PAGE 4Statistic II - Seminariile 11-12

CORELAIA

1. Corelaia Braivais Pearson 2. Corelaia Spearman (de ranguri)3. Coeficientul Kendall4. Corelaia punct biserial 5. Corelaia ntre 2 serii de date dihotomice6. Matricea varian covarian7. Corelaia tetrachoric8. Corelaia parial1. Corelaia Braivais Pearson- pentru date de interval (cantitative i continue);

Are formula: ;

sem1 (X)

sem2 (Y)

x

y

x2

y2

XY

6.93

5.36

-0.60

-1.66

0.36

2.76

1.00

6.13

5.40

-1.40

-1.62

1.96

2.62

2.27

5.90

5.53

-1.63

-1.49

2.66

2.22

2.43

6.23

5.80-1.30

-1.22

1.69

1.49

1.59

5.96

6.13

-1.57

-0.89

2.46

0.79

1.40

7.26

6.16

-0.27

-0.86

0.07

0.74

0.23

6.83

6.16

-0.70

-0.86

0.49

0.74

0.60

6.80

6.26

-0.73

-0.76

0.53

0.58

0.55

6.26

6.36

-1.27

-0.66

1.61

0.44

0.84

7.26

6.36

-0.27

-0.66

0.07

0.44

0.18

7.03

6.46

-0.50

-0.56

0.25

0.31

0.28

7.53

7.03

0.00

0.01

0.00

0.00

0.00

8.30

7.56

0.77

0.54

0.59

0.29

0.42

8.06

7.56

0.53

0.54

0.28

0.29

0.29

8.76

7.76

1.23

0.74

1.51

0.55

0.91

9.33

8.23

1.80

1.21

3.24

1.46

2.18

9.13

8.46

1.60

1.44

2.56

2.07

2.30

8.70

8.76

1.17

1.74

1.37

3.03

2.04

8.93

9.40

1.40

2.38

1.96

5.66

3.33

9.36

9.56

1.83

2.54

3.35

6.45

4.65

27.03

32.93

27.47

1. ; ;

2.

2. Corelaia Spearman (de ranguri)

- pentru date ordinale;

Are formula: ;

sem1 (X)

sem2 (Y)

rxrydd2

6.93

5.36

13

20

-7

49

6.13

5.40

18

19

-1

1

5.90

5.53

20

18

2

4

6.23

5.80

17

17

0

0

5.96

6.13

19

16

3

9

7.26

6.16

10.5

14.5

-4

16

6.83

6.16

14

14.5

-0.5

0.25

6.80

6.26

15

13

2

4

6.26

6.36

16

11.5

4.5

20.25

7.26

6.36

10.5

11.5

-1

1

7.03

6.46

12

10

2

4

7.53

7.03

9

9

0

0

8.30

7.56

7

8

-1

1

8.06

7.56

8

7

1

1

8.76

7.76

5

6

-1

1

9.33

8.23

2

5

-3

9

9.13

8.46

3

4

-1

1

8.70

8.76

6

3

3

9

8.93

9.40

4

2

2

4

9.36

9.56

1

1

0

0

134.5

Pentru aceleai date, prin cele 2 procedee se pot obine rezultate uor diferite.

Coeficientul de corelaie ia valori ntre +1 i 1. Valoarea +1 semnific o relaie (direct) perfect ntre cele 2 serii de date; valoarea 0 indic faptul c ntre cele 2 serii de date nu exist nici o relaie; valoarea 1 indic i ea o relaia (invers) perfect. O valoare pozitiv a corelaiei arat o relaie direct, iar una negativ arat o relaie invers.

Pentru a vedea semnificaia unei corelaii se calculeaz un test t dup formula: sau

; la = 19 grade de liberate gasim cea mai apropiat valoare 3,883 creia i corespunde p< 0,0005 < 0,01 deci corelaia este semnificativ;

; aceleai concluzii

Prin urmare, putem spune c ntre performanele colare pe cele 2 semestre exist o legtur (direct) puternic.

3. Coeficientul Kendall

sem1(X)

sem2(Y)

rxryd

D +

9.36

9.56

11

0

19

9.33

8.23

2

5

-3

15

9.13

8.46

3

4

-2

15

8.93

9.4

4

2

0

16

8.76

7.76

5

6

-1

14

8.7

8.76

6

3

0

14

8.3

7.56

7

8

-1

12

8.06

7.56

8

7

0

12

7.53

7.03

9

9

0

11

7.26

6.16

10.5

14.5

-5

5

7.26

6.36

10.5

11.5

-3

7

7.03

6.46

12

10

0

8

6.93

5.36

13

20

-7

0

6.83

6.16

14

14.5

-3

4

6.8

6.26

15

13

-1

4

6.26

6.36

16

11.5

-1

4

6.23

5.8

17

17

-1

2

6.13

5.4

18

19

-2

0

5.96

6.13

19

16

0

1

5.9

5.53

20

18

0

0

-30

163

; T =( suma d-) + (suma d+)

T= -30 + 163 = 133

K= 0,7

Coeficientul K ia valori ntre 1(indicnd valori inverse) i +1(concordan).

Deci putem spune c exist o foarte bun concordan ntre cele evalurile pe cele 2 semestre.

4. Corelaia punct biserial - pentru o variabil dihotomic i una continu.it1

it2

x

x2

1

9

1

0.5

0.25

2

8

1

-0.5

0.25

3

5

0

-3.5

12.25

4

10

1

1.5

2.25

5

11

1

2.5

6.25

6

12

1

3.5

12.25

7

7

0

-1.5

2.25

8

8

0

-0.5

0.25

9

6

0

-2.5

6.25

10

9

1

0.5

0.25

85

42.5

sau , unde:

este media notelor subiecilor la it1 (cel continuu) care au rezolvat corect it2;

este media notelor subiecilor la it1 care nu au rezolvat corect it2;

este media pentru it1 (toi subiecii)

este abaterea standard pentru it1 (continuu)

p este proporia subiecilor care au rezolvat corect it2 (dihotomic);

q= 1 p este proporia subiecilor care nu au rezolvat corect it2;

p = 6/10 = 0,6; q= 1 0,6 = 0,4;

= 85/10 = 8,5; ;

= (8+9+10+11+12+9)/6 =9,83; = (5+7+8+6)/4 = 6,5;

sau

5. Corelaia ntre 2 serii de date dihotomice

it1

it2

1

1

1

2

1

1

3

1

0

4

0

1

5

0

1

6

1

1

7

0

0

it1

it2

8

1

0

9

1

0

10

1

1

fi

7

6

pi

0.7

0.6

qi

0.3

0.4

s2i

0.21

0.24

Pij este proporia rspunsurilor corecte comune itemilor it1 i it2;

Observaie: = pi(1-pi) = pi*qi = s2iPij = 4/10 = 0,4;

, adic ntre cei doi itemi nu exist nici o corelaie (valoarea fiind foarte apropiat de 0);

6. Matricea varian covarian

1

2

3

1

C12

C13

2

C21

C233

C31C32

reprezint variana ansamblului celor 3 itemi.it1

it2

it3

1

1

1

1

2

1

1

1

3

1

0

0

4

0

1

0

5

0

1

1

6

1

1

1

7

0

0

0

8

1

0

1

9

1

0

0

10

1

1

1

fi

7

6

6

pi

0.7

0.6

0.6

qi

0.3

0.4

0.4

s2i0.21

0.24

0.24

0.69

1

2

3

1

0,21

-0,020,08

2

-0,020,24

0,14

3

0,24

0,14

0,24

7. Corelaia tetrachoric - ntre 2 variabile artificial dihotomice (mprirea n 2 clase este artificial ex: inteligen medie inteligen ridicat )Inteligen medie

Inteligen ridicat

Note sub 8

43

16

Note peste 8

17

39

;

8. Corelaia parial - cnd vrem s tim corelaia ntre 2 variabile cu excluderea influenei celei de a treia:;r12=0,63;r13=0,56;

r23=0,70;

;

deci, dup excluderea influenei celei de-a treia variabile, observm o corelaie medie ntre cele 2 variabile (fa de corelaia mare iniial)

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