2-The Normal & Standard Normal Distribution

7/23/2019 2-The Normal & Standard Normal Distribution

http://slidepdf.com/reader/full/2-the-normal-standard-normal-distribution 1/115

Anthony J Greene 1

The Normal Distribution &

Standard Normal DistributionI The Normal Distribution

A What is it?

B Why is it everywhere?

robability Theory is why

! The S"ewed Normal Distribution

D #urtosis

II The Standard Normal Distribution

A Standardi$in% a Normal DistributionB !omutin% roortions usin% Table B'1



Anthony J Greene (

A Normal Distribution)

!hest Si$es o* S+ottish ,ilitia ,en



Anthony J Greene -


.isto%ram o* .uman Gestation



Anthony J Greene /

The Normal Distribution) .ei%ht



Anthony J Greene 0


A%e At etirement



Anthony J Greene 2

Normally Distributed 3ariables

4 The most +ommon +ontinuous 5interval6ratio7 variable tye

4 8++urs redominantly in nature 5biolo%y9 sy+holo%y9 et+'74 Determined by the rin+iles o* robability



Anthony J Greene :

robability and the Normal

Distributionrobability is the ;nderlyin% !ause o* the

Normal Distribution



Anthony J Greene <

ossible out+omes

*or *our +oin tosses

.... ...T ..T. ..TT

.T.. .T.T .TT. .TTT

T... T..T T.T. T.TTTT.. TT.T TTT. TTTT

There are 12 ossibilities be+ause there are (

ossible out+omes *or ea+h toss and / tosses) (/

In %eneral the ossible out+omes are mn where m is

the number o* out+omes er event and n is the

number o* events



Anthony J Greene =

robability distribution o* the number

o* heads obtained in / +oin tosses

No' o* .eads

x

robability

P(X=x)

> >'>2(0 1612

1 >'(0>> /612

( >'-:0> 2612

- >'(0>> /612/ >'>2(0 1612

1'>>>> 1



Anthony J Greene 1>

robability distribution o* the

number o* heads obtained in /

+oin tosses



Anthony J Greene 11

@reuen+ies *or the numbers o*

heads obtained in / tosses *or

1>>> observations No' o* .eads

x

robability P(X=x)

8bserved@reAuen+y

> >'>2(0 2/

1 >'(0>> (/<

( >'-:0> -=(

- >'(0>> (2<

/ >'>2(0 (<

1'>>>> 1



Anthony J Greene 1(

5a7 robability *or / +oin *lis vs'

5b7 1>>> observations



Interretation o* a Normal

Distribution in terms o* robabilityConsider what would happen if there were only 4 genes for height

(there are more), each of which has only 2 possible states (like heads

versus tails for a coin), call the states T for tall and S for short. The

distributions would be identical to that for the coin tosses (see leftbelow) with the possibility of 0, 1, 2, 3, and 4 T’s. In reality height

is controlled by many genes so that more than 5 outcomes are

possible (see right below).



Anthony J Greene 1/

And *or 2 +oins instead o* /?



Anthony J Greene 10

Another Camle

( Di+e

ossible out+omes)

191 19( 19- 19/ 190 192(91 (9( (9- (9/ (90 (92

-91 -9( -9- -9/ -90 -92

/91 /9( /9- /9/ /90 /92

091 09( 09- 09/ 090 092

291 29( 29- 29/ 290 292



Anthony J Greene 12

Another Camle

x f 5 x7

( 1

- (

/ -

0 /

2 0

: 2

< 0

= /

1> -

11 (

1( 1



Anthony J Greene 1:

Camles o* the Normal

Distribution4 A%e

4 .ei%ht

4 Wei%ht

4 I''

4 Si+" Days er Eear

4 .ours Slee er Ni%ht4 Words ead er

,inute

4 !alories aten er Day

4 .ours o* Wor" Done

er Day4 yeblin"s er .our

4 Insultin% emar"s er

Wee"

4 Number o* airs o*

So+"s 8wned



Anthony J Greene 1<

The S"ewed Normal Distribution



Anthony J Greene 1=

Camles o* S"ewed Normal

Distributions4 In+ome

4 Number o* mty

Soda !ans in !ar 4 Dru% ;se er Wee"

4 !ar A++idents er Eear

4 Fi*etime.ositali$ations

4 Number o* Guitars

8wned

4 !onse+utive Days;nemloyed

4 .andWashin%s er

Day

4 Number o* Fan%ua%esSo"en @luently

4 .ours o* T'3' er Day



Anthony J Greene (>

Grah o* a Normal Distribution

-/'1-H

1-'0=H

('(<H

-/'1-H

1-'0=H

('(<H



Anthony J Greene (1

Shaes o* the Normal

Distribution#urtosis

4 Feto"urti+

4 lato"urti+



Anthony J Greene ((

@reuen+y and relative*reuen+y

distributions *or hei%hts



Anthony J Greene (-

What do we do with Normal

Distributions?1' Determine the osition o* a %iven s+ore

relative to all other s+ores'

(' !omare distributions'



Anthony J Greene (/

elative*reuen+y

histo%ram *or hei%hts



Anthony J Greene (0

!omarin% Two Distributions

Two distributions o* eCam s+ores' @or both distributions9 :>9

but *or one distribution9 1(' The osition o* X :2 is very

di**erent *or these two distributions'



Anthony J Greene (2

Data Trans*ormations are

eversible and Do not Alter the

elations Amon% Items

17 Add or Subtra+t a !onstant @rom a+h

S+ore(7 ,ultily a+h S+ore By a !onstant

4 e'%'9 i* you wanted to +onvert a %rou o*

@ahrenheit temeratures to !enti%rade youwould subtra+t -( *rom ea+h s+ore then multily

by 06=ths



Anthony J Greene (:

Trans*ormin% a distribution does not +han%e the

shae o* the distribution9 only its units



Anthony J Greene (<

.ei%ht a7 in in+hes b7 in +entimeters

in+hes K ('0/ +entimeters

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

5 6

6 0

6 4

6 8

7 2

7 6

8 0

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

1 4 2

1 5 2

1 6 2

1 7 2

1 8 2

1 9 2

2 0 2



Anthony J Greene (=

Trans*ormations



Anthony J Greene ->

Standard Normal Distribution

A normally distributed variable having mean 0 and standard

deviation 1 is said to have the standard normal distribution.

Its associated normal curve is called the standard normal

curve.



Anthony J Greene -1

Transformation to Standard Units

The idea is to trans*orm 5reversibly7 any normal distribution

into a STANDARD NORMAL distribution with μ > and σ

1



Anthony J Greene -(

Standardi$ed Normally Distributed

3ariableA normally distributed variable9 x, is +onverted to a standard

normal distribution9 z 9 with the *ollowin% *ormula

σ

µ −= x

z



Anthony J Greene --

Standardi$in% normal distributions



Anthony J Greene -/

Standard Normal Distribution

• For a variable x, the variable ( z-score)

• is called the standardized version of x or the standardized variable corresponding to the

variable x.

• This transformation is standard for any variable

and preserves the exact relationships among the

scores

σ

µ −=

x z



Anthony J Greene -0

Standard Normal Distributions

4 The z s+ore trans*ormation is entirely

reversible but allows any distribution to be

+omared 5e'%'9 I'' and SAT s+oreL does a toI'' s+ore +orresond to a to SAT s+ore?7

4 z s+ores all have a mean o* $ero and a standard

deviation o* 19 whi+h %ives them the simlest ossible mathemati+al roerties'



Anthony J Greene -2

Standard Normal Distributions

An eCamle o* a z trans*ormation *rom a

variable 5 x7 with mean - and standard

deviation (



Anthony J Greene -:

;nderstandin% x and z s+ores



Anthony J Greene -<

Basi+ roerties o* the Standard

Normal !urveProperty 1: The total area under the standard normal curve

is equal to 1.

Property 2: The standard normal curve extends indefinitely

in both directions, approaching, but never touching, thehorizontal axis as it does so.

Property 3: The standard normal curve is symmetric about

0; that is, the left side of the curve should be a mirror image

of the right side of the curve.Property 4: Most of the area under the standard normal

curve lies between –3 and 3.



Anthony J Greene -=

@indin% er+enta%es *or a normally

distributed variable *rom areasunder the standard normal +urve

Be+ause the standard normal distribution is the same *or all

variables9 it is an easy way to determine what roortion o* s+ores

is less than a9 what roortion lies between a and b9 and what

roortion is %reater than b 5*or any distribution and any desired

oints a and b7'



Anthony J Greene />

The relationshi between $s+ore values

and lo+ations in a oulation

distribution'



Anthony J Greene /1

The KaCis is relabeled in $s+ore units' The distan+e that is

euivalent to σ +orresonds to 1 oint on the z s+ore s+ale'



TableB'1

' 2<:



Anthony J Greene /-

TableB'1

A

!loser

Foo"

Th N l Di t ib ti



Anthony J Greene //

dx

d e P

x

x

X

∫ −−=

(

1

(( (675

((

1 σ µ

πσ

The Normal Distribution)

why use a table?



Anthony J Greene /0

rom x or z to P

To determine a er+enta%e or

robability *or a normally

distributed variable

Ste 1 S"et+h the normal +urve asso+iated with the variable

Ste ( Shade the re%ion o* interest and mar" the delimitin% Cvalues

Ste - !omute the z s+ores *or the delimitin% xvalues *ound

in Ste (

Ste / ;se Table B'1 to obtain the area under the standard

normal +urve delimited by the z s+ores *ound in Ste -

;se Geometry and remember that the total area under

the +urve is always 1'>>'



Anthony J Greene /2

rom x or z to P @indin% er+enta%es *or a normally

distributed variable *rom areas under

the standard normal +urve



Anthony J Greene /:

@indin% er+enta%es *or a normally

distributed variable *rom areas under the

standard normal +urve

1' µ 9 σ are %iven'

(' a and b are any two values o* the variable x.

-' !omute $s+ores *or a and b'

/' !onsult table B1

0' ;se %eometry to *ind desired area'



Anthony J Greene /<

Given that a ui$ has a mean s+ore o* 1/

and an s'd' o* -9 what roortion o* the

+lass will s+ore between = & 12?

1' µ 1/ and σ -'

2. a = and b 12.

3. z a 06- 1'2:9 z b (6- >'2:'

/' In table B'19 we see that the area to the le*t o* a is >'>/:0

and that the area to the ri%ht o* b is >'(01/'

0' The area between a and b is there*ore

1 M 5>'>/:0 >'(01/7 >':>1 or :>'>1H



Anthony J Greene /=

@indin% the area under the standard

normal +urve to the le*t o* z 1'(-



Anthony J Greene 0>

What i* you start with x

instead o* z ?

z = 1'0>) ;se !olumn !L P >'>22<

!"at is t"#

$ro%a%i&it' of

s#&#(tin) arandom

st*d#nt +"o

s(or#d a%o,#

650 on t"#SAT-



Anthony J Greene 01


normal +urve to the ri%ht o* z >':2

The easiest way would be to use !olumn !9 but lets use

!olumn B instead



Anthony J Greene 0(


normal +urve that lies between z = M>'2< and z = 1'<(

8ne Strate%y) Start with the area to the le*t o* 1'<(9 then

subtra+t the area to the ri%ht o* >'2<'

P = 1 – 0.0344 – 0.2483

= 0.7173

Se+ond Strate%y) Start with 1'>> and subtra+t o** the two

tails



Anthony J Greene 0-

Determination o* the er+enta%e o*

eole havin% Is between 110 and1/>

rom x or z to P



Anthony J Greene 0/

rom x or z to P

eview o* Table B'1 thus *ar ;sin% Table B'1 to *ind the area under the standard normal

+urve that lies

5a7 to the le*t o* a se+i*ied z s+ore9

5b7 to the ri%ht o* a se+i*ied z s+ore9

5+7 between two se+i*ied z s+ores

Then i* x is as"ed *or9 +onvert *rom z to x



Anthony J Greene 00

rom P to z or x Now the other way around

To determine the observations

+orresondin% to a se+i*ied

er+enta%e or robability *or a

normally distributed variable

Ste 1 S"et+h the normal +urve asso+iated the the variable

Ste ( Shade the re%ion o* interest 5%iven as a robability or area

Ste - ;se Table B'1 to obtain the z s+ores delimitin% the re%ion

in Ste (

Ste / 8btain the xvalues havin% the z s+ores *ound in Ste -



Anthony J Greene 02

rom P to z or x

@indin% z- or xs+ores +orresondin%to a %iven re%ion'@indin% the z s+ore havin% area >'>/ to its le*t

;se !olumn !)

The z +orresondin% to >'>/ in the le*t tail is 1':0

x = σ × z + μ

I* μ is (/( σ is 1>>9 then

x = 1>> × 1':0 (/(

x 2:

µ σ

σ

µ

+×=

−=

z x

x z



Anthony J Greene 0:

The z α

Notation

The symbol z O is used to denote the z s+ore havin% area O 5alha7 to its

ri%ht under the standard normal

+urve' We read P z OQ as P z sub OQ or

more simly as P z O'Q



Anthony J Greene 0<

The z α notation ) P(X>x) = α

This is the z s+ore that

demar"s an area under the

+urve with P(X>x)= α

P(X>x)= α



Anthony J Greene 0=

The z α notation ) P(X<x) = α



+urve with P(X<x)= α

P(X<x)= α

Z



Anthony J Greene 2>

The z α notation ) P 5R X|>|x|7 = α



+urve with P 5R X|>|x|7= α

P 5R X|>|x|7= α

/2 /21



Anthony J Greene 21

@indin% z >'>(0

;se !olumn !)

The z +orresondin% to >'>(0 in the ri%ht tail is 1'=2



Anthony J Greene 2(

@indin% z >'>0

;se !olumn !)

The z +orresondin% to >'>0 in the ri%ht tail is 1'2/



Anthony J Greene 2-

@indin% the two z s+ores dividin% thearea under the standard normal +urve

into a middle >'=0 area and two outside

>'>(0 areas

;se !olumn !)

The z +orresondin% to >'>(0 in both tails is 1'=2



Anthony J Greene 2/

@indin% the =>th er+entile *or Is

z >'1> 1'(<

z = 5 x-μ76σ

1'(< 5 x – 1>>7612

120.48 x



Anthony J Greene 20

What you should be able to do

1' Start with z or x-s+ores and +omute re%ions

(' Start with re%ions and +omute z or xs+ores

µ σ

σ

µ

+×=

−=

z x

x z



Anthony J Greene 22

DS!ITI3S

K!IS & 3IW

D i i



Anthony J Greene

Des+ritives

1' Nonarametri+ Statisti+s)

a7 @reuen+y & er+entile

b7 ,edian9 an%e9 Interuartile an%e9 Semi

Interuartile an%e

(' arametri+ Statisti+s)

a7 ,ean9 3arian+e9 Standard Deviation

b7 $s+ore & roortion

NonWeekly Income

540



Non

arametri+

Analysis

540

275

680

8275

425

380

2370

4185

155

0

490

380265

145

755

125

430

675125

155

185

505

425

785

NonWeekly Income Sorted Scores

540 0



Non

arametri+

Analysis

540 0

275 125

680 125

8275 145

425 155

380 155

2370 185

4185 265

155 275

0 380

490 380

380 425265 425

145 430

755 490

125 505

430 540

675 675125 680

155 755

185 785

505 2370

425 4185

785 8275

NonWeekly Income Sorted Scores

540 0



Non

arametri+

Analysis

540 0

275 125

680 125

8275 145

425 155

380 155

2370 185

4185 265

155 275

0 380

490 380

380 425265 425

145 430

755 490

125 505

430 540

675 675125 680

155 755

185 785

505 2370

425 4185

785 8275

an%e .F1

<(:2

or

;FFF

<(:0'05>'07

<(:2

NonWeekly Income Sorted Scores 25%, 50%, 75%

540 0



Non

arametri+

Analysis

540 0

275 125

680 125

8275 145

425 155

380 155 155

2370 185 185

4185 265

155 275

0 380

490 380

380 425 425265 425 425

145 430

755 490

125 505

430 540

675 675 675125 680 680

155 755

185 785

505 2370

425 4185

785 8275

1) (06/ or 2

1) o* the distan+e

between 100 and 1<01 12('0

( /(0 median

-) :06/ or 1<

-) o* the distan+e

between 2:0 and 2<>

- 2:2'(0


540 0



Non

arametri+

Analysis

540 0

275 125

680 125

8275 145

425 155

380 155 155

2370 185 185

4185 265

155 275

0 380

490 380

380 425 425265 425 425

145 430

755 490

125 505

430 540

675 675 675125 680 680

155 755

185 785

505 2370

425 4185

785 8275

1 12('0

( /(0 median

- 2:2'(0

I 01-':0


540 0

Weekly Income Proportion

540 0 000



Non

arametri+

Analysis

540 0

275 125

680 125

8275 145

425 155

380 155 155

2370 185 185

4185 265

155 275

0 380

490 380

380 425 425265 425 425

145 430

755 490

125 505

430 540

675 675 675125 680 680

155 755

185 785

505 2370

425 4185

785 8275

540 0 000

275 125 008

680 125 008

8275 145 013

425 155 021

380 155 155 021

2370 185 185 025

4185 265 029

155 275 033

0 380 042

490 380 042

380 425 425 050265 425 425 050

145 430 054

755 490 058

125 505 063

430 540 067

675 675 675 071125 680 680 075

155 755 079

185 785 083

505 2370 088

425 4185 092

785 8275 096


540 0

Weekly Income Proportion

540 0 000



Non

arametri+

Analysis

540 0

275 125

680 125

8275 145

425 155

380 155 155

2370 185 185

4185 265

155 275

0 380

490 380

380 425 425265 425 425

145 430

755 490

125 505

430 540

675 675 675125 680 680

155 755

185 785

505 2370

425 4185

785 8275

540 0 000

275 125 008

680 125 008

8275 145 013

425 155 021

380 155 155 021

2370 185 185 025

4185 265 029

155 275 033

0 380 042

490 380 042

380 425 425 050265 425 425 050

145 430 054

755 490 058

125 505 063

430 540 067

675 675 675 071125 680 680 075

155 755 079

185 785 083

505 2370 088

425 4185 092

785 8275 096

arametri+ !o"rs Work

48



arametri+

Analysis

5samle7

48

36

72

4

40

36

30

34

40

42

45

60

61

25

29

41

45

55

31

49


48



Analysis48

36

72

4

40

36

30

34

40

42

45

60

61

25

29

41

45

55

31

49

82300Σ


48



Analysis48

36

72

4

40

36

30

34

40

42

45

60

61

25

29

41

45

55

31

49

82300

4115

ΣΣ6n


48



Analysis48

36

72

4

40

36

30

34

40

42

45

60

61

25

29

41

45

55

31

49

82300

4115

ΣΣ6n

M

arametri+



Analysis

ΣΣ6n

arametri+ (



Analysis

ΣΣ6n

ΣΣ65n17

arametri+ (



Analysis

ΣΣ6n

SSΣ

Σ65n17

arametri+ (



Analysis

ΣΣ6n

ΣΣ65n17

arametri+

(



arametri+

Analysis

ΣΣ6n

3arian+e ΣΣ65n17

arametri+ (



Analysis

ΣΣ6n

srt

ΣΣ65n17

arametri+ (



Analysis

ΣΣ6n

srt

ΣΣ65n17s

arametri+ (



Analysis

ΣΣ6n

srt

$

ΣΣ65n17

arametri+ (



Analysis

ΣΣ6n

srt

ΣΣ65n17

0.524

arametri+ (



Analysis

ΣΣ6n

srt

ΣΣ65n17

0.005

arametri+ (



Analysis

ΣΣ6n

srt

ΣΣ65n17

0. 984

arametri+ #$%m Score

76



Analysis

5oulation7

83

81

90

93

88

85

52

90

91

88

95

61

90

100

93

45

8083

74


76



Analysis83

81

90

93

88

85

52

90

91

88

95

61

90

100

93

45

8083

74

163800Σ


76



Analysis83

81

90

93

88

85

52

90

91

88

95

61

90

100

93

45

8083

74

163800

8190

ΣΣ6N


76



Analysis83

81

90

93

88

85

52

90

91

88

95

61

90

100

93

45

8083

74

163800

8190

ΣΣ6N

µ


76 &590

Kµ



Analysis83 110

81 &090

90 810

93 1110

88 610

85 310

52 &2990

90 810

91 910

88 610

95 1310

61 &2090

90 810

100 1810

93 1110

45 &3690

80 &19083 110

74 &790

163800

8190

ΣΣ6N

#$%m Score

76 &590 348100

5Kµ7arametri+ Kµ (



83 110 12100

81 &090 08100

90 810 656100

93 1110 1232100

88 610 372100

85 310 96100

52 &2990 8940100

90 810 656100

91 910 828100

88 610 372100

95 1310 1716100

61 &2090 4368100

90 810 656100

100 1810 3276100

93 1110 1232100

45 &3690 13616100

80 &190 3610083 110 12100

74 &790 624100

163800

8190

Analysis

ΣΣ6N


76 &590 348100

83 1 10 1 2100

5Kµ7Kµ (



Analysis83 110 12100

81 &090 08100

90 810 656100

93 1110 1232100

88 610 372100

85 310 96100

52 &2990 8940100

90 810 656100

91 910 828100

88 610 372100

95 1310 1716100

61 &2090 4368100

90 810 656100

100 1810 3276100

93 1110 1232100

45 &3690 13616100

80 &190 3610083 110 12100

74 &790 624100

163800 390580

8190

ΣΣ6N


76 &590 348100

83 1 10 1 2100

5Kµ7Kµ (




81 &090 08100

90 810 656100

93 1110 1232100

88 610 372100

85 310 96100

52 &2990 8940100

90 810 656100

91 910 828100

88 610 372100

95 1310 1716100

61 &2090 4368100

90 810 656100

100 1810 3276100

93 1110 1232100

45 &3690 13616100

80 &190 3610083 110 12100

74 &790 624100

163800 390580

8190

ΣΣ6N

SS


76 &590 348100

83 1 10 1 2100

5Kµ7Kµ (




81 &090 08100

90 810 656100

93 1110 1232100

88 610 372100

85 310 96100

52 &2990 8940100

90 810 656100

91 910 828100

88 610 372100

95 1310 1716100

61 &2090 4368100

90 810 656100

100 1810 3276100

93 1110 1232100

45 &3690 13616100

80 &190 3610083 110 12100

74 &790 624100

163800 390580

8190 19529

ΣΣ6N


76 &590 348100

83 1 10 1 2100

5Kµ7Kµ (




81 &090 08100

90 810 656100

93 1110 1232100

88 610 372100

85 310 96100

52 &2990 8940100

90 810 656100

91 910 828100

88 610 372100

95 1310 1716100

61 &2090 4368100

90 810 656100

100 1810 3276100

93 1110 1232100

45 &3690 13616100

80 &190 3610083 110 12100

74 &790 624100

163800 390580

8190 19529

ΣΣ6N

3arian+e


76 &590 348100

83 1 10 1 2100

5Kµ7Kµ (




81 &090 08100

90 810 656100

93 1110 1232100

88 610 372100

85 310 96100

52 &2990 8940100

90 810 656100

91 910 828100

88 610 372100

95 1310 1716100

61 &2090 4368100

90 810 656100

100 1810 3276100

93 1110 1232100

45 &3690 13616100

80 &190 3610083 110 12100

74 &790 624100

163800 390580

8190 19529

1397

ΣΣ6N

srt

arametri+

l i

#$%m Score

76 &590 348100

83 1 10 1 2100

5Kµ7Kµ (5Kµ76 σ




81 &090 08100

90 810 656100

93 1110 1232100

88 610 372100

85 310 96100

52 &2990 8940100

90 810 656100

91 910 828100

88 610 372100

95 1310 1716100

61 &2090 4368100

90 810 656100

100 1810 3276100

93 1110 1232100

45 &3690 13616100

80 &190 3610083 110 12100

74 &790 624100

163800 390580

8190 19529

1397

ΣΣ6N

srt

σ

arametri+

A l i

#$%m Score

76 &590 348100 &0422

83 1 10 1 2100 0 079

5Kµ7Kµ (5Kµ76 σ



Analysis83 110 12100 0079

81 &090 08100 &0064

90 810 656100 0580

93 1110 1232100 0794

88 610 372100 0437

85 310 96100 0222

52 &2990 8940100 &2140

90 810 656100 0580

91 910 828100 0651

88 610 372100 0437

95 1310 1716100 0937

61 &2090 4368100 &1496

90 810 656100 0580

100 1810 3276100 1295

93 1110 1232100 0794

45 &3690 13616100 &2641

80 &190 36100 &013683 110 12100 0079

74 &790 624100 &0565

163800 390580

8190 19529

1397

ΣΣ6N

srt

$

arametri+

A l i

#$%m Score

76 &590 348100 &0422

83 1 10 1 2100 0 079

5Kµ7Kµ (5Kµ76 σ



Analysis83 110 12100 0079

81 &090 08100 &0064 0476

90 810 656100 0580

93 1110 1232100 0794

88 610 372100 0437

85 310 96100 0222

52 &2990 8940100 &2140

90 810 656100 0580

91 910 828100 0651

88 610 372100 0437

95 1310 1716100 0937

61 &2090 4368100 &1496

90 810 656100 0580

100 1810 3276100 1295

93 1110 1232100 0794

45 &3690 13616100 &2641

80 &190 36100 &013683 110 12100 0079

74 &790 624100 &0565

163800 390580

8190 19529

1397

ΣΣ6N

srt

arametri+

A l i

#$%m Score

76 &590 348100 &0422

83 1 10 1 2100 0 079

5Kµ7Kµ (5Kµ76 σ



Analysis83 110 12100 0079

81 &090 08100 &0064

90 810 656100 0580

93 1110 1232100 0794

88 610 372100 0437

85 310 96100 0222

52 &2990 8940100 &2140

90 810 656100 0580

91 910 828100 0651

88 610 372100 0437

95 1310 1716100 0937

61 &2090 4368100 &1496

90 810 656100 0580

100 1810 3276100 1295

93 1110 1232100 0794

45 &3690 13616100 &2641 0004

80 &190 36100 &013683 110 12100 0079

74 &790 624100 &0565

163800 390580 000

8190 19529

1397

ΣΣ6N

srt

What roortion

o* s+ores is

below /0?

>'>>/Above?

>'==2

arametri+

A l i

#$%m Score

76 &590 348100 &0422

83 1 10 1 2100 0 079

5Kµ7Kµ (5Kµ76 σ



Analysis83 110 12100 0079

81 &090 08100 &0064

90 810 656100 0580

93 1110 1232100 0794

88 610 372100 0437

85 310 96100 0222

52 &2990 8940100 &2140

90 810 656100 0580

91 910 828100 0651

88 610 372100 0437

95 1310 1716100 0937

61 &2090 4368100 &1496

90 810 656100 0580

100 1810 3276100 1295 0902

93 1110 1232100 0794

45 &3690 13616100 &2641

80 &190 36100 &013683 110 12100 0079

74 &790 624100 &0565

163800 390580 000

8190 19529

1397

ΣΣ6N

srt

arametri+

A l i

#$%m Score

76 &590 348100 &0422

83 1 10 1 2100 0 079

5Kµ7Kµ (5Kµ76 σ



Analysis83 110 12100 0079

81 &090 08100 &0064

90 810 656100 0580

93 1110 1232100 0794

88 610 372100 0437

85 310 96100 0222

52 &2990 8940100 &2140

90 810 656100 0580

91 910 828100 0651

88 610 372100 0437

95 1310 1716100 0937

61 &2090 4368100 &1496

90 810 656100 0580

100 1810 3276100 1295 0902

93 1110 1232100 0794

45 &3690 13616100 &2641 0004

80 &190 36100 &013683 110 12100 0079

74 &790 624100 &0565

163800 390580 000

8190 19529

1397

ΣΣ6N

srt

What roortiono* s+ores is

between 1>> and

/0?

>'=>( M >'>>/

>'<=<



Anthony J Greene 1><

!"at s(or# (orr#s$onds to t"# To$ 103-

!"at s(or#s (orr#s$ond to t"# Midd&# 603-

1'(<

>'</

i f 58 d d Of 10 " i "



Anthony J Greene 1>=

i,#n a m#an of 58 and a st. d#,. Of 10 +"at is t"#

&i#&i"ood of random&' %#in) %#t+##n 55 and 65-

!an use +olumn D) >'11:= >'(0<> >'-:0=

i f 58 d t d Of 10 " t i t"



Anthony J Greene 11>

i,#n a m#an of 58 and a st. d#,. Of 10 +"at is t"#

&i#&i"ood of random&' %#in) %#t+##n 65 and 75-

!an use +olumn !) >'(/(> >'>//2 >'1=:/



Anthony J Greene 111

!"at S(or#s ar# in t"# To$ 153-

z = 1'>/

;se +olumn ! x = z U V

x = 2>/



Anthony J Greene 11(

!"at S(or#s ar# t"# midd&# 803-

!an ;se !olumn D) z = ±1'(<

-:( 92(<

-:( 2(<

0>>1(< 0>>1(<

0>>1>>(<'1

=

==

+−=+=

+×±=

+×=

−=

x

x x

x x

x

z x

x

z

µ σ

σ

µ



Anthony J Greene 11-

!"at is t"# $#r(#nt of t"# $o$*&ation t"at &i#s %#&o+ 114-

;se !olumn B) z = 1'/>L P >'=1=(



Anthony J Greene 11/

!"at is t"# $#r(#nti&# ran of x 92-

;se !olumn !) z = >'<>L P = >'(11=

!"at s(or# (orr#s$onds to t"# %ottom 43-



$

;se !olumn B) z '/1

x = z × σ + μ x = > /1 U 0 2> 0: =0

2-The Normal & Standard Normal Distribution

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Transcript of 2-The Normal & Standard Normal Distribution