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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$in% a Normal DistributionB !omutin% roortions usin% Table B'1
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Anthony J Greene (
A Normal Distribution)
!hest Si$es o* S+ottish ,ilitia ,en
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Anthony J Greene -
A Normal Distribution)
.isto%ram o* .uman Gestation
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Anthony J Greene /
The Normal Distribution) .ei%ht
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Anthony J Greene 0
A Normal Distribution)
A%e At etirement
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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
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Anthony J Greene :
robability and the Normal
Distributionrobability is the ;nderlyin% !ause o* the
Normal Distribution
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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
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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
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Anthony J Greene 1>
robability distribution o* the
number o* heads obtained in /
+oin tosses
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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
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Anthony J Greene 1(
5a7 robability *or / +oin *lis vs'
5b7 1>>> observations
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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).
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Anthony J Greene 1/
And *or 2 +oins instead o* /?
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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
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Anthony J Greene 12
Another Camle
x f 5 x7
( 1
- (
/ -
0 /
2 0
: 2
< 0
= /
1> -
11 (
1( 1
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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
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Anthony J Greene 1<
The S"ewed Normal Distribution
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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
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Anthony J Greene (>
Grah o* a Normal Distribution
-/'1-H
1-'0=H
('(<H
-/'1-H
1-'0=H
('(<H
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Anthony J Greene (1
Shaes o* the Normal
Distribution#urtosis
4 Feto"urti+
4 lato"urti+
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Anthony J Greene ((
@reuen+y and relative*reuen+y
distributions *or hei%hts
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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'
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Anthony J Greene (/
elative*reuen+y
histo%ram *or hei%hts
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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'
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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
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Anthony J Greene (:
Trans*ormin% a distribution does not +han%e the
shae o* the distribution9 only its units
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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
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Anthony J Greene (=
Trans*ormations
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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.
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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
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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
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Anthony J Greene --
Standardi$in% normal distributions
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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
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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'
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Anthony J Greene -2
Standard Normal Distributions
An eCamle o* a z trans*ormation *rom a
variable 5 x7 with mean - and standard
deviation (
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Anthony J Greene -:
;nderstandin% x and z s+ores
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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.
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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'
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Anthony J Greene />
The relationshi between $s+ore values
and lo+ations in a oulation
distribution'
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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'
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TableB'1
' 2<:
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Anthony J Greene /-
TableB'1
A
!loser
Foo"
Th N l Di t ib ti
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Anthony J Greene //
dx
d e P
x
x
X
∫ −−=
(
1
(( (675
((
1 σ µ
πσ
The Normal Distribution)
why use a table?
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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'>>'
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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
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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 $s+ores *or a and b'
/' !onsult table B1
0' ;se %eometry to *ind desired area'
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Anthony J Greene /<
Given that a ui$ 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
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Anthony J Greene /=
@indin% the area under the standard
normal +urve to the le*t o* z 1'(-
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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-
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Anthony J Greene 01
@indin% the area under the standard
normal +urve to the ri%ht o* z >':2
The easiest way would be to use !olumn !9 but lets use
!olumn B instead
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Anthony J Greene 0(
@indin% the area under the standard
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
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Anthony J Greene 0-
Determination o* the er+enta%e o*
eole havin% Is between 110 and1/>
rom x or z to P
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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
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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 -
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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
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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
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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
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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
Z
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Anthony J Greene 2>
The z α notation ) P 5R X|>|x|7 = α
This is the z s+ore that
demar"s an area under the
+urve with P 5R X|>|x|7= α
P 5R X|>|x|7= α
/2 /21
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Anthony J Greene 21
@indin% z >'>(0
;se !olumn !)
The z +orresondin% to >'>(0 in the ri%ht tail is 1'=2
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Anthony J Greene 2(
@indin% z >'>0
;se !olumn !)
The z +orresondin% to >'>0 in the ri%ht tail is 1'2/
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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
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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
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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
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Anthony J Greene 22
DS!ITI3S
K!IS & 3IW
D i i
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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
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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
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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
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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
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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
NonWeekly Income Sorted Scores 25%, 50%, 75%
540 0
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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
NonWeekly Income Sorted Scores 25%, 50%, 75%
540 0
Weekly Income Proportion
540 0 000
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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
NonWeekly Income Sorted Scores 25%, 50%, 75%
540 0
Weekly Income Proportion
540 0 000
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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
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arametri+
Analysis
5samle7
48
36
72
4
40
36
30
34
40
42
45
60
61
25
29
41
45
55
31
49
arametri+ !o"rs Work
48
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Analysis48
36
72
4
40
36
30
34
40
42
45
60
61
25
29
41
45
55
31
49
82300Σ
arametri+ !o"rs Work
48
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Analysis48
36
72
4
40
36
30
34
40
42
45
60
61
25
29
41
45
55
31
49
82300
4115
ΣΣ6n
arametri+ !o"rs Work
48
![Page 78: 2-The Normal & Standard Normal Distribution](https://reader036.vdocuments.us/reader036/viewer/2022062600/5695d1181a28ab9b02951c8e/html5/thumbnails/78.jpg)
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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+
![Page 79: 2-The Normal & Standard Normal Distribution](https://reader036.vdocuments.us/reader036/viewer/2022062600/5695d1181a28ab9b02951c8e/html5/thumbnails/79.jpg)
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Analysis
ΣΣ6n
arametri+ (
![Page 80: 2-The Normal & Standard Normal Distribution](https://reader036.vdocuments.us/reader036/viewer/2022062600/5695d1181a28ab9b02951c8e/html5/thumbnails/80.jpg)
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Analysis
ΣΣ6n
arametri+ (
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Analysis
ΣΣ6n
ΣΣ65n17
arametri+ (
![Page 82: 2-The Normal & Standard Normal Distribution](https://reader036.vdocuments.us/reader036/viewer/2022062600/5695d1181a28ab9b02951c8e/html5/thumbnails/82.jpg)
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Analysis
ΣΣ6n
SSΣ
Σ65n17
arametri+ (
![Page 83: 2-The Normal & Standard Normal Distribution](https://reader036.vdocuments.us/reader036/viewer/2022062600/5695d1181a28ab9b02951c8e/html5/thumbnails/83.jpg)
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Analysis
ΣΣ6n
ΣΣ65n17
arametri+
(
![Page 84: 2-The Normal & Standard Normal Distribution](https://reader036.vdocuments.us/reader036/viewer/2022062600/5695d1181a28ab9b02951c8e/html5/thumbnails/84.jpg)
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arametri+
Analysis
ΣΣ6n
3arian+e ΣΣ65n17
arametri+ (
![Page 85: 2-The Normal & Standard Normal Distribution](https://reader036.vdocuments.us/reader036/viewer/2022062600/5695d1181a28ab9b02951c8e/html5/thumbnails/85.jpg)
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Analysis
ΣΣ6n
srt
ΣΣ65n17
arametri+ (
![Page 86: 2-The Normal & Standard Normal Distribution](https://reader036.vdocuments.us/reader036/viewer/2022062600/5695d1181a28ab9b02951c8e/html5/thumbnails/86.jpg)
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Analysis
ΣΣ6n
srt
ΣΣ65n17s
arametri+ (
![Page 87: 2-The Normal & Standard Normal Distribution](https://reader036.vdocuments.us/reader036/viewer/2022062600/5695d1181a28ab9b02951c8e/html5/thumbnails/87.jpg)
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Analysis
ΣΣ6n
srt
$
ΣΣ65n17
arametri+ (
![Page 88: 2-The Normal & Standard Normal Distribution](https://reader036.vdocuments.us/reader036/viewer/2022062600/5695d1181a28ab9b02951c8e/html5/thumbnails/88.jpg)
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Analysis
ΣΣ6n
srt
ΣΣ65n17
0.524
arametri+ (
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Analysis
ΣΣ6n
srt
ΣΣ65n17
0.005
arametri+ (
![Page 90: 2-The Normal & Standard Normal Distribution](https://reader036.vdocuments.us/reader036/viewer/2022062600/5695d1181a28ab9b02951c8e/html5/thumbnails/90.jpg)
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Analysis
ΣΣ6n
srt
ΣΣ65n17
0. 984
arametri+ #$%m Score
76
![Page 91: 2-The Normal & Standard Normal Distribution](https://reader036.vdocuments.us/reader036/viewer/2022062600/5695d1181a28ab9b02951c8e/html5/thumbnails/91.jpg)
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Analysis
5oulation7
83
81
90
93
88
85
52
90
91
88
95
61
90
100
93
45
8083
74
arametri+ #$%m Score
76
![Page 92: 2-The Normal & Standard Normal Distribution](https://reader036.vdocuments.us/reader036/viewer/2022062600/5695d1181a28ab9b02951c8e/html5/thumbnails/92.jpg)
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Analysis83
81
90
93
88
85
52
90
91
88
95
61
90
100
93
45
8083
74
163800Σ
arametri+ #$%m Score
76
![Page 93: 2-The Normal & Standard Normal Distribution](https://reader036.vdocuments.us/reader036/viewer/2022062600/5695d1181a28ab9b02951c8e/html5/thumbnails/93.jpg)
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Analysis83
81
90
93
88
85
52
90
91
88
95
61
90
100
93
45
8083
74
163800
8190
ΣΣ6N
arametri+ #$%m Score
76
![Page 94: 2-The Normal & Standard Normal Distribution](https://reader036.vdocuments.us/reader036/viewer/2022062600/5695d1181a28ab9b02951c8e/html5/thumbnails/94.jpg)
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Analysis83
81
90
93
88
85
52
90
91
88
95
61
90
100
93
45
8083
74
163800
8190
ΣΣ6N
µ
arametri+ #$%m Score
76 &590
Kµ
![Page 95: 2-The Normal & Standard Normal Distribution](https://reader036.vdocuments.us/reader036/viewer/2022062600/5695d1181a28ab9b02951c8e/html5/thumbnails/95.jpg)
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Analysis83 110
81 &090
90 810
93 1110
88 610
85 310
52 &2990
90 810
91 910
88 610
95 1310
61 &2090
90 810
100 1810
93 1110
45 &3690
80 &19083 110
74 &790
163800
8190
ΣΣ6N
#$%m Score
76 &590 348100
5Kµ7arametri+ Kµ (
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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
arametri+ #$%m Score
76 &590 348100
83 1 10 1 2100
5Kµ7Kµ (
![Page 97: 2-The Normal & Standard Normal Distribution](https://reader036.vdocuments.us/reader036/viewer/2022062600/5695d1181a28ab9b02951c8e/html5/thumbnails/97.jpg)
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Analysis83 110 12100
81 &090 08100
90 810 656100
93 1110 1232100
88 610 372100
85 310 96100
52 &2990 8940100
90 810 656100
91 910 828100
88 610 372100
95 1310 1716100
61 &2090 4368100
90 810 656100
100 1810 3276100
93 1110 1232100
45 &3690 13616100
80 &190 3610083 110 12100
74 &790 624100
163800 390580
8190
ΣΣ6N
arametri+ #$%m Score
76 &590 348100
83 1 10 1 2100
5Kµ7Kµ (
![Page 98: 2-The Normal & Standard Normal Distribution](https://reader036.vdocuments.us/reader036/viewer/2022062600/5695d1181a28ab9b02951c8e/html5/thumbnails/98.jpg)
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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
SS
arametri+ #$%m Score
76 &590 348100
83 1 10 1 2100
5Kµ7Kµ (
![Page 99: 2-The Normal & Standard Normal Distribution](https://reader036.vdocuments.us/reader036/viewer/2022062600/5695d1181a28ab9b02951c8e/html5/thumbnails/99.jpg)
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Analysis83 110 12100
81 &090 08100
90 810 656100
93 1110 1232100
88 610 372100
85 310 96100
52 &2990 8940100
90 810 656100
91 910 828100
88 610 372100
95 1310 1716100
61 &2090 4368100
90 810 656100
100 1810 3276100
93 1110 1232100
45 &3690 13616100
80 &190 3610083 110 12100
74 &790 624100
163800 390580
8190 19529
ΣΣ6N
arametri+ #$%m Score
76 &590 348100
83 1 10 1 2100
5Kµ7Kµ (
![Page 100: 2-The Normal & Standard Normal Distribution](https://reader036.vdocuments.us/reader036/viewer/2022062600/5695d1181a28ab9b02951c8e/html5/thumbnails/100.jpg)
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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 19529
ΣΣ6N
3arian+e
arametri+ #$%m Score
76 &590 348100
83 1 10 1 2100
5Kµ7Kµ (
![Page 101: 2-The Normal & Standard Normal Distribution](https://reader036.vdocuments.us/reader036/viewer/2022062600/5695d1181a28ab9b02951c8e/html5/thumbnails/101.jpg)
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Analysis83 110 12100
81 &090 08100
90 810 656100
93 1110 1232100
88 610 372100
85 310 96100
52 &2990 8940100
90 810 656100
91 910 828100
88 610 372100
95 1310 1716100
61 &2090 4368100
90 810 656100
100 1810 3276100
93 1110 1232100
45 &3690 13616100
80 &190 3610083 110 12100
74 &790 624100
163800 390580
8190 19529
1397
ΣΣ6N
srt
arametri+
l i
#$%m Score
76 &590 348100
83 1 10 1 2100
5Kµ7Kµ (5Kµ76 σ
![Page 102: 2-The Normal & Standard Normal Distribution](https://reader036.vdocuments.us/reader036/viewer/2022062600/5695d1181a28ab9b02951c8e/html5/thumbnails/102.jpg)
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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 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
![Page 103: 2-The Normal & Standard Normal Distribution](https://reader036.vdocuments.us/reader036/viewer/2022062600/5695d1181a28ab9b02951c8e/html5/thumbnails/103.jpg)
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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
#$%m Score
76 &590 348100 &0422
83 1 10 1 2100 0 079
5Kµ7Kµ (5Kµ76 σ
![Page 104: 2-The Normal & Standard Normal Distribution](https://reader036.vdocuments.us/reader036/viewer/2022062600/5695d1181a28ab9b02951c8e/html5/thumbnails/104.jpg)
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Analysis83 110 12100 0079
81 &090 08100 &0064 0476
90 810 656100 0580
93 1110 1232100 0794
88 610 372100 0437
85 310 96100 0222
52 &2990 8940100 &2140
90 810 656100 0580
91 910 828100 0651
88 610 372100 0437
95 1310 1716100 0937
61 &2090 4368100 &1496
90 810 656100 0580
100 1810 3276100 1295
93 1110 1232100 0794
45 &3690 13616100 &2641
80 &190 36100 &013683 110 12100 0079
74 &790 624100 &0565
163800 390580
8190 19529
1397
ΣΣ6N
srt
arametri+
A l i
#$%m Score
76 &590 348100 &0422
83 1 10 1 2100 0 079
5Kµ7Kµ (5Kµ76 σ
![Page 105: 2-The Normal & Standard Normal Distribution](https://reader036.vdocuments.us/reader036/viewer/2022062600/5695d1181a28ab9b02951c8e/html5/thumbnails/105.jpg)
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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
#$%m Score
76 &590 348100 &0422
83 1 10 1 2100 0 079
5Kµ7Kµ (5Kµ76 σ
![Page 106: 2-The Normal & Standard Normal Distribution](https://reader036.vdocuments.us/reader036/viewer/2022062600/5695d1181a28ab9b02951c8e/html5/thumbnails/106.jpg)
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Analysis83 110 12100 0079
81 &090 08100 &0064
90 810 656100 0580
93 1110 1232100 0794
88 610 372100 0437
85 310 96100 0222
52 &2990 8940100 &2140
90 810 656100 0580
91 910 828100 0651
88 610 372100 0437
95 1310 1716100 0937
61 &2090 4368100 &1496
90 810 656100 0580
100 1810 3276100 1295 0902
93 1110 1232100 0794
45 &3690 13616100 &2641
80 &190 36100 &013683 110 12100 0079
74 &790 624100 &0565
163800 390580 000
8190 19529
1397
ΣΣ6N
srt
arametri+
A l i
#$%m Score
76 &590 348100 &0422
83 1 10 1 2100 0 079
5Kµ7Kµ (5Kµ76 σ
![Page 107: 2-The Normal & Standard Normal Distribution](https://reader036.vdocuments.us/reader036/viewer/2022062600/5695d1181a28ab9b02951c8e/html5/thumbnails/107.jpg)
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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 >'>>/
>'<=<
![Page 108: 2-The Normal & Standard Normal Distribution](https://reader036.vdocuments.us/reader036/viewer/2022062600/5695d1181a28ab9b02951c8e/html5/thumbnails/108.jpg)
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Anthony J Greene 1><
!"at s(or# (orr#s$onds to t"# To$ 103-
!"at s(or#s (orr#s$ond to t"# Midd&# 603-
1'(<
>'</
i f 58 d d Of 10 " i "
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Anthony J Greene 1>=
i,#n a m#an of 58 and a st. d#,. Of 10 +"at is t"#
&i#&i"ood of random&' %#in) %#t+##n 55 and 65-
!an use +olumn D) >'11:= >'(0<> >'-:0=
i f 58 d t d Of 10 " t i t"
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Anthony J Greene 11>
i,#n a m#an of 58 and a st. d#,. Of 10 +"at is t"#
&i#&i"ood of random&' %#in) %#t+##n 65 and 75-
!an use +olumn !) >'(/(> >'>//2 >'1=:/
![Page 111: 2-The Normal & Standard Normal Distribution](https://reader036.vdocuments.us/reader036/viewer/2022062600/5695d1181a28ab9b02951c8e/html5/thumbnails/111.jpg)
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Anthony J Greene 111
!"at S(or#s ar# in t"# To$ 153-
z = 1'>/
;se +olumn ! x = z U V
x = 2>/
![Page 112: 2-The Normal & Standard Normal Distribution](https://reader036.vdocuments.us/reader036/viewer/2022062600/5695d1181a28ab9b02951c8e/html5/thumbnails/112.jpg)
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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
![Page 113: 2-The Normal & Standard Normal Distribution](https://reader036.vdocuments.us/reader036/viewer/2022062600/5695d1181a28ab9b02951c8e/html5/thumbnails/113.jpg)
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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=(
![Page 114: 2-The Normal & Standard Normal Distribution](https://reader036.vdocuments.us/reader036/viewer/2022062600/5695d1181a28ab9b02951c8e/html5/thumbnails/114.jpg)
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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-
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$
;se !olumn B) z '/1
x = z × σ + μ x = > /1 U 0 2> 0: =0