Quantilstabellen werden an zahlreichen Stellen in der mathematischen Statistik verwendet. So werden sie beispielsweise für die Bestimmung von Konfidenzintervallen herangezogen. Des Weiteren lassen sich bei normalverteilten Zufallsvariablen mit gegebener Varianz und gegebenem Erwartungswert über die Z-Transformation in Kombination mit der entsprechenden Quantiltabelle direkt Wahrscheinlichkeiten bestimmen.
Als p-Quantil einer Wahrscheinlichkeitsverteilung auf den reellen Zahlen wird eine reelle Zahl bezeichnet, so dass
und
ist. Hierbei ist . Besitzt die Wahrscheinlichkeitsverteilung eine stetige Verteilungsfunktion, so ist die äquivalent zu
.
Ist die Verteilungsfunktion streng monoton wachsend. so ist eindeutig bestimmt. Das p-Quantil trennt dann die reellen Zahlen in zwei Teile: der Teil kleiner als , welcher die Wahrscheinlichkeit erhält, und der Teil größer als , welcher die Wahrscheinlichkeit erhält.
In vielen Anwendungen der Statistik benötigt man häufig die Quantile gewisser Wahrscheinlichkeitsverteilungen. Zu diesen Verteilungen gehören:
Alle Quantile dieser Verteilungen sind eindeutig. Jedoch existiert für manche Verteilungen keine geschlossene Darstellung der Verteilungsfunktion (Normalverteilung) oder diese geschlossene Darstellung ist sehr komplex beziehungsweise das Lösen der Gleichung
ist nicht praktikabel. Daher werden die wichtigen Quantile diese Verteilungen mit der notwendigen Genauigkeit numerisch bestimmt und in Tabellen zusammengefasst. So können sie nachgeschlagen werden, ohne jedes Mal erneut numerisch bestimmt zu werden.
Welche Werte die Tabelle genau enthält und wie viele Nachkommastellen sie enthält, hängt von der jeweiligen Verteilung ab und dem Kontext, in dem diese benötigt wird. So wird für die leicht zu normierende Normalverteilung beispielsweise nur die Standardnormalverteilung tabelliert, hierbei dann jedoch mit auf zwei Nachkommastellen genau und das entsprechende auf vier Nachkommastellen. Für die Student-Verteilung werden dagegen nur die Quantile etc. angegeben, dafür aber mit variabler Anzahl an Freiheitsgraden. Details hierzu und zur Verwendung der einzelnen Tabellen finden sich in den entsprechenden Abschnitten.
Wichtige Quantiltabellen
Nachfolgend sind einige wichtige Quantiltabellen aufgeführt. Die Auswahl der tabellierten Werte folgt dabei[1][2][3]
p-Quantile der Chi-Quadrat-Verteilung mit n Freiheitsgraden
p=0,005
0,010
0,020
0,025
0,050
0,100
0,250
0,500
0,750
0,900
0,950
0,975
0,980
0,990
0,995
n=1
3,927e-5
1,571e-4
6,285e-4
9,820e-4
3,932e-3
1,579e-2
1,015e-1
4,549e-1
1,323
2,706
3,841
5,024
5,412
6,635
7,879
2
1,003e-2
2,010e-2
4,041e-2
5,064e-2
1,026e-1
2,107e-1
5,754e-1
1,386
2,773
4,605
5,991
7,378
7,824
9,210
10,60
3
7,172e-2
1,1480e-1
1,848e-1
2,158e-1
3,518e-1
5,844e-1
1,213
2,366
4,108
6,251
7,815
9,348
9,837
11,34
12,84
4
2,070e-1
2,971e-1
4,294e-1
4,844e-1
7,107e-1
1,064
1,923
3,357
5,385
7,779
9,488
11,14
11,67
13,28
14,86
5
4,117e-1
5,543e-1
7,519e-1
8,312e-1
1,145
1,6100
2,675
4,351
6,626
9,236
11,07
12,83
13,39
15,09
16,75
6
6,757e-1
8,721e-1
1,134
1,237
1,635
2,204
3,455
5,348
7,841
10,64
12,59
14,45
15,03
16,81
18,55
7
0,9893
1,239
1,564
1,6900
2,167
2,833
4,255
6,346
9,037
12,02
14,07
16,01
16,62
18,48
20,28
8
1,344
1,646
2,032
2,180
2,733
3,490
5,071
7,344
10,22
13,36
15,51
17,53
18,17
20,09
21,95
9
1,735
2,088
2,532
2,700
3,325
4,168
5,899
8,343
11,39
14,68
16,92
19,02
19,68
21,67
23,59
10
2,156
2,558
3,059
3,247
3,940
4,865
6,737
9,342
12,55
15,99
18,31
20,48
21,16
23,21
25,19
11
2,603
3,053
3,609
3,816
4,575
5,578
7,584
10,34
13,70
17,28
19,68
21,92
22,62
24,72
26,76
12
3,074
3,571
4,178
4,404
5,226
6,304
8,438
11,34
14,85
18,55
21,03
23,34
24,05
26,22
28,30
13
3,565
4,107
4,765
5,009
5,892
7,042
9,299
12,34
15,98
19,81
22,36
24,74
25,47
27,69
29,82
14
4,075
4,660
5,368
5,629
6,571
7,790
10,17
13,34
17,12
21,06
23,68
26,12
26,87
29,14
31,32
15
4,601
5,229
5,985
6,262
7,261
8,547
11,04
14,34
18,25
22,31
25,00
27,49
28,26
30,58
32,80
16
5,142
5,812
6,614
6,908
7,962
9,312
11,91
15,34
19,37
23,54
26,30
28,85
29,63
32,00
34,27
17
5,697
6,408
7,255
7,564
8,672
10,09
12,79
16,34
20,49
24,77
27,59
30,19
31,00
33,41
35,72
18
6,265
7,015
7,906
8,231
9,390
10,86
13,68
17,34
21,60
25,99
28,87
31,53
32,35
34,81
37,16
19
6,844
7,633
8,567
8,907
10,12
11,65
14,56
18,34
22,72
27,20
30,14
32,85
33,69
36,19
38,58
20
7,434
8,260
9,237
9,591
10,85
12,44
15,45
19,34
23,83
28,41
31,41
34,17
35,02
37,57
40,00
21
8,034
8,897
9,915
10,28
11,59
13,24
16,34
20,34
24,93
29,62
32,67
35,48
36,34
38,93
41,40
22
8,643
9,542
10,60
10,98
12,34
14,04
17,24
21,34
26,04
30,81
33,92
36,78
37,66
40,29
42,80
23
9,260
10,20
11,29
11,69
13,09
14,85
18,14
22,34
27,14
32,01
35,17
38,08
38,97
41,64
44,18
24
9,886
10,86
11,99
12,40
13,85
15,66
19,04
23,34
28,24
33,20
36,42
39,36
40,27
42,98
45,56
25
10,52
11,52
12,70
13,12
14,61
16,47
19,94
24,34
29,34
34,38
37,65
40,65
41,57
44,31
46,93
26
11,16
12,20
13,41
13,84
15,38
17,29
20,84
25,34
30,43
35,56
38,89
41,92
42,86
45,64
48,29
27
11,81
12,88
14,13
14,57
16,15
18,11
21,75
26,34
31,53
36,74
40,11
43,19
44,14
46,96
49,64
28
12,46
13,56
14,85
15,31
16,93
18,94
22,66
27,34
32,62
37,92
41,34
44,46
45,42
48,28
50,99
29
13,12
14,26
15,57
16,05
17,71
19,77
23,57
28,34
33,71
39,09
42,56
45,72
46,69
49,59
52,34
30
13,79
14,95
16,31
16,79
18,49
20,60
24,48
29,34
34,80
40,26
43,77
46,98
47,96
50,89
53,67
35
17,19
18,51
20,03
20,57
22,47
24,80
29,05
34,34
40,22
46,06
49,80
53,20
54,24
57,34
60,27
40
20,71
22,16
23,84
24,43
26,51
29,05
33,66
39,34
45,62
51,81
55,76
59,34
60,44
63,69
66,77
45
24,31
25,90
27,72
28,37
30,61
33,35
38,29
44,34
50,98
57,51
61,66
65,41
66,56
69,96
73,17
50
27,99
29,71
31,66
32,36
34,76
37,69
42,94
49,33
56,33
63,17
67,50
71,42
72,61
76,15
79,49
55
31,73
33,57
35,66
36,40
38,96
42,06
47,61
54,33
61,66
68,80
73,31
77,38
78,62
82,29
85,75
60
35,53
37,48
39,70
40,48
43,19
46,46
52,29
59,33
66,98
74,40
79,08
83,30
84,58
88,38
91,95
70
43,28
45,44
47,89
48,76
51,74
55,33
61,70
69,33
77,58
85,53
90,53
95,02
96,39
100,4
104,2
80
51,17
53,54
56,21
57,15
60,39
64,28
71,14
79,33
88,13
96,58
101,9
106,6
108,1
112,3
116,3
90
59,20
61,75
64,63
65,65
69,13
73,29
80,62
89,33
98,65
107,6
113,1
118,1
119,6
124,1
128,3
100
67,33
70,06
73,14
74,22
77,93
82,36
90,13
99,33
109,1
118,5
124,3
129,6
131,1
135,8
140,2
150
109,1
112,7
116,6
118,0
122,7
128,3
138,0
149,3
161,3
172,6
179,6
185,8
187,7
193,2
198,4
200
152,2
156,4
161,1
162,7
168,3
174,8
186,2
199,3
213,1
226,0
234,0
241,1
243,2
249,4
255,3
250
196,2
200,9
206,2
208,1
214,4
221,8
234,6
249,3
264,7
279,1
287,9
295,7
298,0
304,9
311,3
300
240,7
246,0
251,9
253,9
260,9
269,1
283,1
299,3
316,1
331,8
341,4
349,9
352,4
359,9
366,8
400
330,9
337,2
344,1
346,5
354,6
364,2
380,6
399,3
418,7
436,6
447,6
457,3
460,2
468,7
476,6
600
514,5
522,4
531,0
534,0
544,2
556,1
576,3
599,3
623,0
644,8
658,1
669,8
673,3
683,5
693,0
800
700,7
709,9
720,0
723,5
735,4
749,2
772,7
799,3
826,6
851,7
866,9
880,3
884,3
896,0
906,8
1000
888,6
898,9
910,3
914,3
927,6
943,1
969,5
999,3
1029
1057
1074
1089
1094
1107
1118
Studentsche t-Verteilung
p-Quantile der Studentschen t-Verteilung mit n Freiheitsgraden (einseitiger Test)
p=0,9
0,95
0,96
0,975
0,98
0,99
0,995
0,999
0,9995
n=1
3,078
6,314
7,916
12,71
15,89
31,82
63,66
318,3
636,6
2
1,886
2,920
3,320
4,303
4,849
6,965
9,925
22,33
31,60
3
1,638
2,353
2,605
3,182
3,482
4,541
5,841
10,21
12,92
4
1,533
2,132
2,333
2,776
2,999
3,747
4,604
7,173
8,610
5
1,476
2,015
2,191
2,571
2,757
3,365
4,032
5,893
6,869
6
1,440
1,943
2,104
2,447
2,612
3,143
3,707
5,208
5,959
7
1,415
1,895
2,046
2,365
2,517
2,998
3,499
4,785
5,408
8
1,397
1,860
2,004
2,306
2,449
2,896
3,355
4,501
5,041
9
1,383
1,833
1,973
2,262
2,398
2,821
3,250
4,297
4,781
10
1,372
1,812
1,948
2,228
2,359
2,764
3,169
4,144
4,587
11
1,363
1,796
1,928
2,201
2,328
2,718
3,106
4,025
4,437
12
1,356
1,782
1,912
2,179
2,303
2,681
3,055
3,930
4,318
13
1,350
1,771
1,899
2,160
2,282
2,650
3,012
3,852
4,221
14
1,345
1,761
1,887
2,145
2,264
2,624
2,977
3,787
4,140
15
1,341
1,753
1,878
2,131
2,249
2,602
2,947
3,733
4,073
16
1,337
1,746
1,869
2,120
2,235
2,583
2,921
3,686
4,015
17
1,333
1,740
1,862
2,110
2,224
2,567
2,898
3,646
3,965
18
1,330
1,734
1,855
2,101
2,214
2,552
2,878
3,610
3,922
19
1,328
1,729
1,850
2,093
2,205
2,539
2,861
3,579
3,883
20
1,325
1,725
1,844
2,086
2,197
2,528
2,845
3,552
3,850
21
1,323
1,721
1,840
2,080
2,189
2,518
2,831
3,527
3,819
22
1,321
1,717
1,835
2,074
2,183
2,508
2,819
3,505
3,792
23
1,319
1,714
1,832
2,069
2,177
2,500
2,807
3,485
3,768
24
1,318
1,711
1,828
2,064
2,172
2,492
2,797
3,467
3,745
25
1,316
1,708
1,825
2,060
2,167
2,485
2,787
3,450
3,725
26
1,315
1,706
1,822
2,056
2,162
2,479
2,779
3,435
3,707
27
1,314
1,703
1,819
2,052
2,158
2,473
2,771
3,421
3,690
28
1,313
1,701
1,817
2,048
2,154
2,467
2,763
3,408
3,674
29
1,311
1,699
1,814
2,045
2,150
2,462
2,756
3,396
3,659
30
1,310
1,697
1,812
2,042
2,147
2,457
2,750
3,385
3,646
35
1,306
1,690
1,803
2,030
2,133
2,438
2,724
3,340
3,591
40
1,303
1,684
1,796
2,021
2,123
2,423
2,704
3,307
3,551
45
1,301
1,679
1,791
2,014
2,115
2,412
2,690
3,281
3,520
50
1,299
1,676
1,787
2,009
2,109
2,403
2,678
3,261
3,496
60
1,296
1,671
1,781
2,000
2,099
2,390
2,660
3,232
3,460
70
1,294
1,667
1,776
1,994
2,093
2,381
2,648
3,211
3,435
80
1,292
1,664
1,773
1,990
2,088
2,374
2,639
3,195
3,416
90
1,291
1,662
1,771
1,987
2,084
2,368
2,632
3,183
3,402
100
1,290
1,660
1,769
1,984
2,081
2,364
2,626
3,174
3,390
150
1,287
1,655
1,763
1,976
2,072
2,351
2,609
3,145
3,357
200
1,286
1,653
1,760
1,972
2,067
2,345
2,601
3,131
3,340
250
1,285
1,651
1,758
1,969
2,065
2,341
2,596
3,123
3,330
300
1,284
1,650
1,757
1,968
2,063
2,339
2,592
3,118
3,323
400
1,284
1,649
1,755
1,966
2,060
2,336
2,588
3,111
3,315
500
1,283
1,648
1,754
1,965
2,059
2,334
2,586
3,107
3,310
600
1,283
1,647
1,754
1,964
2,058
2,333
2,584
3,104
3,307
800
1,283
1,647
1,753
1,963
2,057
2,331
2,582
3,100
3,303
1000
1,282
1,646
1,752
1,962
2,056
2,330
2,581
3,098
3,300
100000
1,282
1,645
1,751
1,960
2,054
2,326
2,576
3,090
3,291
Fisher-Verteilung
0.95-Quantile der Fisher-Verteilung mit n Freiheitsgraden im Nenner und m Freiheitsgraden im Zähler
m=1
2
3
4
5
6
7
8
9
10
11
12
14
16
18
20
22
24
30
40
50
60
100
100000
n=1
161,4
199,5
215,7
224,6
230,2
234,0
236,8
238,9
240,5
241,9
243,0
243,9
244,7
245,4
245,9
246,5
246,9
247,3
247,7
248,0
248,3
248,6
248,8
249,1
2
18,51
19,00
19,16
19,25
19,30
19,33
19,35
19,37
19,38
19,40
19,40
19,41
19,42
19,42
19,43
19,43
19,44
19,44
19,44
19,45
19,45
19,45
19,45
19,45
3
10,13
9,552
9,277
9,117
9,013
8,941
8,887
8,845
8,812
8,786
8,763
8,745
8,729
8,715
8,703
8,692
8,683
8,675
8,667
8,660
8,654
8,648
8,643
8,639
4
7,709
6,944
6,591
6,388
6,256
6,163
6,094
6,041
5,999
5,964
5,936
5,912
5,891
5,873
5,858
5,844
5,832
5,821
5,811
5,803
5,795
5,787
5,781
5,774
5
6,608
5,786
5,409
5,192
5,050
4,950
4,876
4,818
4,772
4,735
4,704
4,678
4,655
4,636
4,619
4,604
4,590
4,579
4,568
4,558
4,549
4,541
4,534
4,527
6
5,987
5,143
4,757
4,534
4,387
4,284
4,207
4,147
4,099
4,060
4,027
4,000
3,976
3,956
3,938
3,922
3,908
3,896
3,884
3,874
3,865
3,856
3,849
3,841
7
5,591
4,737
4,347
4,120
3,972
3,866
3,787
3,726
3,677
3,637
3,603
3,575
3,550
3,529
3,511
3,494
3,480
3,467
3,455
3,445
3,435
3,426
3,418
3,410
8
5,318
4,459
4,066
3,838
3,687
3,581
3,500
3,438
3,388
3,347
3,313
3,284
3,259
3,237
3,218
3,202
3,187
3,173
3,161
3,150
3,140
3,131
3,123
3,115
9
5,117
4,256
3,863
3,633
3,482
3,374
3,293
3,230
3,179
3,137
3,102
3,073
3,048
3,025
3,006
2,989
2,974
2,960
2,948
2,936
2,926
2,917
2,908
2,900
10
4,965
4,103
3,708
3,478
3,326
3,217
3,135
3,072
3,020
2,978
2,943
2,913
2,887
2,865
2,845
2,828
2,812
2,798
2,785
2,774
2,764
2,754
2,745
2,737
11
4,844
3,982
3,587
3,357
3,204
3,095
3,012
2,948
2,896
2,854
2,818
2,788
2,761
2,739
2,719
2,701
2,685
2,671
2,658
2,646
2,636
2,626
2,617
2,609
12
4,747
3,885
3,490
3,259
3,106
2,996
2,913
2,849
2,796
2,753
2,717
2,687
2,660
2,637
2,617
2,599
2,583
2,568
2,555
2,544
2,533
2,523
2,514
2,505
13
4,667
3,806
3,411
3,179
3,025
2,915
2,832
2,767
2,714
2,671
2,635
2,604
2,577
2,554
2,533
2,515
2,499
2,484
2,471
2,459
2,448
2,438
2,429
2,420
14
4,600
3,739
3,344
3,112
2,958
2,848
2,764
2,699
2,646
2,602
2,565
2,534
2,507
2,484
2,463
2,445
2,428
2,413
2,400
2,388
2,377
2,367
2,357
2,349
15
4,543
3,682
3,287
3,056
2,901
2,790
2,707
2,641
2,588
2,544
2,507
2,475
2,448
2,424
2,403
2,385
2,368
2,353
2,340
2,328
2,316
2,306
2,297
2,288
16
4,494
3,634
3,239
3,007
2,852
2,741
2,657
2,591
2,538
2,494
2,456
2,425
2,397
2,373
2,352
2,333
2,317
2,302
2,288
2,276
2,264
2,254
2,244
2,235
17
4,451
3,592
3,197
2,965
2,810
2,699
2,614
2,548
2,494
2,450
2,413
2,381
2,353
2,329
2,308
2,289
2,272
2,257
2,243
2,230
2,219
2,208
2,199
2,190
18
4,414
3,555
3,160
2,928
2,773
2,661
2,577
2,510
2,456
2,412
2,374
2,342
2,314
2,290
2,269
2,250
2,233
2,217
2,203
2,191
2,179
2,168
2,159
2,150
19
4,381
3,522
3,127
2,895
2,740
2,628
2,544
2,477
2,423
2,378
2,340
2,308
2,280
2,256
2,234
2,215
2,198
2,182
2,168
2,155
2,144
2,133
2,123
2,114
20
4,351
3,493
3,098
2,866
2,711
2,599
2,514
2,447
2,393
2,348
2,310
2,278
2,250
2,225
2,203
2,184
2,167
2,151
2,137
2,124
2,112
2,102
2,092
2,082
21
4,325
3,467
3,072
2,840
2,685
2,573
2,488
2,420
2,366
2,321
2,283
2,250
2,222
2,197
2,176
2,156
2,139
2,123
2,109
2,096
2,084
2,073
2,063
2,054
22
4,301
3,443
3,049
2,817
2,661
2,549
2,464
2,397
2,342
2,297
2,259
2,226
2,198
2,173
2,151
2,131
2,114
2,098
2,084
2,071
2,059
2,048
2,038
2,028
23
4,279
3,422
3,028
2,796
2,640
2,528
2,442
2,375
2,320
2,275
2,236
2,204
2,175
2,150
2,128
2,109
2,091
2,075
2,061
2,048
2,036
2,025
2,014
2,005
24
4,260
3,403
3,009
2,776
2,621
2,508
2,423
2,355
2,300
2,255
2,216
2,183
2,155
2,130
2,108
2,088
2,070
2,054
2,040
2,027
2,015
2,003
1,993
1,984
25
4,242
3,385
2,991
2,759
2,603
2,490
2,405
2,337
2,282
2,236
2,198
2,165
2,136
2,111
2,089
2,069
2,051
2,035
2,021
2,007
1,995
1,984
1,974
1,964
26
4,225
3,369
2,975
2,743
2,587
2,474
2,388
2,321
2,265
2,220
2,181
2,148
2,119
2,094
2,072
2,052
2,034
2,018
2,003
1,990
1,978
1,966
1,956
1,946
27
4,210
3,354
2,960
2,728
2,572
2,459
2,373
2,305
2,250
2,204
2,166
2,132
2,103
2,078
2,056
2,036
2,018
2,002
1,987
1,974
1,961
1,950
1,940
1,930
28
4,196
3,340
2,947
2,714
2,558
2,445
2,359
2,291
2,236
2,190
2,151
2,118
2,089
2,064
2,041
2,021
2,003
1,987
1,972
1,959
1,946
1,935
1,924
1,915
29
4,183
3,328
2,934
2,701
2,545
2,432
2,346
2,278
2,223
2,177
2,138
2,104
2,075
2,050
2,027
2,007
1,989
1,973
1,958
1,945
1,932
1,921
1,910
1,901
30
4,171
3,316
2,922
2,690
2,534
2,421
2,334
2,266
2,211
2,165
2,126
2,092
2,063
2,037
2,015
1,995
1,976
1,960
1,945
1,932
1,919
1,908
1,897
1,887
31
4,160
3,305
2,911
2,679
2,523
2,409
2,323
2,255
2,199
2,153
2,114
2,080
2,051
2,026
2,003
1,983
1,965
1,948
1,933
1,920
1,907
1,896
1,885
1,875
32
4,149
3,295
2,901
2,668
2,512
2,399
2,313
2,244
2,189
2,142
2,103
2,070
2,040
2,015
1,992
1,972
1,953
1,937
1,922
1,908
1,896
1,884
1,873
1,864
33
4,139
3,285
2,892
2,659
2,503
2,389
2,303
2,235
2,179
2,133
2,093
2,060
2,030
2,004
1,982
1,961
1,943
1,926
1,911
1,898
1,885
1,873
1,863
1,853
34
4,130
3,276
2,883
2,650
2,494
2,380
2,294
2,225
2,170
2,123
2,084
2,050
2,021
1,995
1,972
1,952
1,933
1,917
1,902
1,888
1,875
1,863
1,853
1,843
35
4,121
3,267
2,874
2,641
2,485
2,372
2,285
2,217
2,161
2,114
2,075
2,041
2,012
1,986
1,963
1,942
1,924
1,907
1,892
1,878
1,866
1,854
1,843
1,833
40
4,113
3,259
2,866
2,634
2,477
2,364
2,277
2,209
2,153
2,106
2,067
2,033
2,003
1,977
1,954
1,934
1,915
1,899
1,883
1,870
1,857
1,845
1,834
1,824
45
4,105
3,252
2,859
2,626
2,470
2,356
2,270
2,201
2,145
2,098
2,059
2,025
1,995
1,969
1,946
1,926
1,907
1,890
1,875
1,861
1,848
1,837
1,826
1,816
50
4,098
3,245
2,852
2,619
2,463
2,349
2,262
2,194
2,138
2,091
2,051
2,017
1,988
1,962
1,939
1,918
1,899
1,883
1,867
1,853
1,841
1,829
1,818
1,808
55
4,091
3,238
2,845
2,612
2,456
2,342
2,255
2,187
2,131
2,084
2,044
2,010
1,981
1,954
1,931
1,911
1,892
1,875
1,860
1,846
1,833
1,821
1,810
1,800
60
4,085
3,232
2,839
2,606
2,449
2,336
2,249
2,180
2,124
2,077
2,038
2,003
1,974
1,948
1,924
1,904
1,885
1,868
1,853
1,839
1,826
1,814
1,803
1,793
70
4,079
3,226
2,833
2,600
2,443
2,330
2,243
2,174
2,118
2,071
2,031
1,997
1,967
1,941
1,918
1,897
1,879
1,862
1,846
1,832
1,819
1,807
1,796
1,786
80
4,073
3,220
2,827
2,594
2,438
2,324
2,237
2,168
2,112
2,065
2,025
1,991
1,961
1,935
1,912
1,891
1,872
1,855
1,840
1,826
1,813
1,801
1,790
1,780
90
4,067
3,214
2,822
2,589
2,432
2,318
2,232
2,163
2,106
2,059
2,020
1,985
1,955
1,929
1,906
1,885
1,866
1,849
1,834
1,820
1,807
1,795
1,784
1,773
100
4,062
3,209
2,816
2,584
2,427
2,313
2,226
2,157
2,101
2,054
2,014
1,980
1,950
1,924
1,900
1,879
1,861
1,844
1,828
1,814
1,801
1,789
1,778
1,767
120
4,057
3,204
2,812
2,579
2,422
2,308
2,221
2,152
2,096
2,049
2,009
1,974
1,945
1,918
1,895
1,874
1,855
1,838
1,823
1,808
1,795
1,783
1,772
1,762
150
4,052
3,200
2,807
2,574
2,417
2,304
2,216
2,147
2,091
2,044
2,004
1,969
1,940
1,913
1,890
1,869
1,850
1,833
1,817
1,803
1,790
1,778
1,767
1,756
200
4,047
3,195
2,802
2,570
2,413
2,299
2,212
2,143
2,086
2,039
1,999
1,965
1,935
1,908
1,885
1,864
1,845
1,828
1,812
1,798
1,785
1,773
1,762
1,751
300
4,043
3,191
2,798
2,565
2,409
2,295
2,207
2,138
2,082
2,035
1,995
1,960
1,930
1,904
1,880
1,859
1,840
1,823
1,807
1,793
1,780
1,768
1,757
1,746
400
4,038
3,187
2,794
2,561
2,404
2,290
2,203
2,134
2,077
2,030
1,990
1,956
1,926
1,899
1,876
1,855
1,836
1,819
1,803
1,789
1,775
1,763
1,752
1,742
500
4,034
3,183
2,790
2,557
2,400
2,286
2,199
2,130
2,073
2,026
1,986
1,952
1,921
1,895
1,871
1,850
1,831
1,814
1,798
1,784
1,771
1,759
1,748
1,737
100000
4,030
3,179
2,786
2,553
2,397
2,283
2,195
2,126
2,069
2,022
1,982
1,947
1,917
1,891
1,867
1,846
1,827
1,810
1,794
1,780
1,767
1,754
1,743
1,733
0.99-Quantile der Fisher-Verteilung mit n Freiheitsgraden im Nenner und m Freiheitsgraden im Zähler
m=1
2
3
4
5
6
7
8
9
10
11
12
14
16
18
20
22
24
30
40
50
60
100
100000
n=1
4052,
4999
5403
5624
5763
5859
5928
5981
6022
6055
6083
6106
6125
6142
6157
6170
6181
6191
6200
6208
6216
6222
6229
6234
2
98,50
99,00
99,17
99,25
99,30
99,33
99,36
99,37
99,39
99,40
99,41
99,42
99,42
99,43
99,43
99,44
99,44
99,44
99,45
99,45
99,45
99,45
99,46
99,46
3
34,12
30,82
29,46
28,71
28,24
27,91
27,67
27,49
27,35
27,23
27,13
27,05
26,98
26,92
26,87
26,83
26,79
26,75
26,72
26,69
26,66
26,64
26,62
26,60
4
21,20
18,00
16,69
15,98
15,52
15,21
14,98
14,80
14,66
14,55
14,45
14,37
14,31
14,25
14,20
14,15
14,11
14,08
14,05
14,02
13,99
13,97
13,95
13,93
5
16,26
13,27
12,06
11,39
10,97
10,67
10,46
10,29
10,16
10,05
9,963
9,888
9,825
9,770
9,722
9,680
9,643
9,610
9,580
9,553
9,528
9,506
9,485
9,466
6
13,75
10,92
9,780
9,148
8,746
8,466
8,260
8,102
7,976
7,874
7,790
7,718
7,657
7,605
7,559
7,519
7,483
7,451
7,422
7,396
7,372
7,351
7,331
7,313
7
12,25
9,547
8,451
7,847
7,460
7,191
6,993
6,840
6,719
6,620
6,538
6,469
6,410
6,359
6,314
6,275
6,240
6,209
6,181
6,155
6,132
6,111
6,092
6,074
8
11,26
8,649
7,591
7,006
6,632
6,371
6,178
6,029
5,911
5,814
5,734
5,667
5,609
5,559
5,515
5,477
5,442
5,412
5,384
5,359
5,336
5,316
5,297
5,279
9
10,56
8,022
6,992
6,422
6,057
5,802
5,613
5,467
5,351
5,257
5,178
5,111
5,055
5,005
4,962
4,924
4,890
4,860
4,833
4,808
4,786
4,765
4,746
4,729
10
10,04
7,559
6,552
5,994
5,636
5,386
5,200
5,057
4,942
4,849
4,772
4,706
4,650
4,601
4,558
4,520
4,487
4,457
4,430
4,405
4,383
4,363
4,344
4,327
11
9,646
7,206
6,217
5,668
5,316
5,069
4,886
4,744
4,632
4,539
4,462
4,397
4,342
4,293
4,251
4,213
4,180
4,150
4,123
4,099
4,077
4,057
4,038
4,021
12
9,330
6,927
5,953
5,412
5,064
4,821
4,640
4,499
4,388
4,296
4,220
4,155
4,100
4,052
4,010
3,972
3,939
3,909
3,883
3,858
3,836
3,816
3,798
3,780
13
9,074
6,701
5,739
5,205
4,862
4,620
4,441
4,302
4,191
4,100
4,025
3,960
3,905
3,857
3,815
3,778
3,745
3,716
3,689
3,665
3,643
3,622
3,604
3,587
14
8,862
6,515
5,564
5,035
4,695
4,456
4,278
4,140
4,030
3,939
3,864
3,800
3,745
3,698
3,656
3,619
3,586
3,556
3,529
3,505
3,483
3,463
3,444
3,427
15
8,683
6,359
5,417
4,893
4,556
4,318
4,142
4,004
3,895
3,805
3,730
3,666
3,612
3,564
3,522
3,485
3,452
3,423
3,396
3,372
3,350
3,330
3,311
3,294
16
8,531
6,226
5,292
4,773
4,437
4,202
4,026
3,890
3,780
3,691
3,616
3,553
3,498
3,451
3,409
3,372
3,339
3,310
3,283
3,259
3,237
3,216
3,198
3,181
17
8,400
6,112
5,185
4,669
4,336
4,102
3,927
3,791
3,682
3,593
3,519
3,455
3,401
3,353
3,312
3,275
3,242
3,212
3,186
3,162
3,139
3,119
3,101
3,084
18
8,285
6,013
5,092
4,579
4,248
4,015
3,841
3,705
3,597
3,508
3,434
3,371
3,316
3,269
3,227
3,190
3,158
3,128
3,101
3,077
3,055
3,035
3,016
2,999
19
8,185
5,926
5,010
4,500
4,171
3,939
3,765
3,631
3,523
3,434
3,360
3,297
3,242
3,195
3,153
3,116
3,084
3,054
3,027
3,003
2,981
2,961
2,942
2,925
20
8,096
5,849
4,938
4,431
4,103
3,871
3,699
3,564
3,457
3,368
3,294
3,231
3,177
3,130
3,088
3,051
3,018
2,989
2,962
2,938
2,916
2,895
2,877
2,859
21
8,017
5,780
4,874
4,369
4,042
3,812
3,640
3,506
3,398
3,310
3,236
3,173
3,119
3,072
3,030
2,993
2,960
2,931
2,904
2,880
2,857
2,837
2,818
2,801
22
7,945
5,719
4,817
4,313
3,988
3,758
3,587
3,453
3,346
3,258
3,184
3,121
3,067
3,019
2,978
2,941
2,908
2,879
2,852
2,827
2,805
2,785
2,766
2,749
23
7,881
5,664
4,765
4,264
3,939
3,710
3,539
3,406
3,299
3,211
3,137
3,074
3,020
2,973
2,931
2,894
2,861
2,832
2,805
2,781
2,758
2,738
2,719
2,702
24
7,823
5,614
4,718
4,218
3,895
3,667
3,496
3,363
3,256
3,168
3,094
3,032
2,977
2,930
2,889
2,852
2,819
2,789
2,762
2,738
2,716
2,695
2,676
2,659
25
7,770
5,568
4,675
4,177
3,855
3,627
3,457
3,324
3,217
3,129
3,056
2,993
2,939
2,892
2,850
2,813
2,780
2,751
2,724
2,699
2,677
2,657
2,638
2,620
26
7,721
5,526
4,637
4,140
3,818
3,591
3,421
3,288
3,182
3,094
3,021
2,958
2,904
2,857
2,815
2,778
2,745
2,715
2,688
2,664
2,642
2,621
2,602
2,585
27
7,677
5,488
4,601
4,106
3,785
3,558
3,388
3,256
3,149
3,062
2,988
2,926
2,871
2,824
2,783
2,746
2,713
2,683
2,656
2,632
2,609
2,589
2,570
2,552
28
7,636
5,453
4,568
4,074
3,754
3,528
3,358
3,226
3,120
3,032
2,959
2,896
2,842
2,795
2,753
2,716
2,683
2,653
2,626
2,602
2,579
2,559
2,540
2,522
29
7,598
5,420
4,538
4,045
3,725
3,499
3,330
3,198
3,092
3,005
2,931
2,868
2,814
2,767
2,726
2,689
2,656
2,626
2,599
2,574
2,552
2,531
2,512
2,495
30
7,562
5,390
4,510
4,018
3,699
3,473
3,304
3,173
3,067
2,979
2,906
2,843
2,789
2,742
2,700
2,663
2,630
2,600
2,573
2,549
2,526
2,506
2,487
2,469
31
7,530
5,362
4,484
3,993
3,675
3,449
3,281
3,149
3,043
2,955
2,882
2,820
2,765
2,718
2,677
2,640
2,606
2,577
2,550
2,525
2,502
2,482
2,463
2,445
32
7,499
5,336
4,459
3,969
3,652
3,427
3,258
3,127
3,021
2,934
2,860
2,798
2,744
2,696
2,655
2,618
2,584
2,555
2,527
2,503
2,480
2,460
2,441
2,423
33
7,471
5,312
4,437
3,948
3,630
3,406
3,238
3,106
3,000
2,913
2,840
2,777
2,723
2,676
2,634
2,597
2,564
2,534
2,507
2,482
2,460
2,439
2,420
2,402
34
7,444
5,289
4,416
3,927
3,611
3,386
3,218
3,087
2,981
2,894
2,821
2,758
2,704
2,657
2,615
2,578
2,545
2,515
2,488
2,463
2,440
2,420
2,400
2,383
35
7,419
5,268
4,396
3,908
3,592
3,368
3,200
3,069
2,963
2,876
2,803
2,740
2,686
2,639
2,597
2,560
2,527
2,497
2,470
2,445
2,422
2,401
2,382
2,364
40
7,396
5,248
4,377
3,890
3,574
3,351
3,183
3,052
2,946
2,859
2,786
2,723
2,669
2,622
2,580
2,543
2,510
2,480
2,453
2,428
2,405
2,384
2,365
2,347
45
7,373
5,229
4,360
3,873
3,558
3,334
3,167
3,036
2,930
2,843
2,770
2,707
2,653
2,606
2,564
2,527
2,494
2,464
2,437
2,412
2,389
2,368
2,349
2,331
50
7,353
5,211
4,343
3,858
3,542
3,319
3,152
3,021
2,915
2,828
2,755
2,692
2,638
2,591
2,549
2,512
2,479
2,449
2,421
2,397
2,374
2,353
2,334
2,316
55
7,333
5,194
4,327
3,843
3,528
3,305
3,137
3,006
2,901
2,814
2,741
2,678
2,624
2,577
2,535
2,498
2,465
2,434
2,407
2,382
2,360
2,339
2,319
2,302
60
7,314
5,179
4,313
3,828
3,514
3,291
3,124
2,993
2,888
2,801
2,727
2,665
2,611
2,563
2,522
2,484
2,451
2,421
2,394
2,369
2,346
2,325
2,306
2,288
70
7,296
5,163
4,299
3,815
3,501
3,278
3,111
2,980
2,875
2,788
2,715
2,652
2,598
2,551
2,509
2,472
2,438
2,408
2,381
2,356
2,333
2,312
2,293
2,275
80
7,280
5,149
4,285
3,802
3,488
3,266
3,099
2,968
2,863
2,776
2,703
2,640
2,586
2,539
2,497
2,460
2,426
2,396
2,369
2,344
2,321
2,300
2,281
2,263
90
7,264
5,136
4,273
3,790
3,476
3,254
3,087
2,957
2,851
2,764
2,691
2,629
2,575
2,527
2,485
2,448
2,415
2,385
2,357
2,332
2,310
2,289
2,269
2,251
100
7,248
5,123
4,261
3,778
3,465
3,243
3,076
2,946
2,840
2,754
2,680
2,618
2,564
2,516
2,475
2,437
2,404
2,374
2,346
2,321
2,299
2,278
2,258
2,240
120
7,234
5,110
4,249
3,767
3,454
3,232
3,066
2,935
2,830
2,743
2,670
2,608
2,553
2,506
2,464
2,427
2,393
2,363
2,336
2,311
2,288
2,267
2,248
2,230
150
7,220
5,099
4,238
3,757
3,444
3,222
3,056
2,925
2,820
2,733
2,660
2,598
2,544
2,496
2,454
2,417
2,384
2,353
2,326
2,301
2,278
2,257
2,238
2,220
200
7,207
5,087
4,228
3,747
3,434
3,213
3,046
2,916
2,811
2,724
2,651
2,588
2,534
2,487
2,445
2,408
2,374
2,344
2,316
2,291
2,268
2,247
2,228
2,210
300
7,194
5,077
4,218
3,737
3,425
3,204
3,037
2,907
2,802
2,715
2,642
2,579
2,525
2,478
2,436
2,399
2,365
2,335
2,307
2,282
2,259
2,238
2,219
2,201
400
7,182
5,066
4,208
3,728
3,416
3,195
3,028
2,898
2,793
2,706
2,633
2,571
2,517
2,469
2,427
2,390
2,356
2,326
2,299
2,274
2,251
2,229
2,210
2,192
500
7,171
5,057
4,199
3,720
3,408
3,186
3,020
2,890
2,785
2,698
2,625
2,562
2,508
2,461
2,419
2,382
2,348
2,318
2,290
2,265
2,242
2,221
2,202
2,183
100000
7,159
5,047
4,191
3,711
3,400
3,178
3,012
2,882
2,777
2,690
2,617
2,555
2,500
2,453
2,411
2,374
2,340
2,310
2,282
2,257
2,234
2,213
2,194
2,175
Einzelnachweise
↑Hans-Otto Georgii: Stochastik. Einführung in die Wahrscheinlichkeitstheorie und Statistik. 4. Auflage. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7, S.383–388, doi:10.1515/9783110215274.
↑David Meintrup, Stefan Schäffler: Stochastik. Theorie und Anwendungen. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 3-540-21676-6, S.577–579, doi:10.1007/b137972.
Powiat Wysokie Mazowieckie Lambang Powiat Włodawa Lokasi Powiat Włodawa Informasi Negara Polandia Provinsi Lublin Ibu kota Włodawa (kota) Luas 1.256,27 km² Penduduk 39 148 (30 Juni 2016) Kadar urbanisasi 34,08 % Kode internasional (+48) 82 Pelat nomor kendaraan bermotor LWL Pembagian administratif Kota 1 Kota kecamatan: - Kelurahan 7 Politik (Status: ) Bupati (Starost) Andrzej Romańczuk Alamat al. J. Piłsudskiego 24 22-200 Włodawa Situs web resmi https://powiatwlodawski.pl/ ...
Province in Mimaropa, Philippines For the capital town, see Romblon, Romblon. For other uses, see Romblon (disambiguation). Province in Mimaropa, PhilippinesRomblonProvinceProvince of Romblon(From top, left to right) Romblon Bay and town, Fort San Andres, the Romblon Marble Sign, Mainit Falls in Odiongan, Binucot Beach in Odiongan and Mount Guiting-Guiting. FlagSealNickname: Marble Capital of the PhilippinesLocation in the PhilippinesOpenStreetMapCoordinates: 12°33′N 122°17′E...
Karangampel KidulDesaNegara IndonesiaProvinsiJawa BaratKabupatenIndramayuKecamatanKarangampelKode pos45283Kode Kemendagri32.12.10.2013 Luas... km²Jumlah penduduk... jiwaKepadatan... jiwa/km² Kantor - Desa Karangampel Kidul Karangampel Kidul adalah desa di kecamatan Karangampel, Indramayu, Jawa Barat, Indonesia. Administrasi Desa No Nama Wilayah Administrasi Satuan Jumlah 1 Jumlah Dusun Dusun 0 2 Jumlah Rukun Warga RW 5 3 Jumlah Rukun Tetangga RT 25 Gang Di Karangampel Kidul terdiri ata...
Legacy (Hanzi: 传家) adalah sebuah seri drama Tiongkok tahun 2022 yang tayang di HBO GO pada 20 Juni 2022. Karya dari Wang Wei dan Bai Yun Mo tersebut menampilkan Qin Lang, Han Geng, Wu Jinyan, dan Nie Yuan. Seri tersebut berkisah seputar perebutan hak waris dari keluarga terpandang yang ada di Tiongkok.[1] Sinopsis Kisahnya dimulai pada akhir tahun 1920-an, di mana saat itu bisnis pusat perbelanjaan milik keluarga Yi mengalami krisis keuangan. Putra tunggal Yi Xing Hua yang ber...
American Marine Corps general Eli Kelley ColeEli Kelley Cole in 1917Born(1867-09-01)September 1, 1867New York City, New YorkDiedJuly 4, 1929(1929-07-04) (aged 61)San Francisco, CaliforniaPlace of burialCarmel, New YorkAllegiance United States of AmericaService/branch United States Navy United States Marine CorpsYears of service1888–1890 (USN)1890–1929 (USMC)Rank Major GeneralCommands heldAssistant Commandant of the Marine CorpsDepartment of the Pacific41st Infantr...
Cet article est une ébauche concernant une chanson, le Concours Eurovision de la chanson et l’Albanie. Vous pouvez partager vos connaissances en l’améliorant (comment ?) selon les recommandations des projets correspondants. Ktheju tokës Chanson de Jonida Maliqi auConcours Eurovision de la chanson 2019 Sortie 2019 Langue Albanais Auteur-compositeur Eriona Rushiti Chansons représentant l'Albanie au Concours Eurovision de la chanson Mall(2018) Fall from the Sky(2020)modifier K...
Martial art from the Punjab region For other uses, see Gatka (disambiguation). Not to be confused with Shastar Vidya, the traditional battle system of northern India. This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Gatka – news · newspapers · books · scholar · JSTOR (January 2018) (Learn how and when to remo...
بلد وين الإحداثيات 4°44′09″N 45°12′14″E / 4.7359722222222°N 45.203972222222°E / 4.7359722222222; 45.203972222222 تقسيم إداري البلد الصومال[1] التقسيم الأعلى مقاطعة بلدوين عاصمة لـ هيران خصائص جغرافية المساحة 24 كيلومتر مربع ارتفاع 182 متر معلومات أخرى منطقة زمني...
هذه المقالة تحتاج للمزيد من الوصلات للمقالات الأخرى للمساعدة في ترابط مقالات الموسوعة. فضلًا ساعد في تحسين هذه المقالة بإضافة وصلات إلى المقالات المتعلقة بها الموجودة في النص الحالي. (يناير 2019) تحتاج هذه المقالة كاملةً أو أجزاءً منها إلى تدقيق لغوي أو نحوي. فضلًا ساهم في تح...
Arab Saudi Artikel ini adalah bagian dari seri Politik dan KetatanegaraanKerajaan Arab Saudi Hukum Dasar Monarki Raja dan Penjaga Dua Kota Suci Salman Putra Mahkota Muhammad bin Salman Wakil Putra Mahkota Kosong Dewan Kesetiaan Garis suksesi Pemerintah Perdana Menteri Salman Dewan Menteri Majelis Permusyawaratan Presiden: Abdullah bin Muhammad Al ash-Syaikh Dewan Keamanan Nasional Komite Pendukung Kebajikan dan Pencegah Sifat Buruk Sistem Hukum Al-Qur'an Yudikatif Hukuman mati Kebebasan berag...
American college football season 1959 college football seasonHeisman Trophy winner Billy CannonPreseason AP No. 1LSU[1]Regular seasonSeptember 19 – December 5, 1959Number of bowls8Bowl gamesDecember 19, 1959 – January 2, 1960Champion(s)Syracuse (AP, Coaches, FWAA, NFF)HeismanBilly Cannon (halfback, LSU) ← 1958 · football seasons · 1960 → The 1959 college football season was the 91st season of intercollegiate football in the United States...
يفتقر محتوى هذه المقالة إلى الاستشهاد بمصادر. فضلاً، ساهم في تطوير هذه المقالة من خلال إضافة مصادر موثوق بها. أي معلومات غير موثقة يمكن التشكيك بها وإزالتها. (نوفمبر 2019) كأس كوسوفو 2013–14 تفاصيل الموسم كأس كوسوفو البلد كوسوفو كأس كوسوفو 2014–15 تعديل مصدري - تعديل &...
هذه المقالة يتيمة إذ تصل إليها مقالات أخرى قليلة جدًا. فضلًا، ساعد بإضافة وصلة إليها في مقالات متعلقة بها. (أبريل 2019) مايك غرانت معلومات شخصية الميلاد 27 نوفمبر 1873 مونتريال الوفاة 19 أغسطس 1955 (81 سنة) مونتريال مواطنة كندا الحياة العملية المهنة لاعب هوكي الج...
Georg Detlev von Flemming (Polish: Jerzy Detloff Fleming) (3 March 1699 – 10 December 1771) was a General in Polish-Saxon service, Grand Treasurer of Lithuania in 1746–1764 and Voivode of Pomerelia in 1766. Von Flemming was born on 3 March 1699 in Iven, a town in Swedish Pomerania, then part of the Holy Roman Empire.[1] His family was a renowned magnate family from Pomerania, with long traditions of military service. His father was a Prussian privy councillor, Count Felix Friedri...
For similarly named synagogues, see Beth-El. Congregation Beth ElReligionAffiliationConservative JudaismEcclesiastical or organizational statusSynagogueLeadershipRabbi David EnglanderRabbi Aaron Krupnick (Emeritus)StatusActiveLocationLocation8000 Main Street, Voorhees, Camden County, New JerseyCountryUnited StatesLocation in Camden County, New JerseyGeographic coordinates39°52′03″N 74°56′44″W / 39.867637°N 74.945447°W / 39.867637; -74.945447ArchitectureDate...
Brazilian footballer (born 1986) Flávio Caça-Rato Personal informationFull name Flávio Augusto do NascimentoDate of birth (1986-06-29) 29 June 1986 (age 38)Place of birth Recife, BrazilHeight 1.76 m (5 ft 9 in)Position(s) ForwardTeam informationCurrent team São PauloNumber 7Youth career2005 Sport RecifeSenior career*Years Team Apps (Gls)2005–2008 Sport Recife 2006 → Salgueiro (loan) 2007 → SEV (loan) 2008–2009 Omiš 2009–2011 Cabense 2010 → América de Nata...
Basilika Tubuh TuhanBasilika Minor Tubuh Tuhanbahasa Italia: Basilica del Corpus DominiBasilika Tubuh TuhanWikipedia | Kode sumber | Tata penggunaan Koordinat: 45°28′29.07836″N 9°10′12.78228″E / 45.4747439889°N 9.1702173000°E / 45.4747439889; 9.1702173000LokasiMilanNegara ItaliaDenominasiGereja Katolik RomaArsitekturStatusBasilika minorStatus fungsionalAktifAdministrasiKeuskupan AgungKeuskupan Agung Milan Basilika Tubuh Tuhan (bahasa Italia: Ba...
![{\displaystyle P((-\infty ,x_{p}])\geq p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6b39a1e7e034395a982b9f4c07056422a5e36aa)
