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CORRELATION
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Learning Outcomes
Upon completion of this chapter, you should be able to:
Construct a scatter diagram given two sets of data
Interpret a given scatter plot in terms of strength ofrelationship and direction of relationships
Decide whether the relationship between two sets ofdata is linear/non-linear given the scatter diagram
Calculate earson correlation given the data
Calculate !pearman correlation given the data Interpret given correlation coefficient
Decide on whether to use earson or !pearman
correlation given the data sets
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Correlation
A correlation is showing the degree or strength of
relationship between two variables. "he population correlation, denoted by "he sample correlation, denoted byr Usually, the variables denoted by # and $% &r' can ta(e on any value from ) to )% Three method can be used for the describe the
relation and estimating association between
variablesa) Scatter plotb) Pearsons Correlation Coefficientc) Spearmans Rank Correlation Coefficient
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Scatter Plot / Diagram !catter plots usually consist of a large body of data%
"he closer the data points come when plotted to ma(inga straight line, the higher the correlation between thetwo variables, or the stronger the relationship%
In a positive linear relationship indicates that as the #score increase, the $ also tend to increase%
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Cont.. In a negative linear relationship indicates that as the #
score increases, the $ score tend to decreases%
In a nonlinear relationship denotes that as the # scoresincreases, the $ score do not increases nor decreases%
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erfect positive !trong positive ositivecorrelation r * ) correlation r * +% correlation r * +%+
!trong negative .o correlation .on-linearcorrelation r * +%++correlation r * -+%
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Strength of Relationship
r Relationship Between Two Variables
r * -)%++ indicates a perfect negative linear relationship
r * )%++ indicates a perfect positive linear relationship
-)%++ 0 r 0 -+%1+ indicates a strong negative linear relationship
+%1+ 0 r 0
)%++
indicates a strong positive linear relationship
-+%1+ 0 r 0 + indicates awea( negative linear relationship
+ 0 r 0 +%1+ indicates awea( positive linear relationship
r * + indicates no linear relationship
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The important of a scatter plot
We need a scatter plot to find if the
relationship between # and $ is a linearrelationship%
It can be positive linear relationship ornegative linear relationship%
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Perkara Penting Untuk Melakar
Scatter Diagram
2engenal pasti pembolehubah bersandar danpembolehubah tida( bersandar
2ela(ar scatter diagram "a3u( 4raf
5abel a(si #
5abel a(si $
lot !era(an
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Pearsons correlation coefficient ( r)
Definition
Pearsons correlation coefficient measures the strength or thedegree of the linear relationship between two variables.
It is assumed that both variables (often called X and Y) are of
interval or ratio scale. Data set approximately normally distribute.
Synonyms:product moment correlation coefficient
simple linear correlation coefficient
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Cont Pearsons Correlation Coefficient is usuall signified b r
!rho").
#ormula for computing Pearson correlation is given as:-
))((
!
!
YNYXNX
YXNXYr
YxSSSS
SPr
$here%
X &ean of '
Y &ean of (
number of sample
))(())((
))((
YYNXXN
YXXYNrp
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Spearman ran"
#ormally
distributed
$ida"
Ya
Pearson%orrelation
Interval&ratiodata'
ula
$amat
$ida"
Ya
How to choose the Correlation ?
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Cont
Example 1
A high school guidance is interested in a relationship between pro*imit
to school and participation in e*tracurricular activities. +e collects the
data on the distance from home to school !in miles) and number ofclubs ,oined for a sample of - ,uniors. /sing the following data
compute a Pearsons correlation is significant.
Distance to
school (in miles)
X
Numbers of clubs
Joined
Y
5ee 6 7
8honda 9 )
ess ; 1
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Penelesaian
!tep )
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!tep 9
84.)*.+8(,.+-
.-/
),.4)(,(4+)(+./)(,(4,(
),.4)(+./)(,(**
p
p
p
r
r
r
Interpretation
#ilai pe"ali "orelasi Pearson 0.841 menun2u""an terdapatnya satu hubungan
linear positif diantara 2ara" dari se"olah dengan bilangan penyertaan dalam"elab.
))((
!
!
YNYXNX
YXNXYr
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Pearsons Coefficient Correlation Test
earson>s Coefficient correlation test can be determineby using critical value from earson?s "able or "-test%
"o test the significant of a a measure of correlation, we
usually set up that
3
3
a
o
H
H ull hpothesis
Alternative hpothesis
4
5
0egree of freedom" df 1 n23
,
3
p
pr
nrTujianstatistik
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Contoh soalan 3
daripada contoh -" u,i keertian pekali korelasi Pearson
dengan aras keertian" 41.5.
5ang(ah ):Nyatakan HodanHailai korelasi adalah positif !6.78)" maka u,ian
hipotesis satu hu,ung digunakan.
@o : "ida( terdapat per(aitan antara 3ara( dari se(olahdengan penglibatan dalam a(tiviti (elab
@o :s = +@a : "erdapat per(aitan antara 3ara( dari se(olah dengan
penglibatan dalam a(tiviti (elab%
@a :s > +
pr
pr
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Aeputusan u3ian:0ar,ah kebebasan
d# 1 n23
1 -23
17
Compare the obtained Pearsons with the appropriate value of
Pearsons in Table #
Cari dengan menggunakan ,adual nilai kritikal
Pearsonscriticalr
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0aripada ,adual didapati bahawa
maka" null hpotesis ditolak dan mempunai bukti ang
kukuh untuk membuat kesimpulan bahawa
9esimpulan" ini menun,ukkan bahawa wu,udna
hubungan ang signifikan pada aras signifikan .5 iaitu
,ika ,arak dari sekolah ,auh" penglibatan dalam aktivitikelab meningkat.
criticalp rr
pr
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Contoh soalan : !menggunakan u,ian T)
,
3
p
pr
nrTujianstatistik
+8.4
84.,
884.
T
T
0aripada ,adual T" pada aras signifikan 41.5" T ; t.5" 7 1 -.7 +
Terdapat perhubungan positif
ang signifikan antara keputusan
u,ian aptitud dan hasilan ker,a
),8(8
)/.(-,
kiraansr
**4,.kiraansr
),(-,
nndrkiraans
P l i
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Penyelesaian
#Menggunakan (adual Spearman' Daripada adual !pearman
rs kritikal = +0.643
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Penyelesaian
#Menggunakan (adual Spearman'
5ang(ah 6: MembuatKeputusan
Jleh (erana rs kiraan * +%6 lebih besar daripada rskritikal * +%=67 dan berada di (awasan penola(an%
8e3ection region
+%=67
@o ditola( dan (esimpulannyaterdapat per(aitan antara(eputusan u3ian aptitud dan hasilan(er3a
.on re3ection region
P l i
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Penyelesaian
#Menggunakan (adual Ta+uran,t'
5ang(ah ):NyatakanHodanHa
@o : "ida( terdapat per(aitan antara (eputusan
u3ian aptitud dan hasilan (er3a%@o :s = +
@a : "erdapat per(aitan antara (eputusan u3ianaptitud dan hasilan (er3a%
@a :s > +
P l i
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Penyelesaian
#Menggunakan (adual Ta+uran,t'
5ang(ah 9: Menentukankawasanpenolakandankawasanpenerimaan
Degree of freedom, df * B 9 * =
!ignificant level , E * +%+1tkritikal * )%67 &ru3u( 3adual taburan-t'
.on re3ection region
8e3ection region
)%67
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Penyelesaian
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Penyelesaian
#Menggunakan (adual Ta+uran,t'
5ang(ah 7: Kirakannilaiujianstatistik,T
,
s
s
r
nrT
**4.,
8**4.
T
-.T
Penyelesaian
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Penyelesaian
#Menggunakan (adual Ta+uran,t'
5ang(ah 6: MembuatKeputusan T> tkritikal
99%9= F )%67 @o ditola( dan terdapat per(aitan antara
(eputusan u3ian aptitud dan hasilan (er3a%
.on re3ection region
8e3ection region
)%67
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Latihan 1 !yari(at insuran "a(aful telah men3alan(an
(ursus pen3ualan dan pemasaran yangdire(abentu( untu( mening(at(an prestasiwa(il-wa(il pen3ualan% Dalam usaha untu(
menilai program tersebut, pengurus latihanpemasaran dan pen3ualan ingin melihat samaada terdapat hubungan atau tida( antara
pencapaian program dan pen3anaan pen3ualantahunan selepas itu% adual beri(utmenun3u((an data yang di(umpul(an oleh
pengurus (e atas )) orang graduan program itu%
Latihan 1
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Latihan 1 adual: 8ang(ing restasi encapaian Aursus dan 8ang(ing ualan
"ahunan Gagi )) orang e(er3a Di !yari(at Insuran "a(aful
Gerdasar(an 3adual, (ira(an (olerasi !pearman bagi data yang diberi(an%
Pe!er"a Ran!in' Prestasi
Pencapaian#ursus
Ran'!in' Jualan
Tahunan
!aifudin ) 6
.oraHlina 9 =
shraf 7 )
.or Hura 6 9
.orlaila 1 ;
.ura(ma = )+
4auri ; 7
8ahifa 1.orhidayu
innie )+
.urhaini )) ))
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(aapan Latihan 1
rs kiraan * +%=7= lebih besar daripada rs kritikal *+%17= dan berada di (awasan penola(an%
@o ditola( dan (esimpulannya terdapatperhubungan antara pen(aitan antara prestasipencapaian (ursus dan pen3anaan 3ualan
tahunan%
8 b K
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"he minimum values of rs needed for statisticalsignificance are shown in critical values table of
!pearman for values ofNfrom 1 to 7+%&.otethat when using this table, you need only refer toNB the number of pairs of ran(s B rather than
degrees of freedom%' WhenNis greater than 7+, the critical values for
testing a earson r for statistical significance will
give a very good approimation% "hat is, you canrefer the computed !pearman correlationcoefficient to critical values table of the earson
rwithNB 9 degrees of freedom%
8ememberK