ΣΤΑΤΙΣΤΙΚΗ ΙΙ
DESCRIPTION
...............TRANSCRIPT
-
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