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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
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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 $%cel Demos
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Objectives of Current Lecture
In the current lecture:
#elation b&w central moments and moments about ori!in
Moment #atios
S'ewness
urtosis
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Moments
moment is a *uantitative measure of the shape of a set of points+
,he first moment is called the mean which describes the center of the
distribution+
,he second moment is the variance which describes the spread of the
observations around the center+
ther moments describe other aspects of a distribution such as how the
distribution is s'ewed from its mean or pea'ed+
moment desi!nates the power to which deviations are raised before
avera!in! them+
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Central (or Mean) Moments
In mean moments- the deviations are ta'en from the mean+
For Ungroupe Data!
In .eneral-
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( )r Population Moment about Mean/
r
ith
r
x
N
µ µ
−=
∑
( )r Sample Moment about Mean/
r
ith
r
x xm
n
−=
∑
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Central (or Mean) Moments
Formula for "roupe Data!
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( )
( )
r Population Moment about Mean/
r Sample Moment about Mean/
r
ith
r
r
ith
r
f x
f
f x xm f
µ µ
−=
−=
∑∑
∑∑
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Central (or Mean) Moments
#$ample! Calculate first four moments about the mean for the followin! set
of e%amination mar's:
%olution! 0or solution- move to MS1$%cel+
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X
45
3237
46
39
36
41
4836
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Central (or Mean) Moments
()a$*le: Calculate: frst our moments about mean or theollowing reuenc! "istribution:
Solution: #or solution$ mo%e to &'()*cel+
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&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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Moments about (arbitrar) Origin
1,
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Moments about *ero
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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+
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X
45
3237
46
39
36
41
4836
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Moments about *ero
()a$*le: Calculate: frst our moments about -ero .origin/ orthe ollowing reuenc! "istribution:
Solution: #or solution$ mo%e to &'()*cel+
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&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
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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
= =
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+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+
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=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 @
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%,ewness
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( ) ( )< 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 =
=
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+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
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+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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+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+
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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+
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Review
>et?s review the main concepts:
Moments
Moments about Mean (Central Moments)
Moments about any arbitrary ri!in
Moments about "ero
#elated $%cel Demos
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Review
>et?s review the main concepts:
#elation b&w central moments and moments about ori!in
Moment #atios
S'ewnessurtosis
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1e$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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