Appendix 3. Chi-Square Distribution

Upper tail probabilities for the Chi-square distribution are shown. For larger degrees of freedom, n, the distribution may be approximated as a normal distribution with

χ2n = 1 n = 2 n = 3 n = 4 n = 5 
0.50.479500.778800.918890.973500.99212
1.00.317310.606530.801250.909800.96257
1.50.220670.472370.682270.826640.91307
2.00.157300.367880.572410.735760.84915
2.50.113850.286510.475290.644640.77650
3.00.083260.223130.391630.557830.69999
3.50.061370.173770.320760.477880.62339
4.00.045500.135340.261460.406010.54942
4.50.033900.105400.212290.342550.47988
5.00.025350.082090.171800.287300.41588
5.50.019020.063930.138640.239730.35795
6.00.014310.049790.111610.199150.30622
6.50.010790.038770.089660.164790.26056
7.00.008150.030200.071900.135890.22064
7.50.006170.023520.057560.111710.18603
8.00.004680.018320.046010.091580.15624
8.50.003550.014260.036730.074890.13075
χ2n = 1 n = 2 n = 3 n = 4 n = 5 
9.00.002700.011110.029290.061100.10906
9.50.002060.008650.023330.049750.09071
10.00.001570.006740.018570.040430.07524
10.50.001190.005250.014760.032800.06225
11.00.000910.004090.011730.026560.05138
11.50.000700.003180.009310.021480.04232
12.00.000530.002480.007380.017350.03479
12.50.000410.001930.005850.014000.02854
13.00.000310.001500.004640.011280.02338
13.50.000240.001170.003670.009070.01912
14.00.000180.000910.002910.007300.01561
14.50.000140.000710.002300.005860.01273
15.00.000110.000550.001820.004700.01036
15.50.000080.000430.001440.003770.00843
16.00.000060.000340.001130.003020.00684
16.50.000050.000260.000900.002420.00555
17.00.000040.000200.000710.001930.00450
17.50.000030.000160.000560.001550.00364
χ2n = 1 n = 2 n = 3 n = 4 n = 5 
18.00.000020.000120.000440.001230.00295
18.50.000020.000100.000350.000990.00238
19.00.000010.000080.000270.000790.00192
19.5 0.000060.000220.000630.00155
20.0 0.000050.000170.000500.00125
20.5 0.000040.000130.000400.00101
21.0 0.000030.000110.000320.00081
21.5 0.000020.000080.000250.00065
22.0 0.000020.000070.000200.00052
22.5 0.000010.000050.000160.00042
23.0  0.000040.000130.00034
23.5  0.000030.000100.00027
24.0  0.000030.000080.00022
24.5  0.000020.000060.00017
25.0  0.000020.000050.00014
χ2n = 6n = 7n = 8n = 9n = 10
5.00.543813



6.00.4231900.539749


7.00.3208470.4288800.536633

8.00.2381030.3325940.4334700.534146
9.00.1735780.2526560.3422960.4372740.532104
10.00.1246520.1885740.2650260.3504850.440493
11.00.0883760.1386190.2016990.2757090.357518
12.00.0619690.1005590.1512040.2133090.285057
13.00.0430360.0721080.1118500.1626060.223672
14.00.0296360.0511810.0817650.1223250.172992
15.00.0202570.0359990.0591450.0909360.132062
16.00.0137540.0251160.0423800.0668820.099632
17.00.0092830.0173960.0301090.0487160.074364
18.00.0062320.0119700.0212260.0351740.054964
19.00.0041640.0081870.0148600.0251930.040263
20.00.0027690.0055700.0103360.0179120.029253
21.00.0018350.0037700.0071470.0126500.021094
22.00.0012110.0025400.0049160.0088790.015105
23.00.0007960.0017050.0033640.0061960.010747
24.00.0005220.0011390.0022920.0043010.007600
χ2n = 6n = 7n = 8n = 9n = 10
25.00.0003410.0007590.0015550.0029710.005346
26.00.0002230.0005040.0010500.0020430.003740
27.00.0001450.0003330.0007070.0013990.002604
28.00.0000940.0002200.0004740.0009540.001805
29.00.0000610.0001450.0003170.0006480.001246
30.00.0000390.0000950.0002110.0004390.000857
31.00.0000250.0000620.0001410.0002960.000587
32.00.0000160.0000410.0000930.0001990.000400
33.00.0000100.0000260.0000620.0001340.000272
34.00.0000070.0000170.0000410.0000890.000185
35.00.0000040.0000110.0000270.0000600.000125
36.00.0000030.0000070.0000180.0000400.000084
37.00.0000020.0000050.0000120.0000260.000057
38.00.0000010.0000030.0000080.0000170.000038
39.0
0.0000020.0000050.0000120.000025
40.0
0.0000010.0000030.0000080.000017
χ2n = 11n = 12n = 13n = 14n = 15
10.00.5303870.6159610.6939350.7621830.819740
11.00.4432630.5289190.6108180.6860360.752594
12.00.3636430.4456800.5276440.6063030.679029
13.00.2933250.3690410.4478120.5265240.602298
14.00.2329940.3007080.3738440.4497110.525529
15.00.1824970.2414360.3073530.3781550.451417
16.00.1411310.1912360.2491300.3133740.382052
17.00.1078760.1495970.1993040.2561780.318864
18.00.0815810.1156910.1575190.2067810.262666
19.00.0610940.0885280.1231040.1649490.213734
20.00.0453410.0670860.0952100.1301410.171933
21.00.0333710.0503800.0729290.1016330.136829
22.00.0243730.0375200.0553620.0786140.107804
23.00.0176750.0277260.0416760.0602700.084140
24.00.0127330.0203410.0311300.0458220.065093
25.00.0091170.0148230.0230840.0345670.049943
26.00.0064900.0107340.0170010.0258870.038023
27.00.0045950.0077270.0124410.0192540.028736
28.00.0032370.0055320.0090500.0142280.021569
29.00.0022700.0039400.0065460.0104500.016085
χ2n = 11n = 12n = 13n = 14n = 15
30.00.0015850.0027920.0047100.0076320.011921
31.00.0011020.0019700.0033720.0055440.008785
32.00.0007630.0013840.0024020.0040060.006438
33.00.0005260.0009680.0017040.0028810.004694
34.00.0003620.0006750.0012040.0020620.003405
35.00.0002480.0004680.0008470.0014700.002459
36.00.0001690.0003240.0005930.0010430.001768
37.00.0001150.0002230.0004140.0007380.001266
38.00.0000780.0001540.0002880.0005200.000902
39.00.0000530.0001050.0002000.0003650.000641
40.00.0000360.0000720.0001380.0002550.000453
41.00.0000240.0000490.0000950.0001780.000320
42.00.0000160.0000330.0000650.0001240.000225
43.00.0000110.0000230.0000450.0000860.000157
44.00.0000070.0000150.0000310.0000590.000110
45.00.0000050.0000100.0000210.0000410.000077
46.00.0000030.0000070.0000140.0000280.000053
47.00.0000020.0000050.0000100.0000190.000037
48.00.0000010.0000030.0000070.0000130.000025
χ2n = 16n = 17n = 18n = 19n =20
15.00.5246390.5954820.6619670.7225970.776408
16.00.4529610.5238350.5925470.6572780.716624
17.00.3855970.4543660.5231050.5898680.652974
18.00.3238970.3888410.4556530.5224380.587408
19.00.2686630.3285320.3918230.4568360.521826
20.00.2202210.2742290.3328200.3945780.457930
21.00.1785110.2262900.2794130.3368010.397133
22.00.1431920.1847190.2319850.2842560.340511
23.00.1137350.1492510.1905900.2373420.288795
24.00.0895040.1194350.1550280.1961520.242392
25.00.0698250.0947100.1249160.1605420.201431
26.00.0540280.0744610.0997580.1301890.165812
27.00.0414830.0580680.0789950.1046530.135264
28.00.0316200.0449380.0620550.0834290.109399
29.00.0239360.0345260.0483790.0659850.087759
30.00.0180020.0263450.0374460.0517980.069854
31.00.0134560.0199720.0287870.0403730.055190
32.00.0100000.0150480.0219870.0312550.043298
33.00.0073900.0112720.0166900.0240400.033741
34.00.0054330.0083960.0125960.0183780.026125
35.00.0039740.0062210.0094520.0139670.020104
χ2n = 16n = 17n = 18n = 19n =20
36.00.0028930.0045870.0070560.0105560.015381
37.00.0020970.0033650.0052410.0079350.011702
38.00.0015130.0024580.0038730.0059350.008856
39.00.0010880.0017870.0028500.0044170.006667
40.00.0007790.0012940.0020870.0032720.004995
41.00.0005550.0009330.0015220.0024130.003725
42.00.0003950.0006710.0011060.0017720.002766
43.00.0002790.0004800.0008000.0012960.002044
44.00.0001970.0003430.0005770.0009440.001505
45.00.0001390.0002440.0004140.0006850.001103
46.00.0000970.0001730.0002970.0004960.000806
47.00.0000680.0001220.0002120.0003570.000587
48.00.0000470.0000860.0001510.0002570.000425
49.00.0000330.0000600.0001070.0001840.000307
50.00.0000230.0000420.0000750.0001310.000221
51.00.0000160.0000300.0000530.0000930.000159
52.00.0000110.0000210.0000370.0000660.000114
53.00.0000080.0000140.0000260.0000470.000081
54.00.0000050.0000100.0000180.0000330.000058
55.00.0000040.0000070.0000130.0000230.000041