View
233
Download
6
Category
Preview:
Citation preview
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?
Scoruri standard. Curba normală. 3
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
mxz
−=
25
6070=
−=z 5.0
20
6070+=
−=z
s=20s=5
75.020
6045−=
−=z
45
Scoruri standard. Curba normală. 4
Transformarea z a unei distribuţii
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
38.2
8.1214−=
−=
s
mxz
Scoruri standard. Curba normală. 5
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=5
ΣX=64
m=12.8
s=2.38
N=5
Σz=0
m=0
s=1
s
mxz
−= x=z*s+m
x=-0,75*2.38+12.8=11
x=+0.50*2.38+12.8=14
Scoruri standard. Curba normală. 6
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
Scoruri standard. Curba normală. 7
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
Scoruri standard. Curba normală. 8
Alte tipuri de scoruri standardizate
s
mXz
−=
s
mXT
−+= *1050
s
mXH
−+= *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
mXQI
−+= *16100
QI (Wechsler) QI=100+15*zs
mXQI
−+= *15100
m=100; s=16
m=100; s=15
m=500; s=100SAT SAT=500+100*z s
mXSAT
−+= *100500
Scoruri standard. Curba normală. 9
Observaţii
• Toate variantele sunt obţinute prin transformarea operată pe distribuţia de note z
• La nici una dintre variante nu mai avem valori negative (cu condiţia ca distribuţia să nu aibă o variabilitatea aberantă).
• Zecimalele nu mai sunt semnificative (ele rezultă din calcule, dar pot fi ignorate)
• Distribuţiile variantelor oscilează în jurul unei valori medii specifice, sub care se află 50% din valori şi peste care se află restul de 50% dintre valori.
Scoruri standard. Curba normală. 10
Inversarea scorului standard
• scor mare = valoare mare
• Uneori, un scor mare are o semnificaţie inversă– timpul de reacţie
– număr de erori
• Într-un astfel de caz se calculează un scor standard “invers” (cu “minus” în formulă)
Scoruri standard. Curba normală. 11
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
Scoruri standard. Curba normală. 12
Scoruri standard. Curba normală. 13
Distribuţia normală (Gauss)
• distribuţie teoretică, caracteristică populaţiilor mari
• formă de „clopot”• perfect simetrică• linia curbei se apropie
la infinit de axa X, fără a o atinge vreodată
• de fiecare parte a mediei se află exact jumătate dintre valorile distribuţiei
mediamodul
mediana
Karl Friedrich Gauss 1777-1855
Scoruri standard. Curba normală. 14
Distribuţia normală=familie de distribuţii
Scoruri standard. Curba normală. 15
Curba normală standardizată(curba normală z)
• valorile sunt exprimate în scoruri z
• utilă pentru a descrie orice distribuţie normală, indiferent de expresia valorilor– o singură tabelă a distribuţiei normale
Scoruri standard. Curba normală. 16
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
Scoruri standard. Curba normală. 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 .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
Tabelul
distribuţiei z
0 z
Scoruri standard. Curba normală. 18
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
Scoruri standard. Curba normală. 19
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
Scoruri standard. Curba normală. 20
Proprietăţile distribuţiei normale permit aflarea răspunsului la unele
întrebări practice...
Scoruri standard. Curba normală. 21
Procentul oamenilor al căror scor
QI este între 100 şi 110?
63.016
100110+=
−=
−=
s
mxz
0 ?
?
Scoruri standard. Curba normală. 22
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
Scoruri standard. Curba normală. 23
Procentul oamenilor al căror scor
QI este între 100 şi 110?
63.016
100110+=
−=
−=
s
mxz
0 0.63
23.57%
Scoruri standard. Curba normală. 24
Procentul oamenilor al căror scor
QI este mai mare de 125?
0 ?
56.116
100125+=
−=
−=
s
mxz
?
Scoruri standard. Curba normală. 25
Procentul oamenilor al căror scor
QI este mai mare de 125?
0 1.56
56.116
100125+=
−=
−=
s
mxz
?
Scoruri standard. Curba normală. 26
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
Scoruri standard. Curba normală. 27
Procentul oamenilor al căror scor
QI este mai mare de 125?
0 1.56
44.06%
50–44.06=5.94%
Scoruri standard. Curba normală. 28
Care este scorul minim pe care trebuie să-l obţină o
persoană pentru a fi între primii 5% din populaţie?
100 ?
5%45%
Scoruri standard. Curba normală. 29
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
Scoruri standard. Curba normală. 30
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
Scoruri standard. Curba normală. 31
Care este scorul care indică cei
mai slabi 33%?
100?
33%
17%
Scoruri standard. Curba normală. 32
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
Scoruri standard. Curba normală. 33
10093
33%
17%Convertim nota z în valoare brută:
X=m+z*s=100+(-0.44)*16= 92.96
Scoruri standard. Curba normală. 34
Scoruri standard. Curba normală. 35
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
Scoruri standard. Curba normală. 36
Exemplificări ale probabilităţilor de sub curba normală z
Pentru z=1.65 → aria=0.95 → p=0.05
1.65
p=0.05p=0.95
Scoruri standard. Curba normală. 37
Pentru z=1.95 → aria=0.975 → p=0.025
1.95
p=0.025p=0.975
Scoruri standard. Curba normală. 38
z=-1 ↔ z=+1 → p=0.68
p=0.68
10-1
Scoruri standard. Curba normală. 39
z=1 → aria=0.84 → p=0.16
p=0.16
p=0.84
10
Scoruri standard. Curba normală. 40
z=-2 ↔ z=+2 → p=0.95
p=0.95
20-2
Scoruri standard. Curba normală. 41
z=-3 ↔ z=+3 → p=0.99
p=0.99
30-3
Scoruri standard. Curba normală. 42
Distribuţii reale şi distribuţii normale z
• distribuţia standardizată z reprezintă transformarea în z a distribuţiei normale teoretice
• distribuţia z a unei distribuţii reale PĂSTREAZĂ FORMA DISTRIBUŢIEI REALE
x z2 -,929635 ,143026 ,500577 ,858123 -,572088 1,215669 1,573212 -,929633 -,572081 -1,28717
curba x curba z
Scoruri standard. Curba normală. 43
Scoruri standard. Curba normală. 44
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 formă distribuţiei (scorurile z nu normalizează distribuţia)
• Indiferent dacă distribuţia este normală sau nu, scorul z exprimă cu precizie poziţia faţă de medie în abateri standard
• Dacă o distribuţie este normală, se pot găsi răspunsuri la mai multe întrebări interesante
Scoruri standard. Curba normală. 45
Recommended