2012

2

1

1.1. .... 4

1.2. ... ...... 5

1.2.1. .. 5

1.2.2. . 9

1.2.3. 2 ............... 9

1.2.4. ........... 13

1.2.5. ..... 14

1.2.5.1. Tukey 14

1.2.6. .. 16

1.2.6.1. Scheffe 17

1.3. . 22

1.3.1. .. 22

1.3.2. 25

1.3.3. 2p 26

1.3.4. .... 32

1.4.

.. 35

1.4.1. . 39

1.4.2. 2p . 39

1.4.3. ............. 40

1.4.4. ... 44

3

2

2.1. .... 46

2.2. ..... 48

2.3. Wilcoxon ..... 52

2.4. Mann-Whitney ...... 56

2.5. . 60

2.6. Spearman 63

2.6.1. Spearman .. 64

2.7. Cramer .. 66

.. 69

70

............. 75

4

1

1.1.

( ), (analysis of variance ANOVA)

. R.A. Fisher (1890-1962). , , , . , , .

, . , , , , , , . , 2mg, 4mg, 6 mg 10-20, 21-30, 31-40, 41-50 , . (factors) (levels)

.

5

1.2.

(one-way ANOVA)

k . ( ) . (between-subjects factor).

k . () n1, n2, .... ,nk . k 1, 2, ....,k. :

k210 ...:H

1 : k

,

1 : k

1.2.1.

:

1) k .

2) k .

3) k . , 2k

22

21 ,, ...., k

12

22 2 2 .......= k

6

2 k .

(t-test, ), . , t-test () . , t-test.

. , (mean squares). (sum of squares) (degrees of freedom).

1) ( ).

:

1N

SSMS tottot

k

1i

n

1j

2

ijtot

i

XXSS

, Xij i () , j . X (grand mean),

7

k21 n...nnN

2) k . X X Xk1 2, ,..., k ,

( )

kN

SSMS ww

SSw

k

1i

n

1j

2

iijw

i

XXSS

3) k X X Xk1 2, ,...,

X . ( )

1k

SSMS bb

SS n X Xb i ii

k

2

1

bMS k

X .

iijiij XXXXXX

ijX

X , iX

.

8

k

1i

n

1j

2

ij

i

XX = n X Xi ii

k

2

1

+

k

1i

n

1j

2

iij

i

XX

SStot = bSS + wSS

.

, .

, ,

(N - 1) = (N - k) + (k - 1)

.

, wMS

. , bMS

.

, (1 = 2 =...= k), wMS

2 k . , 2 ( wMS )

. bMS 2

2.

,

w

b

MS

MSF

9

. , . , F F (k - 1) ( - k) . F ( 1 ) , .

1.2.2.

. , , :

k

1i

n

1j

2ij

i

X ,

k

1i i

2n

1jij

n

Xi

2

k

1i

n

1jij

i

XN

1

, ()

A1N

1MS tot

B1k

1MSb

BAkN

1MSw

1.2.3. 2

2 (eta-squared) (effect size)

wb

b

tot

b2

SSSS

SS

SS

SS

10

() . 2 ANOVA k . . 2 [0, 1] 1.

1.1

. 1, 2, 3 4 . 18 . 1 4 , 2 5 , 3 4 4 5 . ( ) :

E1 E2 E3 E4

37 32 30 27

34 33 29 26

36 34 33 28

40 37 32 30

33 34

4 (completely randomized

11

experimental design).

4 1, 2, 3 4. 4 1 1. 2, 3 4. 4 ( 1

222

42 2 =3

2 ). 1, 2, 3, 4 4

. :

43210 :H

0 , . .

1 :

, , ( ):

k

1i

n

1j

2ij

i

X = 372 + 342 + 362 + .... 302 + 342 = 19247

k

1i i

2n

1jij

n

Xi

= 5

145

4

124

5

169

4

147 2222 = 19163.45

2k

1i

n

1jij

i

XN

1

= 18

5852 = 19012.5

,

SStot = [A - ] = [19247 19012.5] = 234.5

SSb = [B - ] = [19163.45 19012.5] = 150.95

12

SSw = [A - ] = [19247 - 19163.45] = 83.55

SStot = bSS + wSS

234.5 = 150.95 + 83.55

1k

SSMS bb

= 14

95.150

= 50.317

kN

SSMS ww

= 418

55.83

= 5.968

, F

w

b

MS

MSF =

968.5

317.50 = 8.431

(k - 1) = (4 - 1) = 3 (N - k) = (18 - 4) = 14 , = 0.05, F ( 1 ) 3.34. 8.431 . .

. , , (), F.

13

F

3 150.95 50.317 8.431

14 83.55 5.968

17 234.50

2

tot

b2

SS

SS =

5.234

95.150 = 0.644

64.4% . 4 . 4 X1 = 36.75, X2 = 33.80, X3 = 31 X4 = 29 .

.

1.2.4.

. , . . , . , .

14

1.2.5.

, , . .

( ) . , post hoc,

. , , Tukey .

1.2.5.1. Tukey

Tukey k(k-1)/2 . (1.2.1.).

H0 : ii i,i

(n1 = n2 = .... = nk = n).

iX iX , q

0 : i = i

n

MS

XXq

w

iiii

.

, q

ii

w

n

1

n

1

2

MS

15

in in iX iX .

.

1.2

Tukey, , 1, 4 : X1 = 36.75, X2 = 33.80, X3 = 31, X4 = 29. n1 =

4, n2 = 5, n3 = 4 n4 = 5. k = 4 k(k-1)/2 = 4(4-1)/2 = 6.

0 : 1 = 2

41

w

412,1

n

1

n

1

2

MS

XXq =

5

1

4

1

2

968.5

80.3375.36 = 2.55

q, -k, k 2 . , N-k , k . = 0.05, N-k = 14 k = 4, q0.05,14,4 = 4.11. q1,2 < q0.05,14,4 (2.55 < 4.11) .

, 0 : 1 = 3

3,1q

4

1

4

1

2

968.5

00.3175.36 = 4.71, 4.71 > 4.11.

q1,4 = 6.69, q2,3 = 2.42, q2,4 = 6.22, q3,4 = 1.73

, q1,4, q2,4, q1,3 :

1 : 1 4

1 : 2 4

16

1 : 1 3

4 1 2, 3 1. , . .

1.2.6.

(contrast) 1, 2, ..., k k

kk2211 W...WW

W1, W2, ..., Wk

W W Wk1 2 0 ...

, W ( 0) 0.

kk2211 XW...XWXW

.

, = (+1)1 + (-1)4 = 1- 4 . W1 = 1 W2 = -1 W1 + W2 = 1 + (-1) = 1 1 = 0.

. , = (1)1 + (-1)2 + (1)3 + (-1)4

= 1 2 + 3 4 = (1/2)1 + (1/2)2 + (-1)3 = (1/2)1 + (1/2)2 3

17

. W .

H ,

0: = 0

wk

2k

2

22

1

21

MSn

W...

n

W

n

W

t

, wMS

.

. , 1 2

0: = 0 0: 1 2 = 0 0: 1 = 2

w21

212,1

MSn

1

n

1

XXt

t

. k 1

222 2 2 .......= k

2 wMS .

1.2.6.1. Scheffe

. . ( Tukey)

18

. (0 : = 0)

1.2.6.

wk

2k

2

22

1

21

MSn

W...

n

W

n

W

t

S

S k Fc 1

k Fc F (k-1) (-k) .

t S

Scheffe S - - . Tukey (1.2.4.1.). , Scheffe, () . (1 ).

1.3

1.1, , ,

i) post hoc .

19

ii) :

1) 1 : [(1/2)1 + (1/2)2)] [(1/2)3 + (1/2)4)]

2) 2 : (1/3)1 + (1/3)2 + (1/3)3 (1)4

i) 4 :

1X = 147/4 = 36.75, 2X = 169/5 = 33.80, 3X = 124/4 = 31 4X = 145/5 = 29.

( ) (1.2.6.)

968.55

1

4

1

80.3375.36t 2,1

= 1.80

968.54

1

4

1

3175.36t 3,1

= 3.33

968.55

1

4

1

2975.36t 4,1

= 4.73

968.54

1

5

1

3180.33t 3,2

= 1.71

968.55

1

5

1

2980.33t 4,2

= 3.11

968.55

1

4

1

2931t 4,3

=1.22

Scheffe

cF1kS = )34.3(3 = 3.17

, (1 4) (1 3). 1 3 4. Tukey (1.2.5.1.) 2 4.

ii) . . , X1 = 36.75, X2 = 33.80, X3 = 31, X4 = 29. n1 = n3 = 4 n2 = n4 = 5.

20

.

]292/1312/1[]80.332/175.362/1[

275.5292/1312/180.332/175.362/1

SE

wk

2k

2

22

1

21

MSn

W...

n

W

n

W

t

,

SE

1588.1968.55

2/1

4

2/1

5

2/1

4

2/1 2222

t = 552.41588.1275.5

.

,

SE t

1) 5.275 1.1588 4.552

2) 4.850 1.2875 3.767

= 0.05 Fc = 3.34 (3, 14 ) , Scheffe S k Fc 1 = 4 1 334 . = 3.17. t .

, :

1 : (1/2)1 + (1/2)2 (1/2)3 + (1/2)4 2

2

4321

, . ,

21

( ) .

1H : (1/3)1 + (1/3)2 + (1/3)3 4 4321

3

, , , .

22

1.3.

(two-way analysis of variance ANOVA)

(1.2.) . . k () l () B. . k x l (factorial) k x l .

, 3 4 3 x 4 . k l kl (cells). ..

2 x 3 6 , 3 x 3 9 . n (n > 1) kl n . , kl kl . , , n , , , = nkl.

1.3.1.

.

1) kl .

2) kl .

3) kl () 2.

23

() . (main effects) (interaction)

. . , , , .

Xi. i , X j. j B. .i j.

X i. X j. ,

H k0 1 2 ... : . . .

H l0 1 2'

. . .: ...

''0H :

: totSS ,

ASS , B BSS , ABSS WSS .

:

SS SS SS SS SStot A B AB W

.

, , , .

24

N k l k l N kl 1 1 1 1 1

( MS)

:

1N

SSMS tottot

, 1k

SSMS AA

, 1l

SSMS BB

, 1l1k

SSMS ABAB

,

klN

SSMS Ww

, ( ) .

, 2 kl 2.

,

w

AA MS

MSF

F (k - 1) ( - kl) .

B,

w

BB MS

MSF

25

F (l - 1) ( - kl) .

w

ABAB MS

MSF

F (k - 1)(l - 1) ( - kl) .

1.3.2.

.

CN

Xijmm

n

j

l

i

k

1

111

2

. , SS

SS X Ctot ijmm

n

j

l

i

k

2

111

SSn

X Cijmm

n

j

l

i

k

1

1

2

11

SS SS SSw tot

SS X CA ijmm

n

j

l

i

k

1

11

2

1ln

B

26

SSkn

X CB ijmm

n

i

k

j

l

1

11

2

1

SS SS SS SSAB A B

1.3.3. 2p

2p (partial eta-squared)

(effect size)

.

weffect

effect2p SSSS

SS

(effect) , . , effectSS ASS , BSS ABSS .

2p [0, 1]

1 .

1.4

, 80 . . 80 40 ( ) 40 / ( ). 40 1, 2, 3, 4 . 40 .

27

. ( ).

. (, /) I 1, 2, 3, 4. 2 x 4 . .

1 2 3 4

54 53 51 46 48 53 51 45

49 55 55 54 57 46 52 46

51 51 43 45 45 47 47 49

45 47 49 46 48 49 48 43

47 45 43 53 50 44 52 46

47 41 43 39 43 45 47 51

35 38 42 44 56 39 51 45

43 44 44 49 44 47 49 48

44 35 43 48 48 56 51 43

43 37 46 41 46 40 47 44

28

,

H II0: I

0 .

.

H0 2 3 4: 1

0H

.

. . , .

.

1 2 3 4

I 497 485 487 479 1948

II 407 439 464 476 1786

904 924 951 955 3734

29

k = 2, l = 4, n = 10, N = 80.

C Xijmmji

1

80 1

10

1

4

1

22

= 3734

80

2

= 174284.45

SS X Ctot ijmmji

2

1

10

1

4

1

2

= (542 + 492 + 512 + ... + 432 + 442) - 174284.45 =

1773.55

SS X Cijmmji

1

10 1

10 2

1

4

1

2

= 1

10(4972 + 4852 + 4872 + 4792 + 4072 + 4392 +

4642 + 4762) - 174284.45 = 624.15

SS SS SSw tot = 1773.55 - 624.15 = 1149.4

( )

SS X CA ijmmji

1

4 10 1

10

1

42

1

2

= 1

40(19482 + 17862) - 174284.45 = 328.05

B ( )

SS X CB ijmmij

1

2 10 1

10

1

2 2

1

4

= 1

20(9042 + 9242 + 9512 + 9552) - 174284.45 =

86.45

SS SS SS SSAB A B = 624.15 - 328.05 - 86.45 = 209.65

30

, ( )

1k

SSMS AA

= 12

05.328

= 328.05

1l

SSMS BB

= 14

45.86

= 28.817

1l1k

SSMS ABAB

= 209 65

1 3

. = 69.883

klN

SSMS Ww

= 1149 4

80 2 4

.

= 15.964

F

w

AA MS

MSF =

328 05

15 964

.

. = 20.55

(2 - 1) (80 - 8) 1 72 F ( = 0.05) 3.98 ( 1 ). 20.55 > 3.98 . . .

w

BB MS

MSF

28817

15 964

.

. = 1.805

(4 - 1) (80 - 8) 3 72 F 2.74. 1.805 < 2.74 . . .

,

31

w

ABAB MS

MSF =

69 883

15 964

.

. = 4.378

(2 - 1)(4 - 1) (80 - 8) 3 72 F 2.74. 4.378 > 2.74 .

F

1 328.05 328.050 20.550+

3 86.45 28.817 1.805

3 209.65 69.883 4.378++

72 1149.40 15.964

79 1773.55

. + , ++ .

= , = .

2p

( ).

wA

A2p SSSS

SS

=

40.114905.328

05.328

= 0.222.

.

wAB

AB2p SSSS

SS

=

40.114965.209

65.209

= 0.154.

, , .

32

1.3.4.

, . , . . .

1 2 3 4

I 49.7 48.5 48.7 47.9

II 40.7 43.9 46.4 47.6

.

35

40

45

50

55

1 2 3 4

33

4 . . 4 , . .

, (simple effects)

. . . SSB I SSB II .

. F

wMS .

1 ( ).

SSB I 497

10

485

10

487

10

479

10

1948

40

2 2 2 2 2

= 16.8 3

8.16MS )I(B = 5.6

w

IBIB MS

MSF =

56

15964

.

. = 0.35. = 0.05

3 72 2.74. 0.35 < 2.74 .

34

2 ( ) .

SSB II 407

10

439

10

464

10

476

10

1786

40

2 2 2 2 2

= 184.2 3

2.184MS )II(B = 61.4

w

IIBIIB MS

MSF =

614

15964

.

. = 3.846. 2.74.

3.846 > 2.74 .

. , / . .

35

1.4.

1.2. () . . , , . . (within-subjects factor) (repeated-measures factor).

( ) (randomization) .

. , . , , , .

n k . ,

36

N = nk. .

, Xij i (i = 1,2,...,n) j (j = 1,2,...,k) . Si (i = 1,2,...,n) Xij i Pj (j = 1,2,...,k) Xij j.

S P Xii

n

jj

k

ijj

k

i

n

1 1 11

1 2 . . j . . k

1 X11 X12 X1j X1k S1

2 X21 X22 X2j X2k S2

. . . . .

. . . . .

i Xi1 Xi2 Xij Xik Si

. . . . .

. . . . .

n Xn1 Xn2 Xnj Xnk Sn

P1 P2 Pj Pk

k

k210 ...:H

1 : k .

37

, (SS). SStot, SScol SSw. .

X ( X =

n

1i

k

1jij N/X )

n

1i

k

1j

2

ijtot XXSS

k21 P...,,P,P k

( n/XPn

1iijj

),

n

1i

k

1j

2

jijw PXSS

k X .

k

1j

2

jcol XPnSS

(1.2.)

SStot = colSS + wSS

, () SSrow (residual variance SSres).

( ) .

38

SSw = rowSS + resSS

SStot = colSS + rowSS + resSS

.

(N - 1) = (k - 1) + (n - 1) + (k - 1)(n - 1)

()

n

1i

2

irow XSkSS

S S Sn1 2, ... ( k/XSk

1jiji

)

n .

resSS = SStot - colSS - rowSS

1k

SSMS colcol

1n1k

SSMS resres

( ) () () , , =0.05 =0.01,

res

col

MS

MSF

F (k - 1) (k - 1)(n - 1) ( 1 ).

39

1.4.1.

:

[] :

2n

1i

k

1jijXN

1]I[

, (SS) :

n

1i

k

1j

2ijtot ]I[XSS

]I[n

P

SS

k

1j

2j

col

]I[k

SSS

n

1i

2i

row

resSS , ,

resSS = SStot - colSS - rowSS

, ( )

1k

SSMScol

col 1n1k

SSMS resres

, res

col

MS

MSF

1.4.2. 2p

2p (partial eta-squared)

(effect size)

.

rescol

col2p SSSS

SS

40

2p [0, 1]

1 .

1.4.3.

, , post hoc . Tukey Scheffe (1.2.5.1, 1.2.6.1) wMS resMS .

( 1.4.4.).

Tukey q 0 : j = j (1.2.5.1.)

n

MS

XXq

res

jjjj

.

Scheffe, t 0 : j = j (1.2.6.1.)

n

MS2

XXt

res

jjjj

S k Fc 1 (1.2.6.1.) k

Fc . St

'jj .

41

1.5

7 . test : (1), (2) (3).

test 3 1, 2 3 .

1 2 3 iS 2iS

1 10 15 12 37 1369

2 15 20 17 52 2704

3 16 18 15 49 2401

4 16 20 17 53 2809

5 18 23 18 59 3481

6 20 21 19 60 3600

7 21 24 22 67 4489

jP 116 141 120 377 20853 2jP 13456 19881 14400 47737

:

0 : 321

1 : .

:

42

2n

1i

k

1jijXN

1]I[

= 21

3772= 6768.05

n

1i

k

1j

2ijtot ]I[XSS = 102 + 152 + 162 + ... + 192 + 222 - 6768.05 = 244.95

]I[n

P

SS

k

1j

2j

col =

7

47737 - 6768.05 = 51.52

]I[k

SSS

n

1i

2i

row =

3

20853 - 6768.05 = 182.95

resSS = SStot - colSS - rowSS = 244.95 - 51.52 - 182.95 = 10.48

1k

SSMS colcol

= 13

52.51

=

2

52.51 = 25.76

1n1kSS

MS resres =

171348.10

=

12

48.10 = 0.87

res

col

MS

MSF =

87.0

76.25 = 29.61

= 0.05 2 12 , 1 , 3.89. 29.61 > 3.89 . .

43

F

2 51.52 25.76 29.61

12 10.48 0.87

6 182.95

20 244.95

2p

rescol

col2p SSSS

SS

=

48.1052.51

52.51

= 0.831.

.

, , post hoc . Scheffe . , X1 = 16.57, X2 = 20.14, X3 = 17.14 . , resMS = 0.87.

t

16.7

7

87.02

14.2057.16

n

MS2

XXt

res

2112

02.6

7

87.02

14.1714.20

n

MS2

XXt

res

3223

14.1

7

87.02

14.1757.16

n

MS2

XXt

res

3113

= 0.05 Fc = 3.89, Scheffe

44

S k Fc 1 = 89.313 = 2.79

7.16 > 2.79 6.01 > 2.79 1.14 < 2.79 (1, 2) (2, 3) 1 2 2 3. , , , ( 1 = 3). . .

1.4.4.

, (1.2.). , , (sphericity).

. , , 1 2, 1 3 2 3 , . .

, (compound symmetry)

. , , . ( ), , cov(X, Y)

YXXY ssr)Y,Xcov(

45

XYr Pearson , Xs Ys

. , . , , . . , .

. , F F. . F F,

. F ,

. SPSS .

46

2

2.1.

, .

. , , , . . . .

.

. , (nonparametric statistical methods).

.

.

47

. . .

. 1- . - - . , . , .

48

2.2.

(sign test)

.

2.1

, 12 . 12 1 5 . . :

() () d

1 5 3 + 2 5 5 3 1 2 - 4 3 2 + 5 4 1 + 6 5 1 + 7 1 2 - 8 4 2 + 9 5 4 + 10 2 1 + 11 4 1 + 12 2 3 -

49

. , .. 1 2 4 5 . . .

(+) (-) di

iii YXd

i =1, 2, ..., N N .

, . , , . , (, ) d (+) (-). .

:

0 : P(X > Y) = P(X < Y) = 0.5

(- > 0 > ) (- < 0 < ). d 0.

, (+) (-). , .

50

, :

1 : P(X > Y) P(X < Y)

1 : P(X > Y) > P(X < Y)

1 : P(X > Y) < P(X < Y)

S(+) N .

3 =0.05. :

0.025 0.975

0.050 0.950

. . .. =13 3 10. [0 3] [10 13] [4 9].

S(+) . S(+) S(-) .

. 12 =11. S(+) = 8 S(-) = 3. =0.05

51

, ( ) :

1 : P(X > Y) > P(X < Y)

3 2 9. S(+)=8 . S(-)=3. .

52

2.3. Wilcoxon

Wilcoxon . ( ) d = - . Wilcoxon Wilcoxon .

: Xi i (i =1, 2,..., N) ,

iii YXd

( ) 0. , id . ,

1, 2, . . d:

2 -6 -3 9 -11 2 -3 5 1 -3

:

2 6 3 9 11 2 3 5 1 3

:

2.5 8 5 9 10 2.5 5 7 1 5

53

, 1 1. 2 . 2 3 (2+3)/2 2.5. 3 . 4, 5 6. (4+5+6)/3=5. 5 7, 6 8, 9 9 11 10.

d, (+). , (-). (+) (-)

= min{(+), (-)}

(+) = 22, (-) = 33 = 22

Wilcoxon . , (+) (-) . , . 4 0.05 0.01 . .

2.2

2.2 Wilcoxon .

54

d, , :

() () d |d|

1 5 3 + 2 7.5 2 5 5 0 3 1 2 - 1 3.5 4 3 2 + 1 3.5 5 4 1 + 3 9.5 6 5 1 + 4 11.0 7 1 2 - 1 3.5 8 4 2 + 2 7.5 9 5 4 + 1 3.5 10 2 1 + 1 3.5 11 4 1 + 3 9.5 12 2 3 - 1 3.5

(+) = 55.5 (-) = 10.5, = 10.5

4 , =0.05 =11 ( d=0) 10 13. 10.5 < 13. . Wilcoxon.

. Wilcoxon .

55

>20 Wilcoxon (0, 1). ( ). , (+)

= 4

1NN

d

=

24

1N21NN

=

T

(0, 1). . (+) (-).

56

2.4. Mann-Whitney

Mann-Whitney (t-test, ) .

Mann-Whitney , . .

. . . .

. + . 1 2 . . . Wilcoxon (2.3.).

: R1 R2 . :

R1+ R2 =

2

1NMNM

:

57

U1 = MN +

1R2

1MM

U2 = MN +

2R2

1NN

U1 U2 :

U1 + U2 =

U U1 U2 . 5 U (==10) =0.05 =0.01 . U . . Wilcoxon Mann-Whitney .

2.3

stress , =5 =7 . stress ( ), ( ). . . :

() 7 4 6 6 8

() 3 2 5 4 2 1 4

. ,

58

.

12 () () . . . 1 1, 2 (2+3)/2 = 2.5, 3 4, 4 3 (5+6+7)/3 = 6, 5 8, 6 2 (9+10)/2 = 9.5, 7 11 8 12.

1 2 2 3 4 4 4 5 6 6 7 8

1 2.5 2.5 4 6 6 6 8 9.5 9.5 11 12

E E E E E E

,

() 11 6 9.5 9.5 12

() 4 2.5 8 6 2.5 1 6

, R1=48, R2=30.

R1 + R2= 78

2

1NMNM =

2

)13(12 78,

.

U1 = MN +

1R2

1MM

= 35 + 48

2

)6(52

U2 = MN +

2R2

1NN

= 35 +

30

2

87 = 33

59

U1 + U2 = =35 U1 U2 .

U = min{2, 33} = 2. =0.05 =5 =7 5 5. 2 < 5 . stress .

10 U

= 2

MN

=

12

1NMMN

(0,1)

=U

(0,1) .

60

2.5.

McNemar , , . . .

2.4

50 . . 50 . 2 x 2

13 6 19

17 14 31

30 20 50

McNemar . 17 ( ), ( ), 6 ( ) ( ).

61

McNemar

f11 f12 f11+f12

f21 f22 f21+f22

f11+f21 f12+f22 n

f11, f12, f21, f22 () .

A ( B) .

0 : PA() = PA()

(f11+f12)/n (f11+f21)/n. f12 = f21 (1, 2) (2, 1). f12 + f21 (f12 + f21)/2 . .

2 ( )

2

2

1

2

1

2

fij ij

ijji

fij () ij ().

62

(1, 2) (1 , 2 ) (2, 1) (2 , 1 ) ,

2

12 21

2

12 21

1

f f

f f

Yates 2 x 2 , .

, 2 1 .

2

, . 2 , McNemar 5 (f12 + f21)/2 5

,

0 : P () = P ()

.

1 :

2

12 21

2

12 21

1

f f

f f =

6 17 16 17

2

= 4.35

2 1 3.84, ( 7,

) . . (f12 + f21)/2 = (6+17)/2 = 11.5 5.

63

2.6. Spearman

Spearman . Pearson ( ) , Spearman .

Spearman , - - . . . Pearson Spearman.

. - -

rd

n nsi

i

n

16

1

2

12

di d X Yi i i

' ' n .

Pearson Spearman [-1, 1].

2.5

AIDS , () () .

64

Spearman. 8 , .

Y X Y d d2

18 30 7 4 3 9 15 28 5 3 2 4 16 32 6 5 1 1 12 33 3 6 -3 9 14 25 4 2 2 4 10 34 1 7 -6 36 11 35 2 8 -6 36 19 24 8 1 7 49

148

1nnd6

1r 2

n

1i

2i

s

=

1648

14861

= 1 - 1.76 = - 0.76

. .

2.6.1. Spearman

s Spearman ,

0 : s = 0

6 n < 30 =0.05 =0.01. Spearman rs

65

, . n > 30

2s

s r1

2nrt

t n - 2 .

, n = 8 = 0.05 6 0.738. rs 0.760 .

66

2.7. Cramer

. Cramer. . Cramer

Cnm

2

, 2 ( ), n m r-1 c-1 r c , .

Cramer 0 1. C=0 C=1 . r = c C=1 .

2.6

400 ) () ) . 1 (), 2 (), 3 (), 1 , 2 , 3. .

67

1 2 3

1 23 (36.8) 34 (33.8) 52 (38.4) 109

2 29 (29.7) 28 (27.3) 31 (31.0) 88

3 83 (68.5) 62 (62.9) 58 (71.6) 203

135 124 141 400

.

:

0 :

1 :

ij ( )

11109 135

400368

x. , 12

109 124

400338

x. , 13

109 141

40038 4

x. ,

2188 135

40029 7

x. , 22

88 124

40027 3

x. , 23

88 141

400310

x. ,

31203 135

400685

x. , 32

203 124

40062 9

x. , 33

203 141

400716

x. ,

2

r

1i

c

1j ij

2ijij2

f

fij () ij ().

68

2

2

1

3

1

3

fij ij

ijji

=

23 368

368

34 338

338

52 38 4

38 4

2 2 2.

.

.

.

.

.

29 29 729 7

28 27 3

27 3

2 2

.

.

.

.

31 310310

83 685

685

62 62 9

62 9

58 716

716

2 2 2 2.

.

.

.

.

.

.

. = 5.175 + 0.001 + 4.817 +

+ 0.016 + 0.018 + 0.000 + 3.069 + 0.013 + 2.583 = 15.6.

= 0.05 (r-1)(c-1) = (3-1)(3-1) = 4 2 7 9.488. 2 . 2 5.

Cramer

Cnm

2

= 2x400

6.15 = 0.14

.

, Cramer m = 1

2

n

.

69

Howell, D. (2008). Fundamental Statistics for the Behavioral Sciences (6th edition), Belmont, CA: Thomson Wadsworth. Howitt, D. and Cramer, D. (2003). An Introduction to Statistics in Psychology (Revised 2nd edition), Essex: Pearson.

Siegel, S. and Castellan, N.J. (1988). Nonparametric Statistics for the Behavioral Sciences (2nd edition), New York: McGraw-Hill.

70

71

1) 12 , . .

5 3 2 7 5 4 8 4 3 9 5 3

( =0.05) . , . 2 .

2) 1), ( =0.05) .

3) 12 6 () () . 6 3 , () , () . . .

5 7

6 6 7 8 4 7

3 7 2 5

( =0.05) .

4) 7 ( ) , .

72

3 6 4 2 7 2 3 8 3 4 5 2 1 9 5 2 7 3 3 8 4

( =0.05) . , .

5) 4), Wilcoxon, ( =0.05) .

6) 8 () (). :

: 7 9 12 5 6 8 10 13 : 10 7 13 4 9 8 11 16

.

7) 6), , ( =0.05) .

8) 200 . ;

60 80 40 20

73

1) . . bSS = 35.717, wSS = 15.950, totSS =51.667.

bMS = 17.858, wMS = 1.772, F = 10.077, . . = (2, 9).

= 4.2. 10.077 > 4.2 . 2 = 0.691. Scheffe: AX = 7.25, BX = 4, X = 3.4.

. , , . .

2) Mann-Whitney. AR = 29.50, R = 15.50.

U = 0.5. =4 =5. = 1. 0.5 < 1 .

3) 2x2 .

2 ( ): F(1, 8) = 13. 5.3. 13 > 5.3 . X =4.50, X =

6.67. .

2 ( ): F(1, 8) = 9.31. 5.3. 9.31 > 5.3 . X =6.50, X = 4.67.

() ().

: F(1, 8) = 3.77. 5.3. 3.77 < 5.3 .

4) , , . ( mn). colSS = 84.667, resSS = 18, colMS = 42.333,

resMS = 1.5, F = 28.22, . . = (2, 12). = 3.8.

28.222 > 3.8

74

. Scheffe: XX = 2.57, YX = 7.14, ZX = 3.29.

. , , . , .

5) Wilcoxon. (+) = 0 (-) = 28, = 0. =7, = 2. 0 < 2 .

6) Spearman sr = 0.786.

7) . S(+) = 2, S(-) = 5. = 0. =7. = 0. 2 ( 5) (0, 7). .

8) McNemar. 2 = 12.675. = 3.841. 12.675 >

3.841 . (f12 + f21)/2 = 60 > 5.

75

76

1. F ( = 0.05)

: (1).

: (2).

1 2 3 4 5 6 7 8 9 10 12 15 20 24 30 40 60 120

1 161 200 216 225 230 234 237 239 241 242 244 246 248 249 250 251 252 253 254 2 18. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19. 19.

3 10. 9.5 9.2 9.1 9.0 8.9 8.8 8.8 8.8 8.7 8.7 8.7 8.6 8.6 8.6 8.5 8.5 8.5 8.5

4 7.7 6.9 6.5 6.3 6.2 6.1 6.0 6.0 6.0 5.9 5.9 5.8 5.8 5.7 5.7 5.7 5.6 5.6 5.6

5 6.6 5.7 5.4 5.1 5.0 4.9 4.8 4.8 4.7 4.7 4.6 4.6 4.5 4.5 4.5 4.4 4.4 4.4 4.3

6 5.9 5.1 4.7 4.5 4.3 4.2 4.2 4.1 4.1 4.0 4.0 3.9 3.8 3.8 3.8 3.7 3.7 3.7 3.6

7 5.5 4.7 4.3 4.1 3.9 3.8 3.7 3.7 3.6 3.6 3.5 3.5 3.4 3.4 3.3 3.3 3.3 3.2 3.2

8 5.3 4.4 4.0 3.8 3.6 3.5 3.5 3.4 3.3 3.3 3.2 3.2 3.1 3.1 3.0 3.0 3.0 2.9 2.9

9 5.1 4.2 3.8 3.6 3.4 3.3 3.2 3.2 3.1 3.1 3.0 3.0 2.9 2.9 2.8 2.8 2.7 2.7 2.7

10 4.9 4.1 3.7 3.4 3.3 3.2 3.1 3.0 3.0 2.9 2.9 2.8 2.7 2.7 2.7 2.6 2.6 2.5 2.5

11 4.8 3.9 3.5 3.3 3.2 3.0 3.0 2.9 2.9 2.8 2.7 2.7 2.6 2.6 2.5 2.5 2.4 2.4 2.4

12 4.7 3.8 3.4 3.2 3.1 3.0 2.9 2.8 2.8 2.7 2.6 2.6 2.5 2.5 2.4 2.4 2.3 2.3 2.3

13 4.6 3.8 3.4 3.1 3.0 2.9 2.8 2.7 2.7 2.6 2.6 2.5 2.4 2.4 2.3 2.3 2.3 2.2 2.2

14 4.6 3.7 3.3 3.1 2.9 2.8 2.7 2.7 2.6 2.6 2.5 2.4 2.3 2.3 2.3 2 2.2 2.1 2.1

15 4.5 3.6 3.2 3.0 2.9 2.7 2.7 2.6 2.5 2.5 2.4 2.4 2.3 2.2 2.2 2.2 2.1 2.1 2.0

16 4.4 3.6 3.2 3.0 2.8 2.7 2.6 2.5 2.5 2.4 2.4 2.3 2.2 2.2 2.1 2.1 2.1 2.0 2.0

17 4.4 3.5 3.2 2.9 2.8 2.7 2.6 2.5 2.4 2.4 2.3 2.3 2.2 2.1 2.1 2.1 2.0 2.0 1.9

18 4.4 3.5 3.1 2.9 2.7 2.6 2.5 2.5 2.4 2.4 2.3 2.2 2.1 2.1 2.1 2.0 2.0 1.9 1.9

19 4.3 3.5 3.1 2.9 2.7 2.6 2.5 2.4 2.4 2.3 2.3 2.2 2.1 2.1 2.0 2.0 1.9 1.9 1.8

20 4.3 3.4 3.1 2.8 2.7 2.6 2.5 2.4 2.3 2.3 2.2 2.2 2.1 2.0 2.0 1.9 1.9 1.9 1.8

21 4.3 3.4 3.0 2.8 2.6 2.5 2.4 2.4 2.3 2.3 2.2 2.1 2.1 2.0 2.0 1.9 1.9 1.8 1.8

22 4.3 3.4 3.0 2.8 2.6 2.5 2.4 2.4 2.3 2.3 2.2 2.1 2.0 2.0 1.9 l.94 1.8 1.8 1.7

23 4.2 3.4 3.0 2.8 2.6 2.5 2.4 2.3 2.3 2.2 2.2 2.1 2.0 2.0 1.9 1.9 1.8 1.8 1.7

24 4.2 3.4 3.0 2.7 2.6 2.5 2.4 2.3 2.3 2.2 2.1 2.1 2.0 1.9 1.9 1.8 1.8 1.7 1.7

25 4.2 3.3 2.9 2.7 2.6 2.4 2.4 2.3 2.2 2.2 2.1 2.0 2.0 1.9 1.9 1.8 1.8 1.7 1.7

26 4.2 3.3 2.9 2.7 2.5 2.4 2.3 2.3 2.2 2.2 2.1 2.0 1.9 1.9 1.9 1.8 1.8 1.7 1.6

27 4.2 3.3 2.9 2.7 2.5 2.4 2.3 2.3 2.2 2.2 2.1 2.0 1.9 1.9 1.8 1.8 1.7 1.7 1.6

28 4.2 3.3 2.9 2.7 2.5 2.4 2.3 2.2 2.2 2.1 2.1 2.0 1.9 1.9 1.8 1.8 1.7 1.7 1.6

29 4.1 3.3 2.9 2.7 2.5 2.4 2.3 2.2 2.2 2.1 2.1 2.0 1.9 1.9 1.8 1.8 1.7 1.7 1.6

30 4.1 3.3 2.9 2.6 2.5 2.4 2.3 2.2 2.2 2.1 2.0 2.0 1.9 1.8 1.8 1.7 1.7 1.6 1.6

40 4.0 3.2 2.8 2.6 2.4 2.3 2.2 2.1 2.1 2.0 2.0 1.9 1.8 1.7 1.7 1.6 1.6 1.5 1.5

60 4.0 3.1 2.7 2.5 2.3 2.2 2.1 2.1 2.0 1.9 1.9 1.8 1.7 1.7 1.6 1.5 1.5 1.4 1.3

120 3.9 3.0 2.6 2.4 2.2 2.1 2.0 2.0 1.9 1.9 1.8 1.7 1.6 1.6 1.5 1.5 1.4 1.3 1.2

3.8 3.0 2.6 2.3 2.2 2.1 2.0 1.9 1.8 1.8 1.7 1.6 1.5 1.5 1.4 1.3 1.3 1.2 1.0

77

1. (). F ( = 0.01)

: (1).

: (2).

1 2 3 4 5 6 7 8 9 10 12 15 20 24 30 40 60 120

1 405 500 540 562 576 585 592 598 602 605 610 615 620 623 626 628 631 633 6362 98.5 99.0 99.2 99.2 99.3 99.3 99.4 99.4 99.4 99.4 99.4 99.4 99.4 99.5 99.5 99.5 99.5 99.5 99.5

3 34.1 30.8 29.5 28.7 28.2 27.9 27.7 27.5 27.3 27.2 27.1 26.9 26.7 26.6 26.5 26.4 26.3 26.2 26.1

4 21.2 18.0 16.7 16.0 15.5 15.2 15.0 14.8 14.7 14.5 14.4 14.2 14.0 13.9 13.8 13.7 13.7 13.6 13.5

5 16.3 13.3 12.1 11.4 11.0 10.7 10.5 10.3 10.2 10.1 9.89 9.72 9.55 9.47 9.38 9.29 9.20 9.11 9.02

6 13.7 10.9 9.78 9.15 8.75 8.47 8.26 8.10 7.98 7.87 7.72 7.56 7.40 7.31 7.23 7.14 7.06 6.97 6.88

7 12.2 9.55 8.45 7.85 7.46 7.19 6.99 6.84 6.72 6.62 6.47 6.31 6.16 6.07 5.99 5.91 5.82 5.74 5.65

8 11.3 8.65 7.59 7.01 6.63 6.37 6.18 6.03 5.91 5.81 5.67 5.52 5.36 5.28 5.20 5.12 5.03 4.95 4.86

9 10.6 8.02 6.99 6.42 6.06 5.80 5.61 5.47 5.35 5.26 5.11 4.96 4.81 4.73 4.65 4.57 4.48 4.40 4.31

10 10.0 7.56 6.55 5.99 5.64 5.39 5.20 5.06 4.94 4.85 4.71 4.56 4.41 4.33 4.25 4.17 4.08 4.00 3.91

11 9.65 7.21 6.22 5.67 5.32 5.07 4.89 4.74 4.63 4.54 4.40 4.25 4.10 4.02 3.94 3.86 3.78 3.69 3.60

12 9.33 6.93 5.95 5.41 5.06 4.82 4.64 4.50 4.39 4.30 4.16 4.01 3.86 3.78 3.70 3.62 3.54 3.45 3.36

13 9.07 6.70 5.74 5.21 4.86 4.62 4.44 4.30 4.19 4.10 3.96 3.82 3.66 3.59 3.51 3.43 3.34 3.25 3.17

14 8.86 6.51 5.56 5.04 4.70 4.46 4.28 4.14 4.03 3.94 3.80 3.66 3.51 3.43 3.35 3.27 3.18 3.09 3.00

15 8.68 6.36 5.42 4.89 4.56 4.32 4.14 4.00 3.89 3.80 3.67 3.52 3.37 3.29 3.21 3.13 3.05 2.96 2.87

16 8.53 6.23 5.29 4.77 4.44 4.20 4.03 3.89 3.78 3.69 3.55 3.41 3.26 3.18 3.10 3.02 2.93 2.84 2.75

17 8.40 6.11 5.19 4.67 4.34 4.10 3.93 3.79 3.68 3.59 3.46 3.31 3.16 3.08 3.00 2.92 2.83 2.75 2.65

18 8.29 6.01 5.09 4.58 4.25 4.01 3.84 3.71 3.60 3.51 3.37 3.23 3.08 3.00 2.92 2.84 2.75 2.66 2.57

19 8.18 5.93 5.01 4.50 4.17 3.94 3.77 3.63 3.52 3.43 3.30 3.15 3.00 2.92 2.84 2.76 2.67 2.58 2.49

20 8.10 5.85 4.94 4.43 4.10 3.87 3.70 3.56 3.46 3.37 3.23 3.09 2.94 2.86 2.78 2.69 2.61 2.52 2.42

21 8.02 5.78 4.87 4.37 4.04 3.81 3.64 3.51 3.40 3.31 3.17 3.03 2.88 2.80 2.72 2.64 2.55 2.46 2.36

22 7.95 5.72 4.82 4.31 3.99 3.76 3.59 3.45 3.35 3.26 3.12 2.98 2.83 2.75 2.67 2.58 2.50 2.40 2.31

23 7.88 5.66 4.76 4.26 3.94 3.71 3.54 3.41 3.30 3.21 3.07 2.93 2.78 2.70 2.62 2.54 2.45 2.35 2.26

24 7.82 5.61 4.72 4.22 3.90 3.67 3.50 3.36 3.26 3.17 3.03 2.89 2.74 2.66 2.58 2.49 2.40 2.31 2.21

25 7.77 5.57 4.68 4.18 3.86 3.63 3.46 3.32 3.22 3.13 2.99 2.85 2.70 2.62 2.54 2.45 2.36 2.27 2.17

26 7.72 5.53 4.64 4.14 3.82 3.59 3.42 3.29 3.18 3.09 2.96 2.82 2.66 2.58 2.50 2.42 2.33 2.23 2.13

27 7.68 5.49 4.60 4.11 3.78 3.56 3.39 3.26 3.15 3.06 2.93 2.78 2.63 2.55 2.47 2.38 2.29 2.20 2.10

28 7.64 5.45 4.57 4.07 3.75 3.53 3.36 3.23 3.12 3.03 2.90 2.75 2.60 2.52 2.44 2.35 2.26 2.17 2.06

29 7.60 5.42 4.54 4.04 3.73 3.50 3.33 3.20 3.09 3.00 2.87 2.73 2.57 2.49 2.41 2.33 2.23 2.14 2.03

30 7.56 5.39 4.51 4.02 3.70 3.47 3.30 3.17 3.07 2.98 2.84 2.70 2.55 2.47 2.39 2.30 2.21 2.11 2.01

40 7.31 5.18 4.31 3.83 3.51 3.29 3.12 2.99 2.89 2.80 2.66 2.52 2.37 2.29 2.20 2.11 2.02 1.92 1.80

60 7.08 4.98 4.13 3.65 3.34 3.12 2.95 2.82 2.72 2.63 2.50 2.35 2.20 2.12 2.03 1.94 1.84 1.73 1.60

120 6.85 4.79 3.95 3.48 3.17 2.96 2.79 2.66 2.56 2.47 2.34 2.19 2.03 1.95 1.86 1.76 1.66 1.53 1.38

6.63 4.61 3.78 3.32 3.02 2.80 2.64 2.51 2.41 2.32 2.18 2.04 1.88 1.79 1.70 1.59 1.47 1.32 1.00

78

2. Tukey

k 2 3 4 5 6 7 8 9 10 11

5 0.05 3.64 4.60 5.22 5.67 6.03 6.33 6.58 6.80 6.99 7.17

0.01 5.70 6.98 7.80 8.42 8.91 9.32 9.67 9.97 10.24 10.48

6 0.05 3.46 4.34 4.90 5.30 5.63 5.90 6.12 6.32 6.49 6.65

0.01 5.24 6.33 7.03 7.56 7.97 8.32 8.61 8.87 9.10 9.30

7 0.05 3.34 4.16 4.68 5.06 5.36 5.61 5.82 6.00 6.16 6.30

0.01 4.95 5.92 6.54 7.01 7.37 7.68 7.94 8.17 8.37 8.55

8 0.05 3.26 4.04 4.53 4.89 5.17 5.40 5.60 5.77 5.92 6.05

0.01 4.75 5.64 6.20 6.62 6.96 7.24 7.47 7.68 7.86 8.03

9 0.05 3.20 3.95 4.41 4.76 5.02 5.24 5.43 5.59 5.74 5.87

0.01 4.60 5.43 5.96 6.35 6.66 6.91 7.13 7.33 7.49 7.65

10 0.05 3.15 3.88 4.33 4.65 4.91 5.12 5.30 5.46 5.60 5.72

0.01 4.48 5.27 5.77 6.14 6.43 6.67 6.87 7.05 7.21 7.36

11 0.05 3.11 3.82 4.26 4.57 4.82 5.03 5.20 5.35 5.49 5.61

0.01 4.39 5.15 5.62 5.97 6.25 6.48 6.67 6.84 6.99 7.13

12 0.05 3.08 3.77 4.20 4.51 4.75 4.95 5.12 5.27 5.39 5.51

0.01 4.32 5.05 5.50 5.84 6.10 6.32 6.51 6.67 6.81 6.94

13 0.05 3.06 3.73 4.15 4.45 4.69 4.88 5.05 5.19 5.32 5.43

0.01 4.26 4.96 5.40 5.73 5.98 6.19 6.37 6.53 6.67 6.79

14 0.05 3.03 3.70 4.11 4.41 4.64 4.83 4.99 5.13 5.25 5.36

0.01 4.21 4.89 5.32 5.63 5.88 6.08 6.26 6.41 6.54 6.66

15 0.05 3.01 3.67 4.08 4.37 4.59 4.78 4.94 5.08 5.20 5.31

0.01 4.17 4.84 5.25 5.56 5.80 5.99 6.16 6.31 6.44 6.55

16 0.05 3.00 3.65 4.05 4.33 4.56 4.74 4.90 5.03 5.15 5.26

0.01 4.13 4.79 5.19 5.49 5.72 5.92 6.08 6.22 6.35 6.46

17 0.05 2.98 3.63 4.02 4.30 4.52 4.70 4.86 4.99 5.11 5.21

0.01 4.10 4.74 5.14 5.43 5.66 5.85 6.01 6.15 6.27 6.38

18 0.05 2.97 3.61 4.00 4.28 4.49 4.67 4.82 4.96 5.07 5.17

0.01 4.07 4.70 5.09 5.38 5.60 5.79 5.94 6.08 6.20 6.31

19 0.05 2.96 3.59 3.98 4.25 4.47 4.65 4.79 4.92 5.04 5.14

0.01 4.05 4.67 5.05 5.33 5.55 5.73 5.89 6.02 6.14 6.25

79

2. (). Tukey

k 2 3 4 5 6 7 8 9 10 11

20 0.05 2.95 3.58 3.96 4.23 4.45 4.62 4.77 4.90 5.01 5.11

0.01 4.02 4.64 5.02 5.29 5.51 5.69 5.84 5.97 6.09 6.19

24 0.05 2.92 3.53 3.90 4.17 4.37 4.54 4.68 4.81 4.92 5.01

0.01 3.96 4.55 4.91 5.17 5.37 5.54 5.69 5.81 5.92 6.02

30 0.05 2.89 3.49 3.85 4.10 4.30 4.46 4.60 4.72 4.82 4.92

0.01 3.89 4.45 4.80 5.05 5.24 5.40 5.54 5.65 5.76 5.85

40 0.05 2.86 3.44 3.79 4.04 4.23 4.39 4.52 4.63 4.73 4.82

0.01 3.82 4.37 4.70 4.93 5.11 5.26 5.39 5.50 5.60 5.69

60 0.05 2.83 3.40 3.74 3.98 4.16 4.31 4.44 4.55 4.65 4.73

0.01 3.76 4.28 4.59 4.82 4.99 5.13 5.25 5.36 5.45 5.53

120 0.05 2.80 3.36 3.68 3.92 4.10 4.24 4.36 4.47 4.56 4.64

0.01 3.70 4.20 4.50 4.71 4.87 5.01 5.12 5.21 5.30 5.37

0.05 2.77 3.31 3.63 3.86 4.03 4.17 4.29 4.39 4.47 4.55

0.01 3.64 4.12 4.40 4.60 4.76 4.88 4.99 5.08 5.16 5.23

80

3. (=0.05)

N 0.025 0.050 0.950 0.975 5 0 5 6 0 0 6 6 7 0 0 7 7 8 0 1 7 8 9 1 1 8 8

10 1 1 9 9 11 1 2 9 10 12 2 2 10 10 13 2 3 10 11 14 2 3 11 12 15 3 3 12 12 16 3 4 12 13 17 4 4 13 13 18 4 5 13 14 19 4 5 14 15 20 5 5 15 15 21 5 6 15 16 22 5 6 16 17 23 6 7 16 17 24 6 7 17 18 25 7 7 18 18

81

4. Wilcoxon

0.05 0.025 0.01 0.005

5 0

6 2 0

7 3 2 0

8 5 3 1 0

9 8 5 3 1

10 10 8 5 3

11 13 10 7 5

12 17 13 9 7

13 21 17 12 9

14 25 21 15 12

15 30 25 19 15

16 35 29 23 19

17 41 34 27 23

18 47 40 32 27

19 53 46 37 32

20 60 52 43 37

82

5. Mann-Whitney

0.05 0.025 0.01 0.005 2 5 0 2 6 0 2 7 0 2 8 1 0 2 9 1 0 2 10 1 0 3 3 0 3 4 0 3 5 1 0 3 6 2 1 3 7 2 1 0 3 8 3 2 0 3 9 4 2 1 0 3 10 4 3 1 0 4 4 1 0 4 5 2 1 0 4 6 3 2 1 0 4 7 4 3 1 0 4 8 5 4 2 1 4 9 6 4 3 1 4 10 7 5 3 2 5 5 4 2 1 0 5 6 5 3 2 1 5 7 6 5 3 1 5 8 8 6 4 2 5 9 9 7 5 3 5 10 11 8 6 4 6 6 7 5 3 2 6 7 8 6 4 3 6 8 10 8 6 4 6 9 12 10 7 5 6 10 14 11 8 6 7 7 11 8 6 4 7 8 13 10 7 6 7 9 15 12 9 7 7 10 17 14 11 9 8 8 15 13 9 7 8 9 18 15 11 9 8 10 20 17 13 11 9 9 21 17 14 11 9 10 24 20 16 13

10 10 27 23 19 16

83

6. Spearman

n 0.05 0.025 0.01 0.005

4 1.000

5 0.900 1.000 1.000

6 0.829 0.886 0.943 1.000

7 0.714 0.786 0.893 0.929

8 0.643 0.738 0.833 0.881

9 0.600 0.700 0.783 0.833

10 0.564 0.648 0.745 0.794

11 0.536 0.618 0.709 0.755

12 0.503 0.587 0.671 0.727

13 0.484 0.560 0.648 0.703

14 0.464 0.538 0.622 0.675

15 0.443 0.521 0.604 0.654

16 0.429 0.503 0.582 0.635

17 0.414 0.485 0.566 0.615

18 0.401 0.472 0.550 0.600

19 0.391 0.460 0.535 0.584

20 0.380 0.447 0.520 0.570

21 0.370 0.435 0.508 0.556

22 0.361 0.425 0.496 0.544

23 0.353 0.415 0.486 0.532

24 0.344 0.406 0.476 0.521

25 0.337 0.398 0.466 0.511

26 0.331 0.390 0.457 0.501

27 0.324 0.382 0.448 0.491

28 0.317 0.375 0.440 0.483

29 0.312 0.368 0.433 0.475

30 0.306 0.362 0.425 0.467

84

7. 2

0.10 0.05 0.025 0.01 0.005 0.001

1 2.706 3.841 5.024 6.635 7.879 10.828

2 4.605 5.991 7.378 9.210 10.597 13.816

3 6.251 7.815 9.348 11.345 12.838 16.266

4 7.779 9.488 11.143 13.277 14.860 18.467

5 9.236 11.070 12.833 15.086 16.750 20.515

6 10.645 12.592 14.449 16.812 18.548 22.458

7 12.017 14.067 16.013 18.475 20.278 24.322

8 13.362 15.507 17.535 20.090 21.955 26.124

9 14.684 16.919 19.023 21.666 23.589 27.877

10 15.987 18.307 20.483 23.209 25.188 29.588

11 17.275 19.675 21.920 24.725 26.757 31.264

12 18.549 21.026 23.337 26.217 28.300 32.909

13 19.812 22.362 24.736 27.688 29.819 34.528

14 21.064 23.685 26.119 29.141 31.319 36.123

15 22.307 24.996 27.488 30.578 32.801 37.697

16 23.542 26.296 28.845 32.000 34.267 39.252

17 24.769 27.587 30.191 33.409 35.718 40.790

18 25.989 28.869 31.526 34.805 37.156 42.312

19 27.204 30.144 32.852 36.191 38.582 43.820

20 28.412 31.410 34.170 37.566 39.997 45.315

21 29.615 32.671 35.479 38.932 41.401 46.797

22 30.813 33.924 36.781 40.289 42.796 48.268

23 32.007 35.172 38.076 41.638 44.181 49.728

24 33.196 36.415 39.364 42.980 45.559 51.179

25 34.382 37.652 40.646 44.314 46.928 52.620

26 35.563 38.885 41.923 45.642 48.290 54.052

27 36.741 40.113 43.195 46.963 49.645 55.476

28 37.916 41.337 44.461 48.278 50.993 56.892

29 39.087 42.557 45.722 49.588 52.336 58.301

30 40.256 43.773 46.979 50.892 53.672 59.703

40 51.805 55.758 59.342 63.691 66.766 73.402

60 74.397 79.082 83.298 88.379 91.952 99.607

80 96.578 101.879 106.629 112.329 116.321 124.839

100 118.498 124.342 129.561 135.807 140.169 149.449