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Transcript of ΣΤΑΤΙΣΤΙΚΗ ΙΙ

  • 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