scoruri Z

5/11/2018 scoruri Z - slidepdf.com

http://slidepdf.com/reader/full/scoruri-z 1/45

Scoruri standardCurba normală (Gauss)

M. Popa



Scoruri standard. Curba normală. 2

Scoruri standard• cunoaştere → evaluare, măsurare

– evaluare → comparare (Gh. Zapan)

– comparare → raportare la un sistem de referinţă• Povestea Scufiţei Roşii... – 70 de puncte la un test de limba engleză....

• Certitudinea bunicii

– doar bucuria fetiţei...• Nedumeririle bunicii• ce înseamnă scorul 70?• este mai bun sau mai slab decât ceilalţi colegi?• dacă da, cât de bun sau cât de slab?

• Întrebările bunicii – care este scorul mediu la examen? – care este abaterea standard a scorurilor?




m=60 70 m=60 70

Distribuţia I Distribuţia II

Scorul standard z

(scor normat z, notă standard z)

- distanţa dintre o anumită valoare şi media distribuţiei din

care face parte, măsurată în abateri standard s

m x z

−=

2

5

6070=

−= z 5.0

20

6070+=

−= z

s=20s=5

75.0

20

6045−=

−= z

45




Transformarea z a unei distribuţiiX Z

14 +0.50

11 -0.75

10 -1.17

16 +1.3413 +0.08

N=5

ΣX=64

m=12.8s=2.38

N=5

Σz=0

m=0s=1

38.28.1214−=

−=

sm x z




Calcularea valorii (x)din parametrii scorului z

X z

14 +0.50

11 -0.75

10 -1.17

16 +1.34

13 +0.08

N=5ΣX=64

m=12.8

s=2.38

N=5Σz=0

m=0

s=1

s

m x z

−= x=z *s+m

x=-0,75*2.38+12.8=11

x=+0.50*2.38+12.8=14




Proprietăţile scorurilor z• Media unei distribuţii z este întotdeauna

egală cu 0 – rezultă din proprietăţile mediei

• Abaterea standard a unei distribuţii z este întotdeauna 1 – rezultă din proprietăţile abaterii standard

• Ca urmare: – transformarea în z înseamnă transformarea

într-o distribuţie cu m=0 şi s=1




x z

14 +0.50

11 -0.75

10 -1.17

16 +1.34

13 +0.08

N=5

ΣX=64

m=12.

8

s=2.38

N=5

Σz=0

m=0

s=1

14 este la o jumătate de abatere standard peste medie

10 este la 1.17 abateri standard sub medie

11 este la 0.75 abateri standard sub medie

16 este la 1.34 abateri standard peste medie

13 este la 0.08 abateri standard peste medie

“Interpretarea” scorurilor z




Alte tipuri de scoruri standardizate s

m X z

−=

s

m X T

−+= *1050

s

m X H

−+= *1450

z m=0; s=1

T (Thurstone) T=50+10*z m=50; s=10

H (Hull) H=50+14*z m=50; s=14

QI (Binet) QI=100+16*z s

m X QI

−+= *16100

QI (Wechsler) QI=100+15*z m X QI

−+= *15100

m=100; s=16

m=100; s=15

m=500; s=100SAT SAT=500+100*z

s

m X SAT

−+= *100500




Observaţii• Toate variantele sunt obţinute prin

transformarea operată pe distribuţia de note z • La nici una dintre variante nu mai avem valorinegative (cu condiţia ca distribuţia să nu aibă ovariabilitatea aberantă).

• Zecimalele nu mai sunt semnificative (ele rezultădin calcule, dar pot fi ignorate)• Distribuţiile variantelor oscilează în jurul unei

valori medii specifice, sub care se află 50% dinvalori şi peste care se află restul de 50% dintrevalori.




Inversarea scorului standard• scor mare = valoare mare

• Uneori, un scor mare are o semnificaţieinversă

– timpul de reacţie – număr de erori

• Într-un astfel de caz se calculează un scor

standard “invers” (cu “minus” în formulă)




Inversarea scorului standard• Exemplu: un test de timp de reacţie

• x=0.15• m=0.11 sec

• s=0.05

• T(direct)=50+10*(0,15-0,11)/0,05=58• T(invers)=50-10*(0,15-0,11)/0,05=42

• atât 58 cât şi 42 exprimă acelaşi lucru• 8 unităţi standard faţă de medie (+ respectiv -)

• Scoruri T → MMPI







Distribuţia normală(Gauss)

• distribuţie teoretică,

caracteristicăpopulaţiilor mari

• formă de „clopot”

• perfect simetrică• linia curbei se apropiela infinit de axa X, fărăa o atinge vreodată

• de fiecare parte amediei se află exact jumătate dintre valoriledistribuţiei

media

modulmediana

Karl Friedrich Gauss 1777-1855




Distribuţia normală=familie de distribuţii




Curba normală standardizată(curba normală z)

• valorile sunt exprimate în scoruri z

• utilă pentru a descrie orice distribuţienormală, indiferent de expresia valorilor – o singură tabelă a distribuţiei normale




Z Aria - frc(1)

- ∞ .000

-3.00 .001

-2.33 .01

-2.00 .02

-1.65 .05

-1.28 .10

-1.00 .16

-.84 .20

-.52 .30

-.25 .40

.00 0.50.25 .60

.52 .70

.84 .80

1.00 .84

1.28 .90

1.65 .95

2.00 .98

2.33 .99

3.00 .999

+ ∞ 1.00




z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09

0.0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .0359

0.1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .0753

0.2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .1141

0.3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .1517

0.4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1879

0.5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224

0.6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2517 .2549

0.7 .2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852

0.8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133

0.9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389

1.0 .3413 .3438 .3461 .3485 .3508 .3531 .3554 .3577 .3599 .3621

1.1 .3643 .3665 .3686 .3708 .3729 .3749 .3770 .3790 .3810 .3830

1.2 .3849 .3869 .3888 .3907 .3925 .3944 .3962 .3980 .3997 .4015

1.3 .4032 .4049 .4066 .4082 .4099 .4115 .4131 .4147 .4162 .4177

1.4 .4192 .4207 .4222 .4236 .4251 .4265 .4279 .4292 .4306 .4319

1.5 .4332 .4345 .4357 .4370 .4382 .4394 .4406 .4418 .4429 .4441

1.6 .4452 .4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535 .4545

1.7 .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 .4625 .4633

1.8 .4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 .4699 .4706

1.9 .4713 .4719 .4726 .4732 .4738 .4744 .4750 .4756 .4761 .4767

2.0 .4772 .4778 .4783 .4788 .4793 .4798 .4803 .4808 .4812 .4817

Tabeluldistribuţiei z

0 z




z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09

0.0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .0359

0.1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .0753

0.2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .1141

0.3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .1517

0.4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1879

0.5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224

0.6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2517 .2549

0.7 .2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852

0.8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133

0.9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389

1.0 .3413 .3438 .3461 .3485 .3508 .3531 .3554 .3577 .3599 .3621

1.1 .3643 .3665 .3686 .3708 .3729 .3749 .3770 .3790 .3810 .3830

1.2 .3849 .3869 .3888 .3907 .3925 .3944 .3962 .3980 .3997 .4015

1.3 .4032 .4049 .4066 .4082 .4099 .4115 .4131 .4147 .4162 .4177

1.4 .4192 .4207 .4222 .4236 .4251 .4265 .4279 .4292 .4306 .4319

1.5 .4332 .4345 .4357 .4370 .4382 .4394 .4406 .4418 .4429 .4441

1.6 .4452 .4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535 .4545

1.7 .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 .4625 .4633

1.8 .4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 .4699 .4706

1.9 .4713 .4719 .4726 .4732 .4738 .4744 .4750 .4756 .4761 .4767

2.0 .4772 .4778 .4783 .4788 .4793 .4798 .4803 .4808 .4812 .4817

pentru z=1.65




z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09

0.0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .0359

0.1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .0753

0.2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .1141

0.3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .1517

0.4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1879

0.5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224

0.6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2517 .2549

0.7 .2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852

0.8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133

0.9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389

1.0 .3413 .3438 .3461 .3485 .3508 .3531 .3554 .3577 .3599 .3621

1.1 .3643 .3665 .3686 .3708 .3729 .3749 .3770 .3790 .3810 .3830

1.2 .3849 .3869 .3888 .3907 .3925 .3944 .3962 .3980 .3997 .4015

1.3 .4032 .4049 .4066 .4082 .4099 .4115 .4131 .4147 .4162 .4177

1.4 .4192 .4207 .4222 .4236 .4251 .4265 .4279 .4292 .4306 .4319

1.5 .4332 .4345 .4357 .4370 .4382 .4394 .4406 .4418 .4429 .4441

1.6 .4452 .4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535 .4545

1.7 .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 .4625 .4633

1.8 .4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 .4699 .4706

1.9 .4713 .4719 .4726 .4732 .4738 .4744 .4750 .4756 .4761 .4767

2.0 .4772 .4778 .4783 .4788 .4793 .4798 .4803 .4808 .4812 .4817

pentru z=1.96




Proprietăţile distribuţiei normalepermit aflarea răspunsului la unele

întrebări practice...




Procentul oamenilor al căror scor

QI este între 100 şi 110 ?

63.016

100110+=

−=

−=

s

m x z

0 ?

?




z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09

0.0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .0359

0.1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .0753

0.2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .1141

0.3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .1517

0.4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1879

0.5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224

0.6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2517 .2549

0.7 .2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852

0.8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133

0.9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389

1.0 .3413 .3438 .3461 .3485 .3508 .3531 .3554 .3577 .3599 .3621

1.1 .3643 .3665 .3686 .3708 .3729 .3749 .3770 .3790 .3810 .3830

1.2 .3849 .3869 .3888 .3907 .3925 .3944 .3962 .3980 .3997 .4015

1.3 .4032 .4049 .4066 .4082 .4099 .4115 .4131 .4147 .4162 .4177

1.4 .4192 .4207 .4222 .4236 .4251 .4265 .4279 .4292 .4306 .4319

1.5 .4332 .4345 .4357 .4370 .4382 .4394 .4406 .4418 .4429 .4441

1.6 .4452 .4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535 .4545

1.7 .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 .4625 .4633

1.8 .4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 .4699 .4706

1.9 .4713 .4719 .4726 .4732 .4738 .4744 .4750 .4756 .4761 .4767

2.0 .4772 .4778 .4783 .4788 .4793 .4798 .4803 .4808 .4812 .4817





QI este între 100 şi 110?

63.016

100110+=

−=

−=

s

m x z

0 0.63

23.57%





QI este mai mare de 125?

0 ?

56.1

16

100125+=

−=

−=

s

m x z

?






0 1.56

56.1

16

100125+=

−=

−=

s

m x z

?




z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09

0.0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .0359

0.1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .0753

0.2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .1141

0.3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .1517

0.4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1879

0.5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224

0.6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2517 .2549

0.7 .2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852

0.8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133

0.9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389

1.0 .3413 .3438 .3461 .3485 .3508 .3531 .3554 .3577 .3599 .3621

1.1 .3643 .3665 .3686 .3708 .3729 .3749 .3770 .3790 .3810 .3830

1.2 .3849 .3869 .3888 .3907 .3925 .3944 .3962 .3980 .3997 .4015

1.3 .4032 .4049 .4066 .4082 .4099 .4115 .4131 .4147 .4162 .4177

1.4 .4192 .4207 .4222 .4236 .4251 .4265 .4279 .4292 .4306 .4319

1.5 .4332 .4345 .4357 .4370 .4382 .4394 .4406 .4418 .4429 .4441

1.6 .4452 .4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535 .4545

1.7 .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 .4625 .4633

1.8 .4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 .4699 .4706

1.9 .4713 .4719 .4726 .4732 .4738 .4744 .4750 .4756 .4761 .4767

2.0 .4772 .4778 .4783 .4788 .4793 .4798 .4803 .4808 .4812 .4817






0 1.56

44.06%

50–44.06=5.94%




Care este scorul minim pe care trebuie să-l obţină o

persoană pentru a fi între primii 5% din populaţie?

100 ?

5%45%




z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09

0.0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .0359

0.1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .07530.2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .1141

0.3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .1517

0.4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1879

0.5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224

0.6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2517 .2549

0.7 .2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852

0.8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133

0.9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389

1.0 .3413 .3438 .3461 .3485 .3508 .3531 .3554 .3577 .3599 .3621

1.1 .3643 .3665 .3686 .3708 .3729 .3749 .3770 .3790 .3810 .3830

1.2 .3849 .3869 .3888 .3907 .3925 .3944 .3962 .3980 .3997 .4015

1.3 .4032 .4049 .4066 .4082 .4099 .4115 .4131 .4147 .4162 .4177

1.4 .4192 .4207 .4222 .4236 .4251 .4265 .4279 .4292 .4306 .4319

1.5 .4332 .4345 .4357 .4370 .4382 .4394 .4406 .4418 .4429 .4441

1.6 .4452 .4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535 .4545

1.7 .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 .4625 .4633

1.8 .4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 .4699 .4706

1.9 .4713 .4719 .4726 .4732 .4738 .4744 .4750 .4756 .4761 .4767

2.0 .4772 .4778 .4783 .4788 .4793 .4798 .4803 .4808 .4812 .4817




100 126.24

5%45%

Convertim scorul z=1.64 în valoare brută

X=m+z*s=100+ (+1.64)*16=126.24

Pentru a fi în primii 5%QI=126.24




Care este scorul care indică cei

mai slabi 33%?

100?

33%

17%




z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09

0.0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .0359

0.1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .07530.2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .1141

0.3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .1517

0.4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1879

0.5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224

0.6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2517 .25490.7 .2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852

0.8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133

0.9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389

1.0 .3413 .3438 .3461 .3485 .3508 .3531 .3554 .3577 .3599 .3621

1.1 .3643 .3665 .3686 .3708 .3729 .3749 .3770 .3790 .3810 .3830

1.2 .3849 .3869 .3888 .3907 .3925 .3944 .3962 .3980 .3997 .4015

1.3 .4032 .4049 .4066 .4082 .4099 .4115 .4131 .4147 .4162 .4177

1.4 .4192 .4207 .4222 .4236 .4251 .4265 .4279 .4292 .4306 .4319

1.5 .4332 .4345 .4357 .4370 .4382 .4394 .4406 .4418 .4429 .4441

1.6 .4452 .4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535 .4545

1.7 .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 .4625 .4633

1.8 .4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 .4699 .4706

1.9 .4713 .4719 .4726 .4732 .4738 .4744 .4750 .4756 .4761 .4767

2.0 .4772 .4778 .4783 .4788 .4793 .4798 .4803 .4808 .4812 .4817




10093

33%

17%Convertim nota z în valoare brută:

X=m+z*s=100+(-0.44)*16= 92.96







Aria de sub curba normală văzută ca probabilitate

• definiţii ale probabilităţii

1. şansa teoretică a unui eveniment într-o serie posibilă (1/2=0.5)

2. şansa reală în cazul repetării “extragerii” (tinde spre 0.5 când n → ∞)3. “şansa de apariţie a unui eveniment” – complementară şansei de

neapariţie

• valorile de pe curba normală nu rezultă dintr-un proces de

măsurare• ele sunt valori teoretice

• pot rezulta ca unui proces aleatoriu

• extrăgând la infinit numere aleatoare → distribuţie normală• frecvenţa relativă a fiecărui număr este similară cu

probabilitatea sa de apariţie - fr(1)=p




Exemplificări ale probabilităţilor de

sub curba normală zPentru z=1.65 → aria=0.95 → p=0.05

1.65

p=0.05p=0.95




Pentru z=1.95 → aria=0.975 → p=0.025

1.95

p=0.025p=0.975




z=-1 ↔ z=+1 → p=0.68

p=0.68

10-1




z=1 → aria=0.84 → p=0.16

p=0.16

p=0.84

10




z=-2 ↔ z=+2 → p=0.95

p=0.95

20-2




z=-3 ↔ z=+3 → p=0.99

p=0.99

30-3




Distribuţii reale şi distribuţii

normale z• distribuţia standardizată z reprezintă transformarea în z

a distribuţiei normale teoretice

• distribuţia z a unei distribuţii reale PĂSTREAZĂ FORMADISTRIBUŢIEI REALE

x z

2 -,929635 ,143026 ,500577 ,858123 -,57208

8 1,215669 1,573212 -,929633 -,572081 -1,28717

curba x curba z







Concluzii importante• O distribuţie de scoruri z

– are întotdeauna media egală cu 0 – are întotdeauna abaterea standard egală cu 1

– convertirea unei distribuţii reale în scoruri z nu schimbă formădistribuţiei (scorurile z nu normalizează distribuţia)

• Indiferent dacă distribuţia este normală sau nu, scorul zexprimă cu precizie poziţia faţă de medie în abateristandard

• Dacă o distribuţie este normală, se pot găsi răspunsuri lamai multe întrebări interesante




Date post:	12-Jul-2015
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scoruri Z

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