µ µ

= =

9

< 57 9< 9

9 9

-m m

b b

m m

= =


16/29

+urtosis

arl Pearson introduced the term urtosis (literally the amount of hump) for

the de!ree of pea'edness or flatness of a unimodal fre*uency curve+

16

=hen the pea' of a curve

becomes relatively hi!h then that

curve is called >epto'urtic+

=hen the curve is flat1topped-

then it is called Platy'urtic+

Since normal curve is neithervery pea'ed nor very flat topped-

so it is ta'en as a basis for

comparison+

,he normal curve is called

Meso'urtic+


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%,ewness

distribution in which the values e*uidistant from the mean have e*ualfre*uencies and is called %mmetric Distribution+

ny departure from symmetry is called s,ewness+

In a perfectly smmetric istribution- Mean.Meian.Moe and the twotails of the distribution are e*ual in len!th from the mean+ ,hese values are

pulled apart when the distribution departs from symmetry and conse*uentlyone tail become lon!er than the other+

If ri!ht tail is lon!er than the left tail then the distribution is said to have

positive s,ewness+ In this case- Mean/Meian/Moe

If left tail is lon!er than the ri!ht tail then the distribution is said to havenegative s,ewness+ In this case- Mean0Meian0Moe

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%,ewness

=hen the distribution is symmetric- the value of s'ewness should be ;ero+

arl Pearson defined coefficient of S'ewness as:

Since in some cases- Mode doesn?t e%ist- so usin! empirical relation-

=e can write-

(it ran!es b&w 1< to @


19/29

%,ewness

19

( ) ( )< 9 9 7 7 9 < 7 <

< 7 < 7 < 7

9 9Q Q Q Q Q Q Q Q Median Q sk

Q Q Q Q Q Q

− − − − + − += = =− − −

<

<

<

<

- for population data

- for sample data

sk

m sk

s

µ

σ =

=


20/29

+urtosis

2,

59 9

9

5

9 99

- for population data

- for sample data

Kurt

m Kurt b

m

µ β

µ = =

= =

0or a normal distribution- 'urtosis is e*ual to


21/29

+urtosis

#$cess +urtosis (#+)! It is defined

as:

$/urtosis1<

0or a normal distribution- $/8+

=hen $A8- then the curve is saidto be >epto'urtic+

=hen $B8- then the curve is said

to be Platy'urtic+

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22/29

+urtosis

nother measure of urtosis- 'nown as Percentile coefficient of 'urtosis is:

=here-

+D is semi1inter*uartile ran!e/+D/(


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Describing a Freuenc Distribution

,o describe the maor characteristics of a fre*uency distribution- we need to

calculate the followin! five *uantities:

,he total number of observations in the data+

measure of central tendency (e+!+ mean- median etc+) that provides theinformation about the center or avera!e value+

measure of dispersion (e+!+ variance- SD etc+) that indicates the spread of

the data+

measure of s'ewness that shows lac' of symmetry in fre*uency

distribution+ measure of 'urtosis that !ives information about its pea'edness+

23


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Describing a Freuenc Distribution

It is interestin! to note that all these *uantities can be derived from the first

four moments+

0or e%ample-

,he first moment about ;ero is the arithmetic mean,he second moment about mean is the variance+

,he third standardi;ed moment is a measure of s'ewness+

,he fourth standardi;ed moment is used to measure 'urtosis+

,hus first four moments play a 'ey role in describin! fre*uency distributions+

24


25/29

Review

>et?s review the main concepts:

Moments

Moments about Mean (Central Moments)

Moments about any arbitrary ri!in

Moments about "ero

#elated $%cel Demos

25


26/29

Review

>et?s review the main concepts:

#elation b&w central moments and moments about ori!in

Moment #atios

S'ewnessurtosis

26


27/29

1e$t Lecture

In ne%t lecture- we will study:

2ntrouction to Probabilit

Definition an 3asic concepts of probabilit

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4uestions

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Lecture 9 Moments

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