1 D'Agostino (1971, p. 343)

N 0.5 1 2.5 5 10 90 95 97.5 99 99.5
50 -3.949 -3.442 -2.757 -2.220 -1.661 0.759 0923 1.038 1.140 1.192
60 -3.846 -3.360 -2.699 -2.179 -1.634 0.807 0.986 1.115 1.236 1.301
70 -3.762 -3.293 -2.652 -2.146 -1.612 0.844 1.036 1.176 1.312 1.388
80 -3.693 -3.237 -2.613 -2.118 -1.594 0.874 1.076 1.226 1.374 1.459
90 -3.635 -3.100 -2.580 -2.095 -1.579 0.899 1.109 1.268 1.426 1.518
100 -3.584 -3.150 -2.552 -2.075 -1.566 0.920 1.137 1.303 1.470 1.569
150 -3.409 -3.009 -2.452 -2.004 -1.520 0.990 1.233 1.423 1.623 1.746
200 -3.302 -2.922 -2.391 -1.960 -1.491 1.032 1.290 1.496 1.715 1.853
250 -3.227 -2.861 -2.348 -1.926 -1.471 1.060 1.328 1.545 1.779 1.927
300 -3.172 -2.816 -2.316 -1.906 -1.456 1.080 1.357 1.528 1.826 1.983
350 -3.129 -2.781 -2.291 -1.888 -1.444 1.096 1.379 1.610 1.863 2.026
400 -3.094 -2.753 -2.270 -1.873 -1.434 1.108 1.396 1.633 1.893 2.061
450 -3.064 -2.729 -2.253 -1.861 -1.426 1.119 1.411 1.652 1.918 2.090
500 -3.040 -2.709 -2.239 -1.850 -1.419 1.127 1.423 1.668 1.938 2.114
550 -3.019 -2.691 -2.226 -1.841 -1.413 1.135 1.434 1.682 1.957 2.136
600 -3.000 -2.676 -2.215 -1.833 -1.408 1.141 1.443 1.694 1.972 2.154
650 -2.984 -2.663 -2.206 -1.826 -1.403 1.147 1.451 1.704 1.986 2.171
700 -2.969 -2.651 -2.197 -1.820 -1.399 1.152 1.458 1.714 1.999 2.185
750 -2.956 -2.640 -2.189 -1.814 -1.395 1.157 1.465 1.722 2.010 2.199
800 -2.944 -2.630 -2.182 -1.809 -1.392 1.161 1.471 1.730 2.020 2.211
850 -2.933 -2.621 -2.176 -1.804 -1.389 1.165 1.476 1.737 2.029 2.221
900 -2.923 -2.613 -2.170 -1.800 -1.386 1.168 1.481 1.743 2.037 2.231
950 -2.914 -2.605 -2.164 -1.796 -1.383 1.171 1.485 1.749 2.045 2.241
1000 -2.906 -2.599 -2.159 -1.792 -1.381 1.174 1.489 1.754 2.052 2.249

2 Cabana-Cabana (1994, p. 1454)

Upper bound for the level Critical points for the general distribution Sharper critical points for product measures
20% 2.795 2.514
10% 3.057 2.807
5% 3.296 3.014
2.5% 3.515 3.160
1% 3.721 3.344
1.25% 3.784 3.435
0.5% 3.974 3.568

3 Chen-Shapiro (1995, p. 283)

N 0.001 0.005 0.010 0.015 0.020 0.025 0.030 0.035
10 0.34896 0.22821 0.17690 0.14937 0.12967 0.11397 0.10124 0.09054
11 0.34140 0.22249 0.17236 0.14526 0.12598 0.11074 0.09839 0.08803
12 0.33406 0.21707 0.16805 0.14140 0.12252 0.10769 0.09570 0.08563
13 0.32697 0.21193 0.16397 0.13777 0.11928 0.10483 0.09316 0.08337
14 0.32017 0.20707 0.16012 0.13437 0.11625 0.10215 0.09076 0.08122
15 0.31367 0.20247 0.15648 0.13116 0.11339 0.09962 0.08850 0.07918
16 0.30744 0.19811 0.15303 0.12814 0.11071 0.09723 0.08636 0.07725
17 0.30150 0.19398 0.14977 0.12529 0.10818 0.09498 0.08434 0.07542
18 0.29582 0.19006 0.14667 0.12259 0.10578 0.09285 0.08242 0.07368
19 0.29039 0.18633 0.14373 0.12003 0.10352 0.09083 0.08059 0.07203
20 0.28519 0.18278 0.14093 0.11761 0.10137 0.08892 0.07886 0.07046
21 0.28022 0.17940 0.13827 0.11530 0.09933 0.08709 0.07722 0.06896
22 0.27545 0.17618 0.13572 0.11310 0.09738 0.08536 0.07564 0.06753
23 0.27088 0.17310 0.13329 0.11101 0.09553 0.08370 0.07414 0.06616
24 0.26650 0.17015 0.13097 0.10901 0.09376 0.08212 0.07271 0.06485
25 0.26229 0.16733 0.12875 0.10709 0.09207 0.08061 0.07134 0.06360
26 0.25824 0.16462 0.12661 0.10526 0.09046 0.07917 0.07002 0.06240
27 0.25434 0.16202 0.12457 0.10350 0.08891 0.07778 0.06876 0.06125
28 0.25059 0.15952 0.12260 0.10182 0.08742 0.07645 0.06755 0.06014
29 0.24697 0.15712 0.12071 0.10020 0.08599 0.07517 0.06639 0.05907
30 0.24348 0.15481 0.11889 0.09864 0.08462 0.07394 0.06527 0.05805
31 0.24011 0.15258 0.11713 0.09714 0.08330 0.07275 0.06419 0.05706
32 0.23686 0.15042 0.11544 0.09569 0.08202 0.07161 0.06315 0.05611
33 0.23371 0.14835 0.11381 0.09430 0.08079 0.07051 0.06215 0.05519
34 0.23067 0.14634 0.11223 0.09295 0.07961 0.06945 0.06118 0.05430
35 0.22772 0.14440 0.11070 0.09165 0.07846 0.06842 0.06024 0.05344
36 0.22486 0.14252 0.10923 0.09039 0.07736 0.06742 0.05934 0.05261
37 0.22209 0.14071 0.10780 0.08918 0.07629 0.06646 0.05846 0.05180
38 0.21941 0.13895 0.10641 0.08800 0.07525 0.06553 0.05761 0.05103
39 0.21680 0.13724 0.10507 0.08685 0.07424 0.06463 0.05679 0.05027
40 0.21427 0.13558 0.10377 0.08575 0.07327 0.06376 0.05599 0.04954
41 0.21181 0.13397 0.10250 0.08467 0.07233 0.06291 0.05521 0.04883
42 0.20942 0.13241 0.10127 0.08363 0.07141 0.06209 0.05446 0.04814
43 0.20710 0.13089 0.10008 0.08261 0.07052 0.06129 0.05373 0.04747
44 0.20483 0.12941 0.09892 0.08163 0.06965 0.06051 0.05302 0.04681
45 0.20263 0.12798 0.09779 0.08067 0.06881 0.05975 0.05233 0.04618
46 0.20048 0.12658 0.09669 0.07974 0.06799 0.05902 0.05166 0.04556
47 0.19839 0.12522 0.09562 0.07883 0.06719 0.05830 0.05101 0.04496
48 0.19635 0.12389 0.09458 0.07795 0.06642 0.05760 0.05037 0.04438
49 0.19436 0.12260 0.09356 0.07709 0.06566 0.05692 0.04975 0.04381
50 0.19242 0.12134 0.09257 0.07625 0.06492 0.05626 0.04914 0.04325
60 0.17531 0.11024 0.08385 0.06887 0.05845 0.05045 0.04383 0.03837
80 0.14995 0.09386 0.07100 0.05802 0.04894 0.04190 0.03603 0.03119
100 0.13183 0.08220 0.06184 0.05031 0.04219 0.03584 0.03050 0.02611
150 0.10269 0.06349 0.04715 0.03795 0.03138 0.02616 0.02168 0.01803
250 0.07326 0.04453 0.03227 0.02543 0.02045 0.01640 0.01287 0.01001
500 0.04601 0.02673 0.01828 0.01362 0.01015 0.00730 0.00481 0.00279
1000 0.03200 0.01709 0.01067 0.00709 0.00445 0.00244 0.00074 -0.00065
2000 0.02910 0.01424 0.00837 0.00492 0.00257 0.00106 -0.00004 -0.00097
N 0.040 0.045 0.050 0.060 0.070 0.080 0.090 0.100
10 0.08178 0.07392 0.06668 0.05461 0.04433 0.03514 0.02694 0.01981
11 0.07945 0.07180 0.06476 0.05298 0.04297 0.03407 0.02614 0.01922
12 0.07723 0.06977 0.06292 0.05141 0.04165 0.03302 0.02531 0.01859
13 0.07513 0.06784 0.06116 0.04991 0.04037 0.03199 0.02449 0.01795
14 0.07314 0.06601 0.05950 0.04848 0.03916 0.03099 0.02368 0.01730
15 0.07126 0.06428 0.05791 0.04712 0.03799 0.03002 0.02290 0.01666
16 0.06947 0.06264 0.05641 0.04582 0.03688 0.02910 0.02214 0.01604
17 0.06778 0.06108 0.05498 0.04459 0.03583 0.02821 0.02140 0.01543
18 0.06618 0.05959 0.05362 0.04342 0.03482 0.02736 0.02069 0.01484
19 0.06465 0.05818 0.05232 0.04231 0.03386 0.02655 0.02001 0.01427
20 0.06320 0.05684 0.05109 0.04124 0.03294 0.02576 0.01935 0.01371
21 0.06181 0.05556 0.04991 0.04023 0.03206 0.02502 0.01872 0.01318
22 0.06049 0.05434 0.04879 0.03926 0.03122 0.02430 0.01811 0.01266
23 0.05923 0.05317 0.04771 0.03833 0.03042 0.02361 0.01752 0.01216
24 0.05802 0.05206 0.04668 0.03744 0.02965 0.02295 0.01696 0.01168
25 0.05687 0.05099 0.04569 0.03659 0.02891 0.02231 0.01641 0.01122
26 0.05576 0.04996 0.04475 0.03577 0.02820 0.02170 0.01589 0.01077
27 0.05470 0.04898 0.04384 0.03498 0.02752 0.02111 0.01539 0.01033
28 0.05367 0.04803 0.04297 0.03423 0.02687 0.02055 0.01490 0.00992
29 0.05269 0.04712 0.04213 0.03350 0.02624 0.02000 0.01443 0.00951
30 0.05175 0.04624 0.04132 0.03280 0.02563 0.01948 0.01398 0.00912
31 0.05084 0.04540 0.04054 0.03213 0.02504 0.01897 0.01354 0.00874
32 0.04996 0.04459 0.03979 0.03148 0.02448 0.01848 0.01312 0.00838
33 0.04911 0.04380 0.03906 0.03085 0.02393 0.01800 0.01271 0.00802
34 0.04829 0.04304 0.03836 0.03024 0.02341 0.01755 0.01231 0.00768
35 0.04750 0.04231 0.03768 0.02966 0.02290 0.01710 0.01193 0.00735
36 0.04674 0.04160 0.03703 0.02909 0.02241 0.01667 0.01156 0.00702
37 0.04599 0.04091 0.03639 0.02854 0.02193 0.01626 0.01120 0.00671
38 0.04528 0.04025 0.03578 0.02801 0.02147 0.01585 0.01085 0.00641
39 0.21680 0.13724 0.10507 0.08685 0.07424 0.06463 0.05679 0.05027
40 0.21427 0.13558 0.10377 0.08575 0.07327 0.06376 0.05599 0.04954
41 0.21181 0.13397 0.10250 0.08467 0.07233 0.06291 0.05521 0.04883
42 0.20942 0.13241 0.10127 0.08363 0.07141 0.06209 0.05446 0.04814
43 0.20710 0.13089 0.10008 0.08261 0.07052 0.06129 0.05373 0.04747
44 0.20483 0.12941 0.09892 0.08163 0.06965 0.06051 0.05302 0.04681
45 0.20263 0.12798 0.09779 0.08067 0.06881 0.05975 0.05233 0.04618
46 0.20048 0.12658 0.09669 0.07974 0.06799 0.05902 0.05166 0.04556
47 0.19839 0.12522 0.09562 0.07883 0.06719 0.05830 0.05101 0.04496
48 0.19635 0.12389 0.09458 0.07795 0.06642 0.05760 0.05037 0.04438
49 0.19436 0.12260 0.09356 0.07709 0.06566 0.05692 0.04975 0.04381
50 0.19242 0.12134 0.09257 0.07625 0.06492 0.05626 0.04914 0.04325
60 0.17531 0.11024 0.08385 0.06887 0.05845 0.05045 0.04383 0.03837
80 0.14995 0.09386 0.07100 0.05802 0.04894 0.04190 0.03603 0.03119
100 0.13183 0.08220 0.06184 0.05031 0.04219 0.03584 0.03050 0.02611
150 0.10269 0.06349 0.04715 0.03795 0.03138 0.02616 0.02168 0.01803
250 0.07326 0.04453 0.03227 0.02543 0.02045 0.01640 0.01287 0.01001
500 0.04601 0.02673 0.01828 0.01362 0.01015 0.00730 0.00481 0.00279
1000 0.03200 0.01709 0.01067 0.00709 0.00445 0.00244 0.00074 -0.00065
2000 0.02910 0.01424 0.00837 0.00492 0.00257 0.00106 -0.00004 -0.00097

4 Coin (2008, p. 2188)

N 0.9 0.95 0.99 0.995 0.999
10 0.069253 0.096065 0.155307 0.192290 0.258412
20 0.017407 0.024703 0.042435 0.050970 0.077089
30 0.009008 0.012618 0.022509 0.027305 0.038290
40 0.005861 0.008326 0.014124 0.017144 0.023908
50 0.004307 0.006316 0.010719 0.013013 0.019142
60 0.003331 0.004690 0.008182 0.010084 0.013719
70 0.002662 0.003768 0.006582 0.007972 0.011837
80 0.002234 0.003134 0.005332 0.006340 0.009136
90 0.001932 0.002723 0.004816 0.005684 0.008162
100 0.001706 0.002453 0.004280 0.005131 0.007455
150 0.001015 0.001447 0.002495 0.003047 0.004483
200 0.000732 0.001033 0.001836 0.002257 0.003017
250 0.000578 0.000806 0.001400 0.001677 0.002330
300 0.000446 0.000628 0.001102 0.001312 0.002005
350 0.000386 0.000543 0.000904 0.001087 0.001592
400 0.000327 0.000467 0.000848 0.000993 0.001406
450 0.000286 0.000411 0.000712 0.000851 0.001130
500 0.000260 0.000365 0.000644 0.000768 0.001081
600 0.000213 0.000303 0.000525 0.000617 0.000837
700 0.000180 0.000258 0.000445 0.000517 0.000762
800 0.000154 0.000217 0.000375 0.000443 0.000614
900 0.000138 0.000194 0.000335 0.000403 0.000562
1000 0.000125 0.000178 0.000302 0.000368 0.000489

5 Epps-Pulley (1983, p. 725)

α = 0.7??

N 0.025 0.050 0.950 0.975
4 2.90 2.76 0.42 0.33
6 3.07 2.79 0.39 0.24
8 3.16 2.91 0.38 0.22
10 3.24 2.96 0.37 0.19
12 3.29 2.98 0.36 0.17
>12 3.30 3.00 0.35 0.17

α = 1.0??

N 0.025 0.050 0.950 0.975
4 4.22 3.91 1.23 1.15
6 4.34 4.00 1.13 0.90
8 4.39 4.03 1.09 0.86
10 4.43 4.08 1.06 0.84
12 4.44 4.09 1.03 0.82
>12 4.45 4.10 1.00 0.79

6 Filliben (1973, p. 113)

N 0.000 0.005 0.010 0.025 0.050 0.100 0.250 0.500 0.750 0.900 0.950 0.975 0.990 0.995
3 0.866 0.867 0.869 0.872 0.879 0.891 0.924 0.966 0.991 0.999 1.000 1.000 1.000 1.000
4 0.784 0.813 0.822 0.845 0.868 0.894 0.931 0.958 0.979 0.992 0.996 0.998 0.999 1.000
5 0.726 0.803 0.822 0.855 0.879 0.902 0.935 0.960 0.977 0.988 0.992 0.995 0.997 0.998
6 0.683 0.818 0.835 0.868 0.890 0.911 0.940 0.962 0.977 0.986 0.990 0.993 0.996 0.997
7 0.648 0.828 0.847 0.876 0.899 0.916 0.944 0.965 0.978 0.986 0.990 0.992 0.995 0.996
8 0.619 0.841 0.859 0.886 0.905 0.924 0.948 0.967 0.979 0.986 0.990 0.992 0.995 0.996
9 0.595 0.851 0.868 0.893 0.912 0.929 0.951 0.968 0.980 0.987 0.990 0.992 0.994 0.995
10 0.574 0.860 0.876 0.900 0.917 0.934 0.954 0.970 0.981 0.987 0.990 0.992 0.994 0.995
11 0.556 0.868 0.883 0.906 0.922 0.938 0.957 0.972 0.982 0.988 0.990 0.992 0.994 0.995
12 0.539 0.875 0.889 0.912 0.926 0.941 0.959 0.973 0.982 0.988 0.990 0.992 0.994 0.995
13 0.525 0.882 0.895 0.917 0.931 0.944 0.962 0.975 0.983 0.988 0.991 0.993 0.994 0.995
14 0.512 0.888 0.901 0.921 0.934 0.947 0.964 0.976 0.984 0.989 0.991 0.993 0.994 0.995
15 0.500 0.894 0.907 0.925 0.937 0.950 0.965 0.977 0.984 0.989 0.991 0.993 0.994 0.995
16 0.489 0.899 0.912 0.928 0.940 0.952 0.967 0.978 0.985 0.989 0.991 0.993 0.994 0.995
17 0.478 0.903 0.916 0.931 0.942 0.954 0.968 0.979 0.986 0.990 0.992 0.993 0.994 0.995
18 0.469 0.907 0.919 0.934 0.945 0.956 0.969 0.979 0.986 0.990 0.992 0.993 0.995 0.995
19 0.460 0.909 0.923 0.937 0.947 0.958 0.971 0.980 0.987 0.990 0.992 0.993 0.995 0.995
20 0.452 0.912 0.925 0.939 0.950 0.960 0.972 0.981 0.987 0.991 0.992 0.994 0.995 0.995
21 0.445 0.914 0.928 0.942 0.952 0.961 0.973 0.981 0.987 0.991 0.993 0.994 0.995 0.996
22 0.437 0.918 0.930 0.944 0.954 0.962 0.974 0.982 0.988 0.991 0.993 0.994 0.995 0.996
23 0.431 0.922 0.933 0.947 0.955 0.964 0.975 0.983 0.988 0.991 0.993 0.994 0.995 0.996
24 0.424 0.926 0.936 0.949 0.957 0.965 0.975 0.983 0.988 0.992 0.993 0.994 0.995 0.996
25 0.418 0.928 0.937 0.950 0.958 0.966 0.976 0.984 0.989 0.992 0.993 0.994 0.995 0.996
26 0.412 0.930 0.939 0.952 0.959 0.967 0.977 0.984 0.989 0.992 0.993 0.994 0.995 0.996
27 0.407 0.932 0.941 0.953 0.960 0.968 0.977 0.984 0.989 0.992 0.994 0.995 0.995 0.996
28 0.402 0.934 0.943 0.955 0.962 0.969 0.978 0.985 0.990 0.992 0.994 0.995 0.995 0.996
29 0.397 0.937 0.945 0.956 0.962 0.969 0.979 0.985 0.990 0.992 0.994 0.995 0.995 0.996
30 0.392 0.938 0.947 0.957 0.964 0.970 0.979 0.986 0.990 0.993 0.994 0.995 0.996 0.996
31 0.388 0.939 0.948 0.958 0.965 0.971 0.980 0.980 0.986 0.990 0.993 0.995 0.996 0.996
32 0.383 0.939 0.949 0.959 0.966 0.972 0.980 0.980 0.986 0.990 0.993 0.995 0.996 0.996
33 0.379 0.940 0.950 0.960 0.967 0.973 0.981 0.981 0.987 0.991 0.993 0.995 0.996 0.996
34 0.375 0.941 0.951 0.960 0.967 0.973 0.981 0.981 0.987 0.991 0.993 0.995 0.996 0.996
35 0.371 0.943 0.952 0.961 0.968 0.974 0.982 0.982 0.987 0.991 0.993 0.995 0.996 0.997
36 0.367 0.945 0.953 0.962 0.968 0.974 0.982 0.987 0.991 0.994 0.995 0.996 0.996 0.997
37 0.364 0.947 0.955 0.962 0.969 0.975 0.982 0.988 0.991 0.994 0.995 0.996 0.996 0.997
38 0.360 0.948 0.956 0.964 0.970 0.975 0.983 0.988 0.992 0.994 0.995 0.996 0.996 0.997
39 0.357 0.949 0.957 0.965 0.971 0.976 0.983 0.988 0.992 0.994 0.995 0.996 0.996 0.997
40 0.354 0.949 0.958 0.966 0.972 0.977 0.983 0.988 0.992 0.994 0.995 0.996 0.996 0.997
41 0.351 0.950 0.958 0.967 0.972 0.977 0.984 0.989 0.992 0.994 0.995 0.996 0.996 0.997
42 0.348 0.951 0.959 0.967 0.973 0.978 0.984 0.989 0.992 0.994 0.995 0.996 0.997 0.997
43 0.345 0.953 0.959 0.967 0.973 0.978 0.984 0.989 0.992 0.994 0.995 0.996 0.997 0.997
44 0.342 0.954 0.960 0.968 0.973 0.978 0.984 0.989 0.992 0.994 0.995 0.996 0.997 0.997
45 0.339 0.955 0.961 0.969 0.974 0.978 0.985 0.989 0.993 0.994 0.995 0.996 0.997 0.997
46 0.336 0.956 0.962 0.969 0.974 0.979 0.985 0.990 0.993 0.995 0.995 0.996 0.997 0.997
47 0.334 0.956 0.963 0.970 0.974 0.979 0.985 0.990 0.993 0.995 0.995 0.996 0.997 0.997
48 0.331 0.957 0.963 0.970 0.975 0.980 0.985 0.990 0.993 0.995 0.996 0.996 0.997 0.997
49 0.329 0.957 0.964 0.971 0.975 0.980 0.986 0.990 0.993 0.995 0.996 0.996 0.997 0.997
50 0.326 0.959 0.965 0.972 0.977 0.981 0.986 0.990 0.993 0.995 0.996 0.996 0.997 0.997
55 0.315 0.962 0.967 0.974 0.978 0.982 0.987 0.991 0.994 0.995 0.996 0.997 0.997 0.997
60 0.305 0.965 0.970 0.976 0.980 0.983 0.988 0.991 0.994 0.995 0.996 0.997 0.997 0.998
65 0.296 0.967 0.972 0.977 0.981 0.984 0.989 0.992 0.994 0.996 0.996 0.997 0.997 0.998
70 0.288 0.969 0.974 0.978 0.982 0.985 0.989 0.993 0.995 0.996 0.997 0.997 0.998 0.998
75 0.281 0.971 0.975 0.979 0.983 0.986 0.990 0.993 0.995 0.996 0.997 0.997 0.998 0.998
80 0.274 0.973 0.976 0.980 0.984 0.987 0.991 0.993 0.995 0.996 0.997 0.997 0.998 0.998
85 0.268 0.974 0.977 0.981 0.985 0.987 0.991 0.994 0.995 0.997 0.997 0.997 0.998 0.998
90 0.263 0.976 0.978 0.982 0.985 0.988 0.991 0.994 0.996 0.997 0.997 0.997 0.998 0.998
95 0.257 0.977 0.979 0.983 0.986 0.989 0.992 0.994 0.996 0.997 0.997 0.997 0.998 0.998
100 0.252 0.979 0.981 0.984 0.987 0.989 0.992 0.994 0.996 0.997 0.998 0.998 0.998 0.998

7 Glen-Leemis-Barr (2001, p. 212)

N α = 0.10 α = 0.05 α = 0.01
2 4.9 6.1 8.9
3 7.6 9.1 13.4
4 10.1 12.1 17.0
5 12.6 15.3 21.5
6 15.1 18.1 24.4
7 17.7 21.1 28.2
8 20.4 23.9 32.0
9 22.7 26.8 36.5
10 24.9 29.4 39.5
11 27.9 32.2 43.7
12 30.0 35.2 48.0
15 37.5 44.0 59.6
20 50.7 58.7 81.1
25 63.2 76.2 116.5
30 80.0 107.1 218.4
40 445.0 576.5 776.8
50 1025.4 1108.8 1231.6

8 Martinez-Iglewicz (1981, p. 332)

N 90.0 95.0 97.5 99.0
10 1.448 1.969 2.917 5.273
15 1.283 1.516 1.858 2.502
20 1.210 1.351 1.540 1.878
25 1.173 1.280 1.405 1.639
30 1.145 1.232 1.334 1.487
35 1.129 1.200 1.290 1.405
40 1.113 1.174 1.244 1.352
45 1.103 1.160 1.221 1.308
50 1.093 1.145 1.199 1.276
60 1.080 1.122 1.168 1.233
70 1.071 1.109 1.147 1.197
80 1.064 1.097 1.132 1.180
90 1.058 1.087 1.117 1.159
100 1.052 1.079 1.109 1.146
150 1.037 1.056 1.075 1.101
200 1.027 1.043 1.060 1.080
300 1.017 1.030 1.042 1.058

9 Rahman-Govindarajulu (1997, p. 226)

N α = 0.01 0.02 0.05 0.10 0.50 0.90 0.95 0.98 0.99
3 0.754 0.758 0.771 0.793 0.933 0.997 0.999 0.999 1.000
4 0.703 0.722 0.760 0.795 0.911 0.983 0.992 0.996 0.998
5 0.702 0.728 0.770 0.803 0.906 0.975 0.984 0.991 0.994
6 0.722 0.748 0.784 0.816 0.909 0.971 0.980 0.988 0.991
7 0.737 0.763 0.799 0.829 0.911 0.968 0.978 0.985 0.989
8 0.755 0.775 0.809 0.836 0.915 0.967 0.976 0.984 0.987
9 0.768 0.790 0.819 0.843 0.917 0.967 0.975 0.983 0.987
10 0.779 0.800 0.828 0.852 0.920 0.967 0.975 0.982 0.986
11 0.794 0.814 0.839 0.859 0.923 0.966 0.975 0.982 0.986
12 0.805 0.823 0.846 0.866 0.926 0.967 0.975 0.982 0.986
13 0.814 0.830 0.852 0.871 0.928 0.968 0.975 0.982 0.986
14 0.817 0.834 0.856 0.875 0.930 0.968 0.975 0.982 0.985
15 0.827 0.842 0.863 0.881 0.932 0.968 0.975 0.982 0.985
16 0.834 0.848 0.868 0.886 0.935 0.969 0.976 0.982 0.985
17 0.839 0.853 0.872 0.889 0.936 0.969 0.976 0.982 0.985
18 0.842 0.857 0.878 0.893 0.938 0.969 0.976 0.982 0.985
19 0.848 0.862 0.881 0.896 0.940 0.970 0.977 0.982 0.985
20 0.853 0.867 0.884 0.899 0.941 0.971 0.977 0.983 0.986
21 0.857 0.870 0.888 0.902 0.943 0.971 0.977 0.983 0.986
22 0.863 0.875 0.891 0.904 0.944 0.971 0.977 0.983 0.986
23 0.865 0.877 0.893 0.907 0.945 0.972 0.978 0.983 0.986
24 0.869 0.880 0.895 0.909 0.946 0.972 0.978 0.983 0.986
25 0.873 0.884 0.899 0.911 0.947 0.973 0.978 0.983 0.986
26 0.875 0.887 0.902 0.914 0.949 0.974 0.979 0.984 0.986
27 0.880 0.890 0.904 0.916 0.950 0.974 0.979 0.984 0.986
28 0.883 0.892 0.906 0.917 0.950 0.974 0.979 0.984 0.986
29 0.885 0.896 0.909 0.920 0.952 0.974 0.980 0.984 0.987
30 0.887 0.897 0.911 0.921 0.952 0.975 0.980 0.984 0.987
31 0.890 0.900 0.913 0.923 0.953 0.975 0.980 0.984 0.987
32 0.891 0.901 0.913 0.924 0.954 0.976 0.980 0.985 0.987
33 0.894 0.904 0.916 0.926 0.955 0.976 0.980 0.985 0.987
34 0.897 0.906 0.917 0.927 0.956 0.976 0.981 0.985 0.987
35 0.899 0.907 0.919 0.928 0.956 0.976 0.981 0.985 0.988
36 0.901 0.909 0.921 0.930 0.957 0.977 0.981 0.985 0.988
37 0.902 0.910 0.922 0.931 0.958 0.977 0.981 0.985 0.988
38 0.905 0.913 0.924 0.933 0.958 0.977 0.982 0.986 0.988
39 0.906 0.913 0.925 0.934 0.959 0.977 0.982 0.986 0.988
40 0.908 0.916 0.925 0.934 0.960 0.978 0.982 0.986 0.988
41 0.909 0.916 0.927 0.935 0.960 0.978 0.982 0.986 0.988
42 0.912 0.918 0.928 0.936 0.961 0.978 0.982 0.986 0.988
43 0.913 0.920 0.930 0.938 0.961 0.979 0.982 0.986 0.988
44 0.914 0.921 0.931 0.939 0.962 0.979 0.982 0.986 0.988
45 0.915 0.923 0.932 0.939 0.962 0.979 0.983 0.986 0.988
46 0.917 0.923 0.933 0.940 0.963 0.979 0.983 0.987 0.988
47 0.918 0.924 0.934 0.942 0.963 0.979 0.983 0.987 0.989
48 0.919 0.926 0.934 0.942 0.964 0.980 0.983 0.987 0.989
49 0.921 0.927 0.936 0.943 0.964 0.980 0.983 0.987 0.989
50 0.921 0.928 0.937 0.944 0.965 0.980 0.984 0.987 0.989
51 0.922 0.928 0.937 0.944 0.965 0.980 0.984 0.987 0.989
52 0.923 0.930 0.938 0.945 0.966 0.981 0.984 0.987 0.989
53 0.925 0.930 0.939 0.946 0.966 0.981 0.984 0.987 0.989
54 0.925 0.932 0.940 0.947 0.966 0.981 0.984 0.987 0.989
55 0.927 0.933 0.941 0.947 0.967 0.981 0.984 0.988 0.989
56 0.928 0.934 0.942 0.948 0.967 0.981 0.985 0.988 0.989
57 0.928 0.934 0.942 0.949 0.967 0.982 0.985 0.988 0.990
58 0.929 0.935 0.942 0.949 0.968 0.982 0.985 0.988 0.990
59 0.930 0.936 0.943 0.950 0.968 0.982 0.985 0.988 0.990
60 0.931 0.936 0.944 0.950 0.968 0.982 0.985 0.988 0.990
61 0.932 0.937 0.945 0.951 0.969 0.982 0.985 0.988 0.990
62 0.933 0.938 0.946 0.952 0.969 0.982 0.985 0.988 0.990
63 0.934 0.939 0.946 0.952 0.969 0.982 0.985 0.988 0.990
64 0.934 0.939 0.946 0.952 0.970 0.983 0.986 0.989 0.990
65 0.935 0.941 0.947 0.953 0.970 0.983 0.986 0.989 0.990
66 0.936 0.941 0.947 0.953 0.970 0.983 0.986 0.989 0.990
67 0.937 0.941 0.948 0.954 0.971 0.983 0.986 0.989 0.990
68 0.937 0.942 0.949 0.954 0.971 0.983 0.986 0.989 0.990
69 0.937 0.942 0.949 0.955 0.971 0.983 0.986 0.989 0.990
70 0.939 0.944 0.950 0.955 0.971 0.983 0.986 0.989 0.990
71 0.940 0.944 0.950 0.956 0.972 0.984 0.986 0.989 0.991
72 0.939 0.944 0.951 0.956 0.972 0.984 0.986 0.989 0.991
73 0.940 0.945 0.951 0.956 0.972 0.984 0.986 0.989 0.991
74 0.940 0.945 0.952 0.957 0.972 0.984 0.987 0.989 0.991
75 0.940 0.945 0.952 0.957 0.973 0.984 0.987 0.989 0.991
16 0.942 0.946 0.953 0.958 0.973 0.984 0.987 0.989 0.991
77 0.942 0.948 0.953 0.958 0.973 0.984 0.987 0.990 0.991
78 0.943 0.947 0.954 0.959 0.973 0.984 0.987 0.990 0.991
79 0.944 0.948 0.954 0.959 0.974 0.984 0.987 0.990 0.991
80 0.944 0.948 0.954 0.959 0.974 0.985 0.987 0.990 0.991
81 0.945 0.949 0.955 0.960 0.974 0.985 0.987 0.990 0.991
82 0.945 0.950 0.955 0.960 0.974 0.985 0.987 0.990 0.991
83 0.946 0.950 0.956 0.960 0.974 0.985 0.987 0.990 0.991
84 0.946 0.950 0.956 0.961 0.975 0.985 0.987 0.990 0.991
85 0.947 0.951 0.957 0.961 0.975 0.985 0.988 0.990 0.991
86 0.947 0.952 0.957 0.962 0.975 0.985 0.988 0.990 0.991
87 0.947 0.952 0.957 0.962 0.975 0.985 0.988 0.990 0.991
88 0.948 0.953 0.958 0.962 0.975 0.985 0.988 0.990 0.991
89 0.948 0.952 0.958 0.962 0.976 0.986 0.988 0.990 0.992
90 0.948 0.953 0.958 0.963 0.976 0.986 0.988 0.990 0.992
91 0.949 0.954 0.959 0.963 0.976 0.986 0.988 0.990 0.992
92 0.950 0.953 0.959 0.963 0.976 0.986 0.988 0.990 0.992
93 0.951 0.954 0.959 0.963 0.976 0.986 0.988 0.990 0.992
94 0.951 0.955 0.960 0.964 0.976 0.986 0.988 0.991 0.992
95 0.951 0.955 0.960 0.964 0.977 0.986 0.988 0.990 0.992
96 0.951 0.955 0.960 0.965 0.977 0.986 0.988 0.991 0.992
97 0.951 0.955 0.961 0.965 0.977 0.986 0.988 0.991 0.992
98 0.952 0.956 0.961 0.965 0.977 0.986 0.988 0.991 0.992
99 0.952 0.956 0.961 0.965 0.977 0.986 0.989 0.991 0.992
100 0.953 0.956 0.961 0.965 0.977 0.987 0.989 0.991 0.992
110 0.957 0.960 0.964 0.968 0.979 0.987 0.989 0.991 0.992
120 0.959 0.962 0.966 0.970 0.980 0.988 0.990 0.992 0.993
130 0.962 0.964 0.968 0.972 0.981 0.988 0.990 0.992 0.993
140 0.964 0.966 0.970 0.973 0.982 0.989 0.991 0.992 0.993
150 0.966 0.968 0.972 0.975 0.983 0.989 0.991 0.993 0.993
160 0.968 0.970 0.973 0.976 0.984 0.990 0.991 0.993 0.994
170 0.969 0.971 0.974 0.977 0.984 0.990 0.992 0.993 0.994
180 0.970 0.972 0.975 0.978 0.985 0.991 0.992 0.993 0.994
190 0.971 0.973 0.976 0.979 0.986 0.991 0.992 0.994 0.994
200 0.973 0.975 0.977 0.979 0.986 0.991 0.993 0.994 0.995
250 0.977 0.979 0.981 0.983 0.988 0.992 0.993 0.994 0.995
300 0.981 0.982 0.984 0.985 0.990 0.993 0.994 0.995 0.996
350 0.983 0.984 0.985 0.987 0.991 0.994 0.995 0.996 0.996
400 0.984 0.985 0.987 0.988 0.992 0.994 0.995 0.996 0.996
450 0.986 0.987 0.988 0.989 0.992 0.995 0.996 0.996 0.997
500 0.987 0.988 0.989 0.990 0.993 0.995 0.996 0.996 0.997
550 0.988 0.989 0.990 0.991 0.993 0.995 0.996 0.997 0.997
600 0.989 0.990 0.991 0.991 0.994 0.996 0.996 0.997 0.997
650 0.990 0.990 0.991 0.992 0.994 0.996 0.996 0.997 0.997
700 0.990 0.991 0.992 0.992 0.994 0.996 0.997 0.997 0.997
750 0.991 0.991 0.992 0.993 0.995 0.996 0.997 0.997 0.997
800 0.991 0.992 0.992 0.993 0.995 0.997 0.997 0.997 0.998
850 0.992 0.992 0.993 0.993 0.995 0.997 0.997 0.997 0.998
900 0.992 0.993 0.993 0.994 0.995 0.997 0.997 0.997 0.998
950 0.992 0.993 0.993 0.994 0.996 0.997 0.997 0.998 0.998
1000 0.993 0.993 0.994 0.994 0.996 0.997 0.997 0.998 0.998
1500 0.995 0.995 0.995 0.996 0.997 0.998 0.998 0.998 0.998
2000 0.996 0.996 0.996 0.997 0.997 0.998 0.998 0.998 0.999
2500 0.996 0.997 0.997 0.997 0.998 0.998 0.998 0.999 0.999
3000 0.997 0.997 0.997 0.997 0.998 0.998 0.999 0.999 0.999
3500 0.997 0.997 0.998 0.998 0.998 0.999 0.999 0.999 0.999
4000 0.998 0.998 0.998 0.998 0.998 0.999 0.999 0.999 0.999
4500 0.998 0.998 0.998 0.998 0.998 0.999 0.999 0.999 0.999
5000 0.998 0.998 0.998 0.998 0.999 0.999 0.999 0.999 0.999

10 Spiegelhalter (1977, p. 417)

Table 1: Estimated significance points for \(T^\prime\)
N cN 5% 10%
5 0.3310 1.532 1.512
10 0.2678 1.453 1.417
15 0.2445 1.423 1.387
20 0.2321 1.403 1.369
50 0.2070 1.337 1.317
100 0.1971 1.308 1.295

11 Zhang (1999, p. 523)

Table 2: Empirical percentage points of \(Q\) compared with Cornish-Fisher percentage points
N 0.025 0.05 0.10 0.25 0.50 0.75 0.90 0.95 0.975
10 0.028 0.054 0.104 0.255 0.502 0.745 0.891 0.946 0.972
20 0.025 0.045 0.099 0.249 0.498 0.744 0.892 0.947 0.970
30 0.024 0.050 0.101 0.249 0.502 0.745 0.889 0.940 0.968
40 0.025 0.048 0.098 0.249 0.507 0.756 0.904 0.950 0.974
50 0.025 0.049 0.100 0.256 0.505 0.757 0.902 0.947 0.972
100 0.027 0.051 0.103 0.258 0.510 0.756 0.905 0.948 0.973
500 0.025 0.054 0.104 0.248 0.505 0.747 0.892 0.940 0.969
1000 0.024 0.050 0.091 0.239 0.493 0.749 0.901 0.951 0.977
1500 0.020 0.041 0.086 0.239 0.501 0.768 0.912 0.954 0.975
2000 0.015 0.030 0.075 0.223 0.496 0.763 0.904 0.955 0.980

Procedure:

  1. Calculate \(Q\) and \(Q^*\) as already defined
  2. Conduct the hypothesis test based on \(Q\) and \(Q^*\) separately at the level of \(\alpha/2\)
  3. Accept the null hypothesis of normality only when both \(Q\) and \(Q^*\) are non-signicant.

12 Zhang-Wu (2005, p. 712)

Table 3: Percentage points of \(10 \times Z_{A} - 32\) for testing normality
N 0.001 0.01 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.99 0.999
5 -0.351 -0.303 -0.186 -0.080 0.093 0.262 0.435 0.613 0.817 1.083 1.457 2.093 2.720 4.188 5.568
6 -0.133 -0.063 0.066 0.170 0.344 0.503 0.660 0.832 1.038 1.299 1.658 2.278 2.923 4.431 6.231
7 0.032 0.111 0.242 0.344 0.510 0.658 0.809 0.976 1.173 1.420 1.765 2.374 2.994 4.503 6.472
8 0.158 0.241 0.369 0.468 0.625 0.766 0.912 1.073 1.261 1.499 1.831 2.414 3.020 4.484 6.540
9 0.258 0.339 0.465 0.561 0.710 0.845 0.985 1.139 1.319 1.545 1.864 2.419 2.993 4.409 6.442
10 0.335 0.416 0.538 0.632 0.774 0.904 1.038 1.184 1.356 1.575 1.879 2.414 2.970 4.318 6.265
12 0.453 0.530 0.645 0.730 0.862 0.983 1.106 1.242 1.401 1.602 1.885 2.375 2.882 4.117 5.978
14 0.533 0.607 0.716 0.795 0.918 1.029 1.144 1.272 1.420 1.607 1.866 2.316 2.783 3.939 5.641
16 0.592 0.663 0.766 0.840 0.954 1.059 1.166 1.285 1.423 1.596 1.838 2.256 2.689 3.740 5.287
18 0.639 0.707 0.803 0.873 0.981 1.079 1.180 1.291 1.419 1.582 1.810 2.199 2.601 3.568 5.031
20 0.674 0.739 0.831 0.897 0.999 1.092 1.187 1.291 1.414 1.567 1.780 2.146 2.521 3.427 4.780
25 0.737 0.796 0.876 0.935 1.025 1.106 1.190 1.282 1.389 1.524 1.710 2.029 2.353 3.126 4.246
30 0.776 0.829 0.902 0.955 1.036 1.109 1.184 1.266 1.362 1.482 1.648 1.931 2.217 2.902 3.914
40 0.823 0.868 0.929 0.973 1.041 1.102 1.165 1.233 1.313 1.413 1.550 1.783 2.015 2.564 3.380
50 0.847 0.887 0.941 0.979 1.038 1.091 1.145 1.204 1.273 1.358 1.475 1.674 1.873 2.342 3.028
70 0.874 0.906 0.949 0.979 1.026 1.068 1.111 1.157 1.212 1.279 1.371 1.526 1.682 2.046 2.567
100 0.890 0.915 0.949 0.973 1.009 1.042 1.075 1.111 1.152 1.204 1.275 1.394 1.514 1.791 2.193
150 0.900 0.919 0.944 0.962 0.989 1.013 1.038 1.064 1.095 1.133 1.184 1.271 1.359 1.562 1.856
200 0.904 0.919 0.939 0.954 0.976 0.995 1.014 1.036 1.060 1.090 1.132 1.202 1.272 1.435 1.670
300 0.906 0.918 0.932 0.943 0.959 0.973 0.987 1.002 1.020 1.042 1.072 1.122 1.172 1.289 1.465
500 0.906 0.914 0.924 0.931 0.942 0.951 0.960 0.970 0.982 0.996 1.016 1.048 1.081 1.159 1.275
1000 0.905 0.909 0.915 0.919 0.925 0.930 0.935 0.941 0.947 0.955 0.966 0.984 1.002 1.046 1.111
Table 4: Percentage points of \(Z_{C}\) for testing normality
N 0.001 0.01 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.99 0.999
5 0.664 0.724 0.874 1.007 1.252 1.501 1.749 2.002 2.261 2.577 3.002 3.639 4.213 5.460 6.757
6 0.704 0.808 1.003 1.176 1.475 1.756 2.028 2.298 2.602 2.967 3.433 4.158 4.849 6.382 8.280
7 0.745 0.881 1.120 1.322 1.662 1.966 2.255 2.555 2.891 3.288 3.797 4.610 5.385 7.196 9.635
8 0.781 0.945 1.218 1.446 1.819 2.144 2.453 2.778 3.141 3.567 4.123 5.007 5.864 7.908 10.954
9 0.813 0.997 1.306 1.556 1.955 2.299 2.631 2.979 3.362 3.810 4.404 5.350 6.267 8.545 12.102
10 0.842 1.049 1.388 1.658 2.079 2.438 2.789 3.155 3.559 4.035 4.659 5.663 6.650 9.138 13.115
12 0.895 1.138 1.526 1.831 2.290 2.682 3.065 3.463 3.902 4.424 5.105 6.209 7.304 10.156 15.147
14 0.935 1.212 1.645 1.972 2.464 2.888 3.298 3.723 4.195 4.755 5.486 6.669 7.862 11.103 16.969
16 0.976 1.277 1.746 2.096 2.616 3.064 3.495 3.944 4.443 5.036 5.808 7.065 8.352 11.838 18.474
18 1.014 1.334 1.838 2.207 2.754 3.222 3.675 4.147 4.669 5.287 6.099 7.422 8.767 12.493 19.899
20 1.046 1.396 1.924 2.309 2.875 3.361 3.835 4.328 4.869 5.511 6.362 7.752 9.157 13.150 21.149
25 1.120 1.519 2.103 2.519 3.137 3.664 4.176 4.707 5.298 5.994 6.918 8.438 9.984 14.432 23.753
30 1.170 1.618 2.246 2.693 3.349 3.910 4.456 5.023 5.649 6.391 7.375 8.998 10.662 15.580 26.091
40 1.285 1.783 2.483 2.972 3.693 4.307 4.901 5.521 6.209 7.031 8.109 9.888 11.733 17.223 29.333
50 1.366 1.912 2.674 3.193 3.957 4.612 5.248 5.913 6.648 7.522 8.683 10.594 12.583 18.480 31.707
70 1.512 2.131 2.963 3.535 4.367 5.079 5.771 6.499 7.302 8.262 9.540 11.640 13.835 20.399 35.532
100 1.693 2.369 3.279 3.902 4.810 5.590 6.344 7.132 8.011 9.059 10.452 12.758 15.171 22.242 39.126
150 1.891 2.653 3.655 4.339 5.327 6.175 6.999 7.862 8.818 9.970 11.488 14.027 16.628 24.405 42.354
200 2.043 2.867 3.923 4.649 5.696 6.593 7.464 8.376 9.391 10.613 12.244 14.934 17.714 25.839 44.611
300 2.298 3.196 4.338 5.118 6.245 7.209 8.149 9.123 10.220 11.530 13.276 16.179 19.139 27.523 46.663
500 2.609 3.596 4.861 5.702 6.932 7.977 8.990 10.055 11.246 12.674 14.567 17.717 20.927 29.760 49.888
1000 3.072 4.191 5.588 6.526 7.885 9.045 10.169 11.346 12.654 14.224 16.322 19.796 23.301 32.811 53.458
n inline image inline image E p
20% 10% 5% 1% 20% 10% 5% 1% 20% 10% 5% 1%
10 0.033 0.017 0.010 0.002 0.158 0.093 0.057 0.024 0.204 0.101 0.048 0.007
20 0.062 0.036 0.022 0.010 0.165 0.091 0.055 0.018 0.185 0.092 0.046 0.011
50 0.094 0.055 0.035 0.012 0.177 0.095 0.055 0.017 0.178 0.088 0.046 0.011
150 0.138 0.070 0.044 0.018 0.191 0.098 0.055 0.016 0.182 0.092 0.047 0.012
MCSE 0.004 0.003 0.002 0.001 0.004 0.003 0.002 0.001 0.004 0.003 0.002 0.001

This tool provides a web interface for several diagnostic test of distributional shape, normality, and homogeneity of variances.

The About tab is under development to provide description of the some of the functions and requirements (e.g., data.csv file) for use of this tool.

Please also refer to http://www.sfu.ca/psychology/research/mml/resources.html for other tools