Table 1: Critical values (percentiles) for the chi-square distribution.
 For each degree of freedom $(D)$ in the first column, the table entries are the critical values for the upper one-sided significance levels in the column headings or, equivalently, the percentiles for the corresponding percentages.  
  Percentage
  2.5 5 50 75 90 95 97.5 99 99.9
D Upper one-sided $\alpha$
  .975 .95 .50 .25 .10 .05 .025 .01 .001
1 .001 .004 .455 1.32 2.71 3.84 5.02 6.63 10.83
2 .051 .103 1.39 2.77 4.61 5.99 7.38 9.21 13.82
3 .216 .352 2.37 4.11 6.25 7.82 9.35 11.34 16.27
4 .484 .711 3.36 5.39 7.78 9.49 11.14 13.28 18.47
5 .831 1.15 4.35 6.63 9.24 11.07 12.83 15.09 20.52
6 1.24 1.64 5.35 7.84 10.64 12.59 14.45 16.81 22.46
7 1.69 2.17 6.35 9.04 12.02 14.07 16.01 18.47 24.32
8 2.18 2.73 7.34 10.22 13.36 15.51 17.53 20.09 26.12
9 2.70 3.33 8.34 11.39 14.68 16.92 19.02 21.67 27.88
10 3.25 3.94 9.34 12.55 15.99 18.31 20.48 23.21 29.59
11 3.82 4.57 10.34 13.70 17.27 19.68 21.92 24.72 31.26
12 4.40 5.23 11.34 14.85 18.55 21.03 23.34 26.22 32.91
13 5.01 5.89 12.34 15.98 19.81 22.36 24.74 27.69 34.53
14 5.63 6.57 13.34 17.12 21.06 23.68 26.12 29.14 36.12
15 6.26 7.26 14.34 18.25 22.31 25.00 27.49 30.58 37.70
16 6.91 7.96 15.34 19.37 23.54 26.30 28.85 32.00 39.25
17 7.56 8.67 16.34 20.49 24.77 27.59 30.19 33.41 40.79
18 8.23 9.39 17.34 21.60 25.99 28.87 31.53 34.81 42.31
19 8.91 10.12 18.34 22.72 27.20 30.14 32.85 36.19 43.82
20 9.59 10.85 19.34 23.83 28.41 31.41 34.17 37.57 45.31
21 10.28 11.59 20.34 24.93 29.62 32.67 35.48 38.93 46.80
22 10.98 12.34 21.34 26.04 30.81 33.92 36.78 40.29 48.27
23 11.69 13.09 22.34 27.14 32.01 35.17 38.08 41.64 49.73
24 12.40 13.85 23.34 28.24 33.20 36.42 39.36 42.98 51.18
25 13.12 14.61 24.34 29.34 34.38 37.65 40.65 44.31 52.62
26 13.84 15.38 25.34 30.43 35.56 38.89 41.92 45.64 54.05
27 14.57 16.15 26.34 31.53 36.74 40.11 43.19 46.96 55.48
28 15.31 16.93 27.34 32.62 37.92 41.34 44.46 48.28 56.89
29 16.05 17.71 28.34 33.71 39.09 42.56 45.72 49.59 58.30
30 16.79 18.49 29.34 34.80 40.26 43.77 46.98 50.89 59.70
35 20.57 22.47 34.34 40.22 46.06 49.80 53.20 57.34 66.62
40 24.43 26.51 39.34 45.62 51.81 55.76 59.34 63.69 73.40
45 28.37 30.61 44.34 50.98 57.51 61.66 65.41 69.96 80.08
50 32.36 34.76 49.33 56.33 63.17 67.50 71.42 76.15 86.66
55 36.40 38.96 54.33 61.66 68.80 73.31 77.38 82.29 93.17
60 40.48 43.19 59.33 66.98 74.40 79.08 83.30 88.38 99.61
65 44.60 47.45 64.33 72.28 79.97 84.82 89.18 94.42 105.99
70 48.76 51.74 69.33 77.58 85.53 90.53 95.02 100.43 112.32
75 52.94 56.05 74.33 82.86 91.06 96.22 100.84 106.39 118.60
80 57.15 60.39 79.33 88.13 96.58 101.88 106.63 112.33 124.84
85 61.39 64.75 84.33 93.39 102.08 107.52 112.39 118.24 131.04
90 65.65 69.13 89.33 98.65 107.57 113.15 118.14 124.12 137.21
95 69.92 73.52 94.33 103.90 113.04 118.75 123.86 129.97 143.34
100 74.22 77.93 99.33 109.14 118.50 124.34 129.56 135.81 149.45
For more than 100 degrees of freedom chi-square critical values may be found in terms of the degrees of freedom $D$ and the corresponding two-sided critical value for a standard normal deviate $Z$ by the equation $X^{2} = 0.5 \cdot (Z + \sqrt{2 \cdot D - 1})^{2}$.


Table 2: $p$-values for the chi-square distribution divided by degrees of freedom.
 This table gives the $p$-value for a chi-square variable divided by it's degrees of freedom. Values with $p\leq 0.05$ are tabled. The column labelled ``ratio'' is the value of the chi-square statistic divided by it's degrees of freedom. The other columns, to the right, are headed by the degrees of freedom. For 1 degree of freedom the two-sided standard normal $p$-value (Table 2) for the square root of the chi-square statistic should be used. For 2 degrees of freedom, if $Y$ is the value of the chi-square statistic, then $\exp(-Y/2)$ is the $p$-value. For more than 100 degrees of freedom use the two-sided standard normal $p$-value (Table 2) for $Z = \sqrt{Y} - \sqrt{2 \cdot DF - 1}$, where $DF$ is the degrees of freedom.  
  Degrees of freedom   Degrees of freedom
Ratio 3 4 5 Ratio 6 7 8 9
2.2 .0858 .0663 .0514 1.85 .0853 .0733 .0632 .0545
2.3 .0752 .0563 .0423 1.90 .0768 .0651 .0554 .0472
2.4 .0658 .0477 .0348 1.95 .0690 .0578 .0485 .0408
2.5 .0576 .0404 .0285 2.00 .0620 .0512 .0424 .0352
2.6 .0503 .0342 .0234 2.05 .0556 .0453 .0370 .0303
2.7 .0440 .0289 .0191 2.10 .0498 .0400 .0323 .0261
2.8 .0384 .0244 .0156 2.15 .0447 .0354 .0281 .0224
2.9 .0336 .0206 .0127 2.20 .0400 .0312 .0244 .0192
3.0 .0293 .0174 .0104 2.25 .0357 .0275 .0212 .0164
3.1 .0256 .0146 .0084 2.30 .0320 .0242 .0184 .0141
3.2 .0223 .0123 .0068 2.35 .0285 .0213 .0160 .0120
3.3 .0194 .0103 .0056 2.40 .0255 .0187 .0138 .0102
3.4 .0169 .0087 .0045 2.45 .0227 .0165 .0120 .0087
3.5 .0148 .0073 .0036 2.50 .0203 .0144 .0103 .0074
3.6 .0129 .0061 .0029 2.55 .0180 .0127 .0089 .0063
3.7 .0112 .0051 .0024 2.60 .0161 .0111 .0077 .0054
3.8 .0097 .0043 .0019 2.65 .0143 .0097 .0066 .0045
3.9 .0085 .0036 .0016 2.70 .0127 .0085 .0057 .0039
4.0 .0074 .0030 .0012 2.75 .0113 .0074 .0049 .0033
4.1 .0064 .0025 .0010 2.80 .0100 .0065 .0042 .0028
4.2 .0056 .0021 .0008 2.85 .0089 .0057 .0036 .0023
4.3 .0049 .0018 .0007 2.90 .0079 .0050 .0031 .0020
4.4 .0042 .0015 .0005 2.95 .0070 .0043 .0027 .0017
4.5 .0037 .0012 .0004 3.00 .0062 .0038 .0023 .0014
4.6 .0032 .0010 .0003 3.05 .0055 .0033 .0020 .0012
4.7 .0028 .0009 .0003 3.10 .0049 .0029 .0017 .0010
4.8 .0024 .0007 .0002 3.15 .0043 .0025 .0014 .0008
4.9 .0021 .0006 .0002 3.20 .0038 .0022 .0012 .0007
5.0 .0018 .0005 .0001 3.25 .0034 .0019 .0011 .0006
5.1 .0016 .0004 .0001 3.30 .0030 .0016 .0009 .0005
5.2 .0014 .0003 .0001 3.35 .0027 .0014 .0008 .0004
5.3 .0012 .0003 .0001 3.40 .0024 .0012 .0007 .0003
5.4 .0010 .0002 .0001 3.45 .0021 .0011 .0006 .0003
5.5 .0009 .0002 .0000 3.50 .0018 .0009 .0005 .0002
5.6 .0008 .0002 .0000 3.55 .0016 .0008 .0004 .0002
5.7 .0007 .0001 .0000 3.60 .0014 .0007 .0003 .0002
5.8 .0006 .0001 .0000 3.65 .0013 .0006 .0003 .0001
5.9 .0005 .0001 .0000 3.70 .0011 .0005 .0002 .0001
6.0 .0004 .0001 .0000 3.75 .0010 .0005 .0002 .0001
6.1 .0004 .0001 .0000 3.80 .0009 .0004 .0002 .0001
6.2 .0003 .0001 .0000 3.85 .0008 .0003 .0002 .0001
6.3 .0003 .0000 .0000 3.90 .0007 .0003 .0001 .0001
6.4 .0002 .0000 .0000 3.95 .0006 .0003 .0001 .0000
6.5 .0002 .0000 .0000 4.00 .0005 .0002 .0001 .0000
6.6 .0002 .0000 .0000 4.05 .0005 .0002 .0001 .0000
6.7 .0002 .0000 .0000 4.10 .0004 .0002 .0001 .0000
6.8 .0001 .0000 .0000 4.15 .0004 .0001 .0001 .0000
        4.20 .0003 .0001 .0000 .0000
        4.25 .0003 .0001 .0000 .0000
        4.30 .0002 .0001 .0000 .0000
        4.35 .0002 .0001 .0000 .0000
        4.40 .0002 .0001 .0000 .0000
        4.45 .0002 .0001 .0000 .0000
        4.50 .0001 .0001 .0000 .0000


Table 4: continued
  Degrees of freedom
Ratio 10 11 12 13 14 15 16 17 18 19
1.46 .1473 .1389 .1311 .1237 .1169 .1105 .1044 .0988 .0935 .0886
1.48 .1395 .1311 .1232 .1159 .1090 .1027 .0967 .0912 .0860 .0811
1.50 .1321 .1236 .1157 .1084 .1016 .0953 .0895 .0841 .0790 .0743
1.52 .1249 .1164 .1086 .1014 .0947 .0885 .0827 .0774 .0725 .0679
1.54 .1181 .1097 .1019 .0947 .0881 .0820 .0764 .0712 .0664 .0620
1.56 .1117 .1032 .0955 .0884 .0820 .0760 .0705 .0655 .0608 .0566
1.58 .1055 .0971 .0895 .0825 .0762 .0704 .0650 .0602 .0557 .0515
1.60 .0996 .0913 .0838 .0770 .0708 .0651 .0599 .0552 .0509 .0469
1.62 .0940 .0859 .0785 .0718 .0657 .0602 .0552 .0506 .0465 .0427
1.64 .0887 .0807 .0734 .0668 .0609 .0556 .0507 .0464 .0424 .0388
1.66 .0837 .0757 .0686 .0622 .0565 .0513 .0466 .0424 .0386 .0352
1.68 .0789 .0711 .0641 .0579 .0523 .0473 .0428 .0388 .0352 .0319
1.70 .0744 .0667 .0599 .0538 .0484 .0436 .0393 .0355 .0320 .0289
1.72 .0701 .0625 .0559 .0500 .0448 .0402 .0361 .0324 .0291 .0262
1.74 .0660 .0586 .0522 .0465 .0414 .0370 .0330 .0295 .0264 .0237
1.76 .0621 .0549 .0487 .0431 .0383 .0340 .0303 .0269 .0240 .0214
1.78 .0584 .0514 .0454 .0400 .0354 .0313 .0277 .0245 .0217 .0193
1.80 .0550 .0482 .0423 .0371 .0326 .0287 .0253 .0223 .0197 .0174
1.82 .0517 .0451 .0394 .0344 .0301 .0264 .0231 .0203 .0178 .0157
1.84 .0486 .0422 .0366 .0319 .0278 .0242 .0211 .0185 .0161 .0141
1.86 .0456 .0394 .0341 .0295 .0256 .0222 .0193 .0168 .0146 .0127
1.88 .0429 .0368 .0317 .0273 .0236 .0204 .0176 .0152 .0132 .0114
1.90 .0403 .0344 .0295 .0253 .0217 .0186 .0160 .0138 .0119 .0103
1.92 .0378 .0321 .0274 .0234 .0200 .0171 .0146 .0125 .0107 .0092
1.94 .0355 .0300 .0254 .0216 .0184 .0156 .0133 .0113 .0097 .0083
1.96 .0333 .0280 .0236 .0199 .0169 .0143 .0121 .0103 .0087 .0074
1.98 .0312 .0261 .0219 .0184 .0155 .0131 .0110 .0093 .0078 .0066
2.00 .0293 .0244 .0203 .0170 .0142 .0119 .0100 .0084 .0071 .0059
2.02 .0274 .0227 .0189 .0157 .0131 .0109 .0091 .0076 .0063 .0053
2.04 .0257 .0212 .0175 .0145 .0120 .0099 .0082 .0069 .0057 .0047
2.06 .0241 .0197 .0162 .0133 .0110 .0091 .0075 .0062 .0051 .0042
2.08 .0225 .0184 .0150 .0123 .0101 .0083 .0068 .0056 .0046 .0038
2.10 .0211 .0171 .0139 .0113 .0092 .0075 .0061 .0050 .0041 .0034
2.12 .0197 .0159 .0129 .0104 .0084 .0069 .0056 .0045 .0037 .0030
2.14 .0185 .0148 .0119 .0096 .0077 .0062 .0050 .0041 .0033 .0027
2.16 .0173 .0138 .0110 .0088 .0071 .0057 .0046 .0037 .0030 .0024
2.18 .0162 .0128 .0102 .0081 .0065 .0052 .0041 .0033 .0026 .0021
2.20 .0151 .0119 .0094 .0075 .0059 .0047 .0037 .0030 .0024 .0019
2.22 .0141 .0111 .0087 .0069 .0054 .0043 .0034 .0027 .0021 .0017
2.24 .0132 .0103 .0080 .0063 .0049 .0039 .0030 .0024 .0019 .0015
2.26 .0123 .0096 .0074 .0058 .0045 .0035 .0027 .0022 .0017 .0013
2.28 .0115 .0089 .0069 .0053 .0041 .0032 .0025 .0019 .0015 .0012
2.30 .0107 .0082 .0063 .0049 .0038 .0029 .0022 .0017 .0013 .0010
2.32 .0100 .0076 .0058 .0045 .0034 .0026 .0020 .0016 .0012 .0009
2.34 .0094 .0071 .0054 .0041 .0031 .0024 .0018 .0014 .0011 .0008
2.36 .0087 .0066 .0050 .0038 .0028 .0022 .0016 .0012 .0009 .0007
2.38 .0081 .0061 .0046 .0034 .0026 .0020 .0015 .0011 .0008 .0006
2.40 .0076 .0057 .0042 .0032 .0024 .0018 .0013 .0010 .0007 .0006
2.42 .0071 .0052 .0039 .0029 .0021 .0016 .0012 .0009 .0007 .0005
2.44 .0066 .0049 .0036 .0026 .0020 .0014 .0011 .0008 .0006 .0004
2.46 .0062 .0045 .0033 .0024 .0018 .0013 .0010 .0007 .0005 .0004
2.48 .0057 .0042 .0030 .0022 .0016 .0012 .0009 .0006 .0005 .0003
2.50 .0053 .0039 .0028 .0020 .0015 .0011 .0008 .0006 .0004 .0003
2.52 .0050 .0036 .0026 .0019 .0013 .0010 .0007 .0005 .0004 .0003
2.54 .0046 .0033 .0024 .0017 .0012 .0009 .0006 .0005 .0003 .0002
2.56 .0043 .0031 .0022 .0015 .0011 .0008 .0006 .0004 .0003 .0002
2.58 .0040 .0028 .0020 .0014 .0010 .0007 .0005 .0004 .0003 .0002
2.60 .0037 .0026 .0018 .0013 .0009 .0006 .0005 .0003 .0002 .0002
2.62 .0035 .0024 .0017 .0012 .0008 .0006 .0004 .0003 .0002 .0001
2.64 .0032 .0022 .0016 .0011 .0007 .0005 .0004 .0003 .0002 .0001


Table 4: continued
  Degrees of freedom
Ratio 20 21 22 23 24 25 26 27 28 29
1.86 .0111 .0097 .0084 .0074 .0064 .0056 .0049 .0043 .0038 .0033
1.88 .0099 .0086 .0075 .0065 .0056 .0049 .0043 .0037 .0032 .0028
1.90 .0089 .0076 .0066 .0057 .0049 .0043 .0037 .0032 .0028 .0024
1.92 .0079 .0068 .0058 .0050 .0043 .0037 .0032 .0028 .0024 .0021
1.94 .0071 .0060 .0052 .0044 .0038 .0032 .0028 .0024 .0021 .0018
1.96 .0063 .0054 .0046 .0039 .0033 .0028 .0024 .0021 .0018 .0015
1.98 .0056 .0047 .0040 .0034 .0029 .0025 .0021 .0018 .0015 .0013
2.00 .0050 .0042 .0035 .0030 .0025 .0021 .0018 .0015 .0013 .0011
2.02 .0044 .0037 .0031 .0026 .0022 .0018 .0016 .0013 .0011 .0009
2.04 .0040 .0033 .0027 .0023 .0019 .0016 .0013 .0011 .0009 .0008
2.06 .0035 .0029 .0024 .0020 .0017 .0014 .0012 .0010 .0008 .0007
2.08 .0031 .0026 .0021 .0018 .0014 .0012 .0010 .0008 .0007 .0006
2.10 .0028 .0023 .0019 .0015 .0013 .0010 .0009 .0007 .0006 .0005
2.12 .0025 .0020 .0016 .0013 .0011 .0009 .0007 .0006 .0005 .0004
2.14 .0022 .0018 .0014 .0012 .0009 .0008 .0006 .0005 .0004 .0003
2.16 .0019 .0016 .0013 .0010 .0008 .0007 .0005 .0004 .0004 .0003
2.18 .0017 .0014 .0011 .0009 .0007 .0006 .0005 .0004 .0003 .0002
2.20 .0015 .0012 .0010 .0008 .0006 .0005 .0004 .0003 .0003 .0002
2.22 .0013 .0011 .0008 .0007 .0005 .0004 .0003 .0003 .0002 .0002
2.24 .0012 .0009 .0007 .0006 .0005 .0004 .0003 .0002 .0002 .0001
2.26 .0010 .0008 .0006 .0005 .0004 .0003 .0002 .0002 .0002 .0001
2.23 .0009 .0007 .0006 .0004 .0003 .0003 .0002 .0002 .0001 .0001
2.30 .0008 .0006 .0005 .0004 .0003 .0002 .0002 .0001 .0001 .0001
2.32 .0007 .0005 .0004 .0003 .0003 .0002 .0002 .0001 .0001 .0001
2.34 .0006 .0005 .0004 .0003 .0002 .0002 .0001 .0001 .0001 .0001
2.36 .0006 .0004 .0003 .0002 .0002 .0001 .0001 .0001 .0001 .0000
2.38 .0005 .0004 .0003 .0002 .0002 .0001 .0001 .0001 .0001 .0000
2.40 .0004 .0003 .0002 .0002 .0001 .0001 .0001 .0001 .0000 .0000
2.42 .0004 .0003 .0002 .0002 .0001 .0001 .0001 .0001 .0000 .0000
2.44 .0003 .0002 .0002 .0001 .0001 .0001 .0001 .0000 .0000 .0000
2.46 .0003 .0002 .0002 .0001 .0001 .0001 .0000 .0000 .0000 .0000
2.48 .0003 .0002 .0001 .0001 .0001 .0001 .0000 .0000 .0000 .0000
2.50 .0002 .0002 .0001 .0001 .0001 .0000 .0000 .0000 .0000 .0000
2.52 .0002 .0001 .0001 .0001 .0001 .0000 .0000 .0000 .0000 .0000
1.30 .1257 .1102 .0968 .0853 .0754 .0667 .0591 .0525    
1.31 .1192 .1037 .0906 .0793 .0696 .0612 .0539 .0476    
1.32 .1128 .0976 .0846 .0736 .0642 .0561 .0491 .0430    
1.33 .1068 .0917 .0790 .0683 .0591 .0513 .0446 .0389    
1.34 .1010 .0862 .0737 .0633 .0544 .0469 .0405 .0351    
1.35 .0955 .0809 .0687 .0586 .0500 .0429 .0368 .0316    
1.36 .0903 .0759 .0640 .0542 .0460 .0391 .0333 .0284    
1.37 .0853 .0711 .0596 .0501 .0422 .0356 .0301 .0255    
1.38 .0805 .0666 .0554 .0462 .0387 .0324 .0272 .0229    
1.39 .0759 .0624 .0515 .0426 .0354 .0295 .0246 .0205    
1.40 .0716 .0584 .0478 .0393 .0324 .0267 .0221 .0184    
1.41 .0674 .0546 .0444 .0362 .0296 .0243 .0199 .0164    
1.42 .0635 .0510 .0411 .0333 .0270 .0220 .0179 .0146    
1.43 .0598 .0476 .0381 .0306 .0246 .0199 .0161 .0130    
1.44 .0563 .0444 .0353 .0281 .0224 .0180 .0144 .0116    
1.45 .0529 .0415 .0326 .0258 .0204 .0162 .0129 .0103    
1.46 .0497 .0386 .0302 .0236 .0186 .0146 .0116 .0092    
1.47 .0467 .0360 .0279 .0217 .0169 .0132 .0103 .0081    
1.48 .0439 .0335 .0257 .0198 .0153 .0119 .0092 .0072    
1.49 .0412 .0312 .0237 .0181 .0139 .0107 .0082 .0064    
1.50 .0386 .0290 .0219 .0166 .0126 .0096 .0073 .0056    
1.51 .0362 .0269 .0202 .0151 .0114 .0086 .0065 .0050    
1.52 .0339 .0250 .0185 .0138 .0103 .0077 .0058 .0044    
1.53 .0318 .0232 .0171 .0126 .0093 .0069 .0052 .0038    
1.54 .0297 .0215 .0157 .0115 .0084 .0062 .0046 .0034    


Table 4: continued
  Degrees of freedom
Ratio 10 11 12 13 14 15 16 17 18 19
2.66 .0030 .0021 .0014 .0010 .0007 .0005 .0003 .0002 .0002 .0001
2.68 .0028 .0019 .0013 .0009 .0006 .0004 .0003 .0002 .0001 .0001
2.70 .0026 .0018 .0012 .0008 .0006 .0004 .0003 .0002 .0001 .0001
2.72 .0024 .0016 .0011 .0007 .0005 .0003 .0002 .0002 .0001 .0001
2.74 .0023 .0015 .0010 .0007 .0005 .0003 .0002 .0001 .0001 .0001
2.76 .0021 .0014 .0009 .0006 .0004 .0003 .0002 .0001 .0001 .0001
2.78 .0019 .0013 .0009 .0006 .0004 .0002 .0002 .0001 .0001 .0000
2.80 .0018 .0012 .0008 .0005 .0003 .0002 .0001 .0001 .0001 .0000
2.82 .0017 .0011 .0007 .0005 .0003 .0002 .0001 .0001 .0001 .0000
2.84 .0016 .0010 .0007 .0004 .0003 .0002 .0001 .0001 .0001 .0000
2.86 .0014 .0009 .0006 .0004 .0003 .0002 .0001 .0001 .0000 .0000
2.88 .0013 .0009 .0006 .0004 .0002 .0001 .0001 .0001 .0000 .0000
2.90 .0012 .0008 .0005 .0003 .0002 .0001 .0001 .0001 .0000 .0000
2.92 .0012 .0007 .0005 .0003 .0002 .0001 .0001 .0000 .0000 .0000
2.94 .0011 .0007 .0004 .0003 .0002 .0001 .0001 .0000 .0000 .0000
2.96 .0010 .0006 .0004 .0002 .0002 .0001 .0001 .0000 .0000 .0000
2.98 .0009 .0006 .0004 .0002 .0001 .0001 .0001 .0000 .0000 .0000
3.00 .0009 .0005 .0003 .0002 .0001 .0001 .0000 .0000 .0000 .0000
3.02 .0008 .0005 .0003 .0002 .0001 .0001 .0000 .0000 .0000 .0000
3.04 .0007 .0004 .0003 .0002 .0001 .0001 .0000 .0000 .0000 .0000
3.06 .0007 .0004 .0002 .0001 .0001 .0001 .0000 .0000 .0000 .0000
3.08 .0006 .0004 .0002 .0001 .0001 .0000 .0000 .0000 .0000 .0000
3.10 .0006 .0003 .0002 .0001 .0001 .0000 .0000 .0000 .0000 .0000
3.12 .0005 .0003 .0002 .0001 .0001 .0000 .0000 .0000 .0000 .0000
3.14 .0005 .0003 .0002 .0001 .0001 .0000 .0000 .0000 .0000 .0000
3.16 .0005 .0003 .0002 .0001 .0001 .0000 .0000 .0000 .0000 .0000
3.18 .0004 .0002 .0001 .0001 .0000 .0000 .0000 .0000 .0000 .0000
3.20 .0004 .0002 .0001 .0001 .0000 .0000 .0000 .0000 .0000 .0000
3.22 .0004 .0002 .0001 .0001 .0000 .0000 .0000 .0000 .0000 .0000
3.24 .0003 .0002 .0001 .0001 .0000 .0000 .0000 .0000 .0000 .0000
3.26 .0003 .0002 .0001 .0001 .0000 .0000 .0000 .0000 .0000 .0000
3.28 .0003 .0002 .0001 .0001 .0000 .0000 .0000 .0000 .0000 .0000
3.30 .0003 .0002 .0001 .0000 .0000 .0000 .0000 .0000 .0000 .0000
3.32 .0003 .0001 .0001 .0000 .0000 .0000 .0000 .0000 .0000 .0000
3.34 .0002 .0001 .0001 .0000 .0000 .0000 .0000 .0000 .0000 .0000
3.36 .0002 .0001 .0001 .0000 .0000 .0000 .0000 .0000 .0000 .0000
3.38 .0002 .0001 .0001 .0000 .0000 .0000 .0000 .0000 .0000 .0000
3.40 .0002 .0001 .0001 .0000 .0000 .0000 .0000 .0000 .0000 .0000
3.42 .0002 .0001 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000
3.44 .0002 .0001 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000
1.46 .0839 .0795 .0754 .0715 .0678 .0643 .0611 .0580 .0551 .0523
1.48 .0766 .0723 .0683 .0646 .0611 .0577 .0546 .0517 .0489 .0463
1.50 .0699 .0657 .0619 .0583 .0549 .0517 .0488 .0460 .0434 .0409
1.52 .0636 .0597 .0559 .0525 .0493 .0463 .0435 .0408 .0384 .0361
1.54 .0579 .0541 .0505 .0472 .0442 .0413 .0387 .0362 .0339 .0317
1.56 .0526 .0489 .0456 .0424 .0395 .0368 .0343 .0320 .0299 .0279
1.58 .0478 .0443 .0410 .0381 .0353 .0328 .0305 .0283 .0263 .0245
1.60 .0433 .0400 .0369 .0341 .0315 .0292 .0270 .0250 .0231 .0214
1.62 .0392 .0361 .0332 .0305 .0281 .0259 .0239 .0220 .0203 .0187
1.64 .0355 .0325 .0298 .0273 .0250 .0230 .0211 .0193 .0178 .0163
1.66 .0321 .0293 .0267 .0244 .0222 .0203 .0186 .0170 .0155 .0142
1.68 .0290 .0263 .0239 .0217 .0198 .0180 .0164 .0149 .0136 .0124
1.70 .0261 .0236 .0214 .0193 .0175 .0159 .0144 .0130 .0118 .0107
1.72 .0235 .0212 .0191 .0172 .0155 .0140 .0126 .0114 .0103 .0093
1.74 .0212 .0190 .0170 .0153 .0137 .0123 .0111 .0100 .0090 .0081
1.76 .0191 .0170 .0152 .0136 .0121 .0108 .0097 .0087 .0078 .0070
1.78 .0171 .0152 .0135 .0120 .0107 .0095 .0085 .0076 .0067 .0060
1.80 .0154 .0136 .0120 .0107 .0094 .0084 .0074 .0066 .0058 .0052
1.82 .0138 .0121 .0107 .0094 .0083 .0073 .0065 .0057 .0050 .0045
1.84 .0124 .0108 .0095 .0083 .0073 .0064 .0056 .0050 .0044 .0038


Table 4: continued
  Degrees of freedom
Ratio 30 35 40 45 50 55 60 65
1.55 .0278 .0200 .0144 .0104 .0076 .0055 .0041 .0030
1.56 .0260 .0185 .0132 .0095 .0068 .0049 .0036 .0026
1.57 .0243 .0171 .0121 .0086 .0062 .0044 .0032 .0023
1.58 .0227 .0159 .0111 .0078 .0055 .0039 .0028 .0020
1.59 .0212 .0147 .0102 .0071 .0050 .0035 .0025 .0017
1.60 .0198 .0136 .0093 .0065 .0045 .0031 .0022 .0015
1.61 .0185 .0125 .0086 .0059 .0040 .0028 .0019 .0013
1.62 .0173 .0116 .0078 .0053 .0036 .0025 .0017 .0012
1.63 .0161 .0107 .0071 .0048 .0032 .0022 .0015 .0010
1.64 .0150 .0099 .0065 .0043 .0029 .0019 .0013 .0009
1.65 .0140 .0091 .0060 .0039 .0026 .0017 .0011 .0008
1.66 .0130 .0084 .0054 .0035 .0023 .0015 .0010 .0007
1.67 .0121 .0077 .0050 .0032 .0021 .0013 .0009 .0006
1.68 .0113 .0071 .0045 .0029 .0019 .0012 .0008 .0005
1.69 .0105 .0065 .0041 .0026 .0017 .0011 .0007 .0004
1.70 .0097 .0060 .0037 .0023 .0015 .0009 .0006 .0004
1.71 .0091 .0055 .0034 .0021 .0013 .0008 .0005 .0003
1.72 .0084 .0051 .0031 .0019 .0012 .0007 .0004 .0003
1.73 .0078 .0047 .0028 .0017 .0010 .0006 .0004 .0002
1.74 .0072 .0043 .0026 .0015 .0009 .0006 .0003 .0002
1.75 .0067 .0039 .0023 .0014 .0008 .0005 .0003 .0002
1.76 .0062 .0036 .0021 .0012 .0007 .0004 .0003 .0002
1.77 .0058 .0033 .0019 .0011 .0006 .0004 .0002 .0001
1.78 .0054 .0030 .0017 .0010 .0006 .0003 .0002 .0001
1.79 .0050 .0028 .0016 .0009 .0005 .0003 .0002 .0001
1.80 .0046 .0025 .0014 .0008 .0004 .0003 .0001 .0001
1.81 .0043 .0023 .0013 .0007 .0004 .0002 .0001 .0001
1.82 .0039 .0021 .0012 .0006 .0004 .0002 .0001 .0001
1.83 .0036 .0020 .0011 .0006 .0003 .0002 .0001 .0001
1.84 .0034 .0018 .0010 .0005 .0003 .0001 .0001 .0000
1.85 .0031 .0016 .0009 .0005 .0002 .0001 .0001 .0000
1.86 .0029 .0015 .0008 .0004 .0002 .0001 .0001 .0000
1.87 .0027 .0014 .0007 .0004 .0002 .0001 .0001 .0000
1.88 .0025 .0012 .0006 .0003 .0002 .0001 .0000 .0000
1.89 .0023 .0011 .0006 .0003 .0001 .0001 .0000 .0000
1.90 .0021 .0010 .0005 .0003 .0001 .0001 .0000 .0000
1.91 .0019 .0009 .0005 .0002 .0001 .0001 .0000 .0000
1.92 .0018 .0009 .0004 .0002 .0001 .0000 .0000 .0000
1.93 .0016 .0008 .0004 .0002 .0001 .0000 .0000 .0000
1.94 .0015 .0007 .0003 .0002 .0001 .0000 .0000 .0000
1.95 .0014 .0006 .0003 .0001 .0001 .0000 .0000 .0000
1.96 .0013 .0006 .0003 .0001 .0001 .0000 .0000 .0000
1.97 .0012 .0005 .0002 .0001 .0001 .0000 .0000 .0000
1.98 .0011 .0005 .0002 .0001 .0000 .0000 .0000 .0000
1.99 .0010 .0004 .0002 .0001 .0000 .0000 .0000 .0000
2.00 .0009 .0004 .0002 .0001 .0000 .0000 .0000 .0000
2.01 .0008 .0004 .0002 .0001 .0000 .0000 .0000 .0000
2.02 .0008 .0003 .0001 .0001 .0000 .0000 .0000 .0000
2.03 .0007 .0003 .0001 .0001 .0000 .0000 .0000 .0000
2.04 .0007 .0003 .0001 .0000 .0000 .0000 .0000 .0000
2.05 .0006 .0002 .0001 .0000 .0000 .0000 .0000 .0000
2.06 .0006 .0002 .0001 .0000 .0000 .0000 .0000 .0000
2.07 .0005 .0002 .0001 .0000 .0000 .0000 .0000 .0000
2.08 .0005 .0002 .0001 .0000 .0000 .0000 .0000 .0000
2.09 .0004 .0002 .0001 .0000 .0000 .0000 .0000 .0000
2.10 .0004 .0002 .0001 .0000 .0000 .0000 .0000 .0000
2.11 .0004 .0001 .0001 .0000 .0000 .0000 .0000 .0000
2.12 .0003 .0001 .0000 .0000 .0000 .0000 .0000 .0000
2.13 .0003 .0001 .0000 .0000 .0000 .0000 .0000 .0000
2.14 .0003 .0001 .0000 .0000 .0000 .0000 .0000 .0000


Table 4: continued.
  Degrees of freedom
Ratio 30 35 40 45 50 55 60 65
2.15 .0003 .0001 .0000 .0000 .0000 .0000 .0000 .0000
2.16 .0002 .0001 .0000 .0000 .0000 .0000 .0000 .0000
2.17 .0002 .0001 .0000 .0000 .0000 .0000 .0000 .0000
2.18 .0002 .0001 .0000 .0000 .0000 .0000 .0000 .0000
2.19 .0002 .0001 .0000 .0000 .0000 .0000 .0000 .0000
2.20 .0002 .0001 .0000 .0000 .0000 .0000 .0000 .0000
2.21 .0001 .0000 .0000 .0000 .0000 .0000 .0000 .0000
  Degrees of freedom  
Ratio 70 75 80 85 90 95 100  
1.24 .0845 .0778 .0718 .0662 .0611 .0565 .0522  
1.25 .0769 .0704 .0646 .0593 .0544 .0500 .0460  
1.26 .0698 .0636 .0580 .0529 .0483 .0442 .0404  
1.27 .0633 .0573 .0520 .0472 .0428 .0389 .0354  
1.28 .0573 .0516 .0465 .0420 .0379 .0343 .0310  
1.29 .0517 .0463 .0415 .0372 .0334 .0300 .0270  
1.30 .0467 .0415 .0370 .0330 .0294 .0263 .0235  
1.31 .0420 .0372 .0329 .0292 .0259 .0230 .0204  
1.32 .0378 .0332 .0292 .0257 .0227 .0200 .0177  
1.33 .0339 .0296 .0259 .0227 .0198 .0174 .0153  
1.34 .0304 .0264 .0229 .0199 .0173 .0151 .0132  
1.35 .0272 .0234 .0202 .0175 .0151 .0131 .0113  
1.36 .0243 .0208 .0178 .0153 .0131 .0113 .0097  
1.37 .0217 .0184 .0157 .0134 .0114 .0097 .0083  
1.38 .0193 .0163 .0138 .0117 .0099 .0084 .0071  
1.39 .0172 .0144 .0121 .0101 .0085 .0072 .0060  
1.40 .0153 .0127 .0106 .0088 .0074 .0061 .0051  
1.41 .0135 .0112 .0092 .0076 .0063 .0053 .0044  
1.42 .0120 .0098 .0081 .0066 .0054 .0045 .0037  
1.43 .0106 .0086 .0070 .0057 .0047 .0038 .0031  
1.44 .0094 .0075 .0061 .0049 .0040 .0032 .0026  
1.45 .0082 .0066 .0053 .0043 .0034 .0027 .0022  
1.46 .0073 .0058 .0046 .0037 .0029 .0023 .0019  
1.47 .0064 .0050 .0040 .0031 .0025 .0020 .0016  
1.48 .0056 .0044 .0034 .0027 .0021 .0017 .0013  
1.49 .0049 .0038 .0030 .0023 .0018 .0014 .0011  
1.50 .0043 .0033 .0025 .0020 .0015 .0012 .0009  
1.51 .0038 .0029 .0022 .0017 .0013 .0010 .0008  
1.52 .0033 .0025 .0019 .0014 .0011 .0008 .0006  
1.53 .0029 .0022 .0016 .0012 .0009 .0007 .0005  
1.54 .0025 .0019 .0014 .0010 .0008 .0006 .0004  
1.55 .0022 .0016 .0012 .0009 .0006 .0005 .0004  
1.56 .0019 .0014 .0010 .0007 .0005 .0004 .0003  
1.57 .0016 .0012 .0009 .0006 .0005 .0003 .0002  
1.58 .0014 .0010 .0007 .0005 .0004 .0003 .0002  
1.59 .0012 .0009 .0006 .0004 .0003 .0002 .0002  
1.60 .0011 .0008 .0005 .0004 .0003 .0002 .0001  
1.61 .0009 .0006 .0004 .0003 .0002 .0002 .0001  
1.62 .0008 .0005 .0004 .0003 .0002 .0001 .0001  
1.63 .0007 .0005 .0003 .0002 .0002 .0001 .0001  
1.64 .0006 .0004 .0003 .0002 .0001 .0001 .0001  
1.65 .0005 .0003 .0002 .0002 .0001 .0001 .0000  
1.66 .0004 .0003 .0002 .0001 .0001 .0001 .0000  
1.67 .0004 .0002 .0002 .0001 .0001 .0000 .0000  
1.68 .0003 .0002 .0001 .0001 .0001 .0000 .0000  
1.69 .0003 .0002 .0001 .0001 .0000 .0000 .0000  
1.70 .0002 .0001 .0001 .0001 .0000 .0000 .0000  
1.71 .0002 .0001 .0001 .0001 .0000 .0000 .0000  
1.72 .0002 .0001 .0001 .0000 .0000 .0000 .0000  
1.73 .0001 .0001 .0001 .0000 .0000 .0000 .0000