Skip to main content
Statistics LibreTexts

14.1: A | Statistical Tables

  • Page ID
    4631
  • \(F\) Distribution

    .

    Figure A1 Table entry for \(p\) is the critical value \(F^{*}\) with probability \(p\) lying to its right.

        Degrees of freedom in the numerator
    Degrees of freedom 
    in the denominator
    \(p\) 1 2 3 4 5 6 7 8 9
    1 .100 39.86 49.50 53.59 55.83 57.24 58.20 58.91 59.44 59.86
    .050 161.45 199.50 215.71 224.58 230.16 233.99 236.77 238.88 240.54
    .025 647.79 799.50 864.16 899.58 921.85 937.11 948.22 956.66 963.28
    .010 4052.2 4999.5 5403.4 5624.6 5763.6 5859.0 5928.4 5981.1 6022.5
    .001 405284 500000 540379 562500 576405 585937 592873 598144 602284
    2 .100 8.53 9.00 9.16 9.24 9.29 9.33 9.35 9.37 9.38
    .050 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38
    .025 38.51 39.00 39.17 39.25 39.30 39.33 39.36 39.37 39.39
    .010 98.50 99.00 99.17 99.25 99.30 99.33 99.36 99.37 99.39
    .001 998.50 999.00 999.17 999.25 999.30 999.33 999.36 999.37 999.39
    3 .100 5.54 5.46 5.39 5.34 5.31 5.28 5.27 5.25 5.24
    .050 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81
    .025 17.44 16.04 15.44 15.10 14.88 14.73 14.62 14.54 14.47
    .010 34.12 30.82 29.46 28.71 28.24 27.91 27.67 27.49 27.35
    .001 167.03 148.50 141.11 137.10 134.58 132.85 131.58 130.62 129.86
    4 .100 4.54 4.32 4.19 4.11 4.05 4.01 3.98 3.95 3.94
    .050 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00
    .025 12.22 10.65 9.98 9.60 9.36 9.20 9.07 8.98 8.90
    .010 21.20 18.00 16.69 15.98 15.52 15.21 14.98 14.80 14.66
    .001 74.14 61.25 56.18 53.44 51.71 50.53 49.66 49.00 48.47
    5 .100 4.06 3.78 3.62 3.52 3.45 3.40 3.37 3.34 3.32
    .050 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77
    .025 10.01 8.43 7.76 7.39 7.15 6.98 6.85 6.76 6.68
    .010 16.26 13.27 12.06 11.39 10.97 10.67 10.46 10.29 10.16
    .001 47.18 37.12 33.20 31.09 29.75 28.83 28.16 27.65 27.24
    6 .100 3.78 3.46 3.29 3.18 3.11 3.05 3.01 2.98 2.96
    .050 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10
    .025 8.81 7.26 6.60 6.23 5.99 5.82 5.70 5.60 5.52
    .010 13.75 10.92 9.78 9.15 8.75 8.47 8.26 8.10 7.98
    .001 35.51 27.00 23.70 21.92 20.80 20.03 19.46 19.03 18.69
    7 .100 3.59 3.26 3.07 2.96 2.88 2.83 2.78 2.75 2.72
    .050 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68
    .025 8.07 6.54 5.89 5.52 5.29 5.12 4.99 4.90 4.82
    .010 12.25 9.55 8.45 7.85 7.46 7.19 6.99 6.84 6.72
    .001 29.25 21.69 18.77 17.20 16.21 15.52 15.02 14.63 14.33

    Table A1 \(F\) critical values

        Degrees of freedom in the numerator
    Degrees of freedom 
    in the denominator
    \(p\) 10 12 15 20 25 30 40 50 60 120 1000
    1 .100 60.19 60.71 61.22 61.74 62.05 62.26 62.53 62.69 62.79 63.06 63.30
    .050 241.88 243.91 245.95 248.01 249.26 250.10 251.14 251.77 252.20 253.25 254.19
    .025 968.63 976.71 984.87 993.10 998.08 1001.4 1005.6 1008.1 1009.8 1014.0 1017.7
    .010 6055.8 6106.3 6157.3 6208.7 6239.8 6260.6 6286.8 6302.5 6313.0 6339.4 6362.7
    .001 605621 610668 615764 620908 624017 626099 628712 630285 631337 633972 636301
    2 .100 9.39 9.41 9.42 9.44 9.45 9.46 9.47 9.47 9.47 9.48 9.49
    .050 19.40 19.41 19.43 19.45 19.46 19.46 19.47 19.48 19.48 19.49 19.49
    .025 39.40 39.41 39.43 39.45 39.46 39.46 39.47 39.48 39.48 39.49 39.50
    .010 99.40 99.42 99.43 99.45 99.46 99.47 99.47 99.48 99.48 99.49 99.50
    .001 999.40 999.42 999.43 999.45 999.46 999.47 999.47 999.48 999.48 999.49 999.50
    3 .100 5.23 5.22 5.20 5.18 5.17 5.17 5.16 5.15 5.15 5.14 5.13
    .050 8.79 8.74 8.70 8.66 8.63 8.62 8.59 8.58 8.57 8.55 8.53
    .025 14.42 14.34 14.25 14.17 14.12 14.08 14.04 14.01 13.99 13.95 13.91
    .010 27.23 27.05 26.87 26.69 26.58 26.50 26.41 26.35 26.32 26.22 26.14
    .001 129.25 128.32 127.37 126.42 125.84 125.45 124.96 124.66 124.47 123.97 123.53
    4 .100 3.92 3.90 3.87 3.84 3.83 3.82 3.80 3.80 3.79 3.78 3.76
    .050 5.96 5.91 5.86 5.80 5.77 5.75 5.72 5.70 5.69 5.66 5.63
    .025 8.84 8.75 8.66 8.56 8.50 8.46 8.41 8.38 8.36 8.31 8.26
    .010 14.55 14.37 14.20 14.02 13.91 13.84 13.75 13.69 13.65 13.56 13.47
    .001 48.05 47.41 46.76 46.10 45.70 45.43 45.09 44.88 44.75 44.40 44.09
    5 .100 3.30 3.27 3.24 3.21 3.19 3.17 3.16 3.15 3.14 3.12 3.11
    .050 4.74 4.68 4.62 4.56 4.52 4.50 4.46 4.44 4.43 4.40 4.37
    .025 6.62 6.52 6.43 6.33 6.27 6.23 6.18 6.14 6.12 6.07 6.02
    .010 10.05 9.89 9.72 9.55 9.45 9.38 9.29 9.24 9.20 9.11 9.03
    .001 26.92 26.42 25.91 25.39 25.08 24.87 24.60 24.44 24.33 24.06 23.82
    6 .100 2.94 2.90 2.87 2.84 2.81 2.80 2.78 2.77 2.76 2.74 2.72
    .050 4.06 4.00 3.94 3.87 3.83 3.81 3.77 3.75 3.74 3.70 3.67
    .025 5.46 5.37 5.27 5.17 5.11 5.07 5.01 4.98 4.96 4.90 4.86
    .010 7.87 7.72 7.56 7.40 7.30 7.23 7.14 7.09 7.06 6.97 6.89
    .001 18.41 17.99 17.56 17.12 16.85 16.67 16.44 16.31 16.21 15.98 15.77
    7 .100 2.70 2.67 2.63 2.59 2.57 2.56 2.54 2.52 2.51 2.49 2.47
    .050 3.64 3.57 3.51 3.44 3.40 3.38 3.34 3.32 3.30 3.27 3.23
    .025 4.76 4.67 4.57 4.47 4.40 4.36 4.31 4.28 4.25 4.20 4.15
    .010 6.62 6.47 6.31 6.16 6.06 5.99 5.91 5.86 5.82 5.74 5.66
    .001 14.08 13.71 13.32 12.93 12.69 12.53 12.33 12.20 12.12 11.91 11.72

    Table A2 \(F\) critical values (continued)

        Degrees of freedom in the numerator
    Degrees of freedom 
    in the denominator
    \(p\) 1 2 3 4 5 6 7 8 9
    8 .100 3.46 3.11 2.92 2.81 2.73 2.67 2.62 2.59 2.56
    .050 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39
    .025 7.57 6.06 5.42 5.05 4.82 4.65 4.53 4.43 4.36
    .010 11.26 8.65 7.59 7.01 6.63 6.37 6.18 6.03 5.91
    .001 25.41 18.49 15.83 14.39 13.48 12.86 12.40 12.05 11.77
    9 .100 3.36 3.01 2.81 2.69 2.61 2.55 2.51 2.47 2.44
    .050 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18
    .025 7.21 5.71 5.08 4.72 4.48 4.32 4.20 4.10 4.03
    .010 10.56 8.02 6.99 6.42 6.06 5.80 5.61 5.47 5.35
    .001 22.86 16.39 13.90 12.56 11.71 11.13 10.70 10.37 10.11
    10 .100 3.29 2.92 2.73 2.61 2.52 2.46 2.41 2.38 2.35
    .050 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02
    .025 6.94 5.46 4.83 4.47 4.24 4.07 3.95 3.85 3.78
    .010 10.04 7.56 6.55 5.99 5.64 5.39 5.20 5.06 4.94
    .001 21.04 14.91 12.55 11.28 10.48 9.93 9.52 9.20 8.96
    11 .100 3.23 2.86 2.66 2.54 2.45 2.39 2.34 2.30 2.27
    .050 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.90
    .025 6.72 5.26 4.63 4.28 4.04 3.88 3.76 3.66 3.59
    .010 9.65 7.21 6.22 5.67 5.32 5.07 4.89 4.74 4.63
    .001 19.69 13.81 11.56 10.35 9.58 9.05 8.66 8.35 8.12
    12 .100 3.18 2.81 2.61 2.48 2.39 2.33 2.28 2.24 2.21
    .050 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80
    .025 6.55 5.10 4.47 4.12 3.89 3.73 3.61 3.51 3.44
    .010 9.33 6.93 5.95 5.41 5.06 4.82 4.64 4.50 4.39
    .001 18.64 12.97 10.80 9.63 8.89 8.38 8.00 7.71 7.48
    13 .100 3.14 2.76 2.56 2.43 2.35 2.28 2.23 2.20 2.16
    .050 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.71
    .025 6.41 4.97 4.35 4.00 3.77 3.60 3.48 3.39 3.31
    .010 9.07 6.70 5.74 5.21 4.86 4.62 4.44 4.30 4.19
    .001 17.82 12.31 10.21 9.07 8.35 7.86 7.49 7.21 6.98
    14 .100 3.10 2.73 2.52 2.39 2.31 2.24 2.19 2.15 2.12
    .050 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.65
    .025 6.30 4.86 4.24 3.89 3.66 3.50 3.38 3.29 3.21
    .010 8.86 6.51 5.56 5.04 4.69 4.46 4.28 4.14 4.03
    .001 17.14 11.78 9.73 8.62 7.92 7.44 7.08 6.80 6.58
    15 .100 3.07 2.70 2.49 2.36 2.27 2.21 2.16 2.12 2.09
    .050 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59
    .025 6.20 4.77 4.15 3.80 3.58 3.41 3.29 3.20 3.12
    .010 8.68 6.36 5.42 4.89 4.56 4.32 4.14 4.00 3.89
    .001 16.59 11.34 9.34 8.25 7.57 7.09 6.74 6.47 6.26

    Table A3 \(F\) critical values (continued)

        Degrees of freedom in the numerator
    Degrees of freedom 
    in the denominator
    \(p\) 10 12 15 20 25 30 40 50 60 120 1000
    8 .100 2.54 2.50 2.46 2.42 2.40 2.38 2.36 2.35 2.34 2.32 2.30
    .050 3.35 3.28 3.22 3.15 3.11 3.08 3.04 3.02 3.01 2.97 2.93
    .025 4.30 4.20 4.10 4.00 3.94 3.89 3.84 3.81 3.78 3.73 3.68
    .010 5.81 5.67 5.52 5.36 5.26 5.20 5.12 5.07 5.03 4.95 4.87
    .001 11.54 11.19 10.84 10.48 10.26 10.11 9.92 9.80 9.73 9.53 9.36
    9 .100 2.42 2.38 2.34 2.30 2.27 2.25 2.23 2.22 2.21 2.18 2.16
    .050 3.14 3.07 3.01 2.94 2.89 2.86 2.83 2.80 2.79 2.75 2.71
    .025 3.96 3.87 3.77 3.67 3.60 3.56 3.51 3.47 3.45 3.39 3.34
    .010 5.26 5.11 4.96 4.81 4.71 4.65 4.57 4.52 4.48 4.40 4.32
    .001 9.89 9.57 9.24 8.90 8.69 8.55 8.37 8.26 8.19 8.00 7.84
    10 .100 2.32 2.28 2.24 2.20 2.17 2.16 2.13 2.12 2.11 2.08 2.06
    .050 2.98 2.91 2.85 2.77 2.73 2.70 2.66 2.64 2.62 2.58 2.54
    .025 3.72 3.62 3.52 3.42 3.35 3.31 3.26 3.22 3.20 3.14 3.09
    .010 4.85 4.71 4.56 4.41 4.31 4.25 4.17 4.12 4.08 4.00 3.92
    .001 8.75 8.45 8.13 7.80 7.60 7.47 7.30 7.19 7.12 6.94 6.78
    11 .100 2.25 2.21 2.17 2.12 2.10 2.08 2.05 2.04 2.03 2.00 1.98
    .050 2.85 2.79 2.72 2.65 2.60 2.57 2.53 2.51 2.49 2.45 2.41
    .025 3.53 3.43 3.33 3.23 3.16 3.12 3.06 3.03 3.00 2.94 2.89
    .010 4.54 4.40 4.25 4.10 4.01 3.94 3.86 3.81 3.78 3.69 3.61
    .001 7.92 7.63 7.32 7.01 6.81 6.68 6.52 6.42 6.35 6.18 6.02
    12 .100 2.19 2.15 2.10 2.06 2.03 2.01 1.99 1.97 1.96 1.93 1.91
    .050 2.75 2.69 2.62 2.54 2.50 2.47 2.43 2.40 2.38 2.34 2.30
    .025 3.37 3.28 3.18 3.07 3.01 2.96 2.91 2.87 2.85 2.79 2.73
    .010 4.30 4.16 4.01 3.86 3.76 3.70 3.62 3.57 3.54 3.45 3.37
    .001 7.29 7.00 6.71 6.40 6.22 6.09 5.93 5.83 5.76 5.59 5.44
    13 .100 2.14 2.10 2.05 2.01 1.98 1.96 1.93 1.92 1.90 1.88 1.85
    .050 2.67 2.60 2.53 2.46 2.41 2.38 2.34 2.31 2.30 2.25 2.21
    .025 3.25 3.15 3.05 2.95 2.88 2.84 2.78 2.74 2.72 2.66 2.60
    .010 4.10 3.96 3.82 3.66 3.57 3.51 3.43 3.38 3.34 3.25 3.18
    .001 6.80 6.52 6.23 5.93 5.75 5.63 5.47 5.37 5.30 5.14 4.99
    14 .100 2.10 2.05 2.01 1.96 1.93 1.91 1.89 1.87 1.86 1.83 1.80
    .050 2.60 2.53 2.46 2.39 2.34 2.31 2.27 2.24 2.22 2.18 2.14
    .025 3.15 3.05 2.95 2.84 2.78 2.73 2.67 2.64 2.61 2.55 2.50
    .010 3.94 3.80 3.66 3.51 3.41 3.35 3.27 3.22 3.18 3.09 3.02
    .001 6.40 6.13 5.85 5.56 5.38 5.25 5.10 5.00 4.94 4.77 4.62
    15 .100 2.06 2.02 1.97 1.92 1.89 1.87 1.85 1.83 1.82 1.79 1.76
    .050 2.54 2.48 2.40 2.33 2.28 2.25 2.20 2.18 2.16 2.11 2.07
    .025 3.06 2.96 2.86 2.76 2.69 2.64 2.59 2.55 2.52 2.46 2.40
    .010 3.80 3.67 3.52 3.37 3.28 3.21 3.13 3.08 3.05 2.96 2.88
    .001 6.08 5.81 5.54 5.25 5.07 4.95 4.80 4.70 4.64 4.47 4.33

    Table A4 \(F\) critical values (continued)

        Degrees of freedom in the numerator
    Degrees of freedom 
    in the denominator
    \(p\) 1 2 3 4 5 6 7 8 9
    16 .100 3.05 2.67 2.46 2.33 2.24 2.18 2.13 2.09 2.06
    .050 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.54
    .025 6.12 4.69 4.08 3.73 3.50 3.34 3.22 3.12 3.05
    .010 8.53 6.23 5.29 4.77 4.44 4.20 4.03 3.89 3.78
    .001 16.12 10.97 9.01 7.94 7.27 6.80 6.46 6.19 5.98
    17 .100 3.03 2.64 2.44 2.31 2.22 2.15 2.10 2.06 2.03
    .050 4.45 3.59 3.20 2.96 2.81 2.70 2.61 2.55 2.49
    .025 6.04 4.62 4.01 3.66 3.44 3.28 3.16 3.06 2.98
    .010 8.40 6.11 5.19 4.67 4.34 4.10 3.93 3.79 3.68
    .001 15.72 10.66 8.73 7.68 7.02 6.56 6.22 5.96 5.75
    18 .100 3.01 2.62 2.42 2.29 2.20 2.13 2.08 2.04 2.00
    .050 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46
    .025 5.98 4.56 3.95 3.61 3.38 3.22 3.10 3.01 2.93
    .010 8.29 6.01 5.09 4.58 4.25 4.01 3.84 3.71 3.60
    .001 15.38 10.39 8.49 7.46 6.81 6.35 6.02 5.76 5.56
    19 .100 3.36 3.01 2.81 2.69 2.61 2.55 2.51 2.47 2.44
    .050 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18
    .025 7.21 5.71 5.08 4.72 4.48 4.32 4.20 4.10 4.03
    .010 10.56 8.02 6.99 6.42 6.06 5.80 5.61 5.47 5.35
    .001 22.86 16.39 13.90 12.56 11.71 11.13 10.70 10.37 10.11
    20 .100 2.97 2.59 2.38 2.25 2.16 2.09 2.04 2.00 1.96
    .050 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39
    .025 5.87 4.46 3.86 3.51 3.29 3.13 3.01 2.91 2.84
    .010 8.10 5.85 4.94 4.43 4.10 3.87 3.70 3.56 3.46
    .001 14.82 9.95 8.10 7.10 6.46 6.02 5.69 5.44 5.24
    21 .100 2.96 2.57 2.36 2.23 2.14 2.08 2.02 1.98 1.95
    .050 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37
    .025 5.83 4.42 3.82 3.48 3.25 3.09 2.97 2.87 2.80
    .010 8.02 5.78 4.87 4.37 4.04 3.81 3.64 3.51 3.40
    .001 14.59 9.77 7.94 6.95 6.32 5.88 5.56 5.31 5.11
    22 .100 2.95 2.56 2.35 2.22 2.13 2.06 2.01 1.97 1.93
    .050 4.30 3.44 3.05 2.82 2.66 2.55 2.46 2.40 2.34
    .025 5.79 4.38 3.78 3.44 3.22 3.05 2.93 2.84 2.76
    .010 7.95 5.72 4.82 4.31 3.99 3.76 3.59 3.45 3.35
    .001 14.38 9.61 7.80 6.81 6.19 5.76 5.44 5.19 4.99
    23 .100 2.94 2.55 2.34 2.21 2.11 2.05 1.99 1.95 1.92
    .050 4.28 3.42 3.03 2.80 2.64 2.53 2.44 2.37 2.32
    .025 5.75 4.35 3.75 3.41 3.18 3.02 2.90 2.81 2.73
    .010 7.88 5.66 4.76 4.26 3.94 3.71 3.54 3.41 3.30
    .001 14.20 9.47 7.67 6.70 6.08 5.65 5.33 5.09 4.89

    Table A5 \(F\) critical values (continued)

        Degrees of freedom in the numerator
    Degrees of freedom 
    in the denominator
    \(p\) 10 12 15 20 25 30 40 50 60 120 1000
    16 .100 2.03 1.99 1.94 1.89 1.86 1.84 1.81 1.79 1.78 1.75 1.72
    .050 2.49 2.42 2.35 2.28 2.23 2.19 2.15 2.12 2.11 2.06 2.02
    .025 2.99 2.89 2.79 2.68 2.61 2.57 2.51 2.47 2.45 2.38 2.32
    .010 3.69 3.55 3.41 3.26 3.16 3.10 3.02 2.97 2.93 2.84 2.76
    .001 5.81 5.55 5.27 4.99 4.82 4.70 4.54 4.45 4.39 4.23 4.08
    17 .100 2.00 1.96 1.91 1.86 1.83 1.81 1.78 1.76 1.75 1.72 1.69
    .050 2.45 2.38 2.31 2.23 2.18 2.15 2.10 2.08 2.06 2.01 1.97
    .025 2.92 2.82 2.72 2.62 2.55 2.50 2.44 2.41 2.38 2.32 2.26
    .010 3.59 3.46 3.31 3.16 3.07 3.00 2.92 2.87 2.83 2.75 2.66
    .001 5.58 5.32 5.05 4.78 4.60 4.48 4.33 4.24 4.18 4.02 3.87
    18 .100 1.98 1.93 1.89 1.84 1.80 1.78 1.75 1.74 1.72 1.69 1.66
    .050 2.41 2.34 2.27 2.19 2.14 2.11 2.06 2.04 2.02 1.97 1.92
    .025 2.87 2.77 2.67 2.56 2.49 2.44 2.38 2.35 2.32 2.26 2.20
    .010 3.51 3.37 3.23 3.08 2.98 2.92 2.84 2.78 2.75 2.66 2.58
    .001 5.39 5.13 4.87 4.59 4.42 4.30 4.15 4.06 4.00 3.84 3.69
    19 .100 1.96 1.91 1.86 1.81 1.78 1.76 1.73 1.71 1.70 1.67 1.64
    .050 2.38 2.31 2.23 2.16 2.11 2.07 2.03 2.00 1.98 1.93 1.88
    .025 2.82 2.72 2.62 2.51 2.44 2.39 2.33 2.30 2.27 2.20 2.14
    .010 3.43 3.30 3.15 3.00 2.91 2.84 2.76 2.71 2.67 2.58 2.50
    .001 5.22 4.97 4.70 4.43 4.26 4.14 3.99 3.90 3.84 3.68 3.53
    20 .100 1.94 1.89 1.84 1.79 1.76 1.74 1.71 1.69 1.68 1.64 1.61
    .050 2.35 2.28 2.20 2.12 2.07 2.04 1.99 1.97 1.95 1.90 1.85
    .025 2.77 2.68 2.57 2.46 2.40 2.35 2.29 2.25 2.22 2.16 2.09
    .010 3.37 3.23 3.09 2.94 2.84 2.78 2.69 2.64 2.61 2.52 2.43
    .001 5.08 4.82 4.56 4.29 4.12 4.00 3.86 3.77 3.70 3.54 3.40
    21 .100 1.92 1.87 1.83 1.78 1.74 1.72 1.69 1.67 1.66 1.62 1.59
    .050 2.32 2.25 2.18 2.10 2.05 2.01 1.96 1.94 1.92 1.87 1.82
    .025 2.73 2.64 2.53 2.42 2.36 2.31 2.25 2.21 2.18 2.11 2.05
    .010 3.31 3.17 3.03 2.88 2.79 2.72 2.64 2.58 2.55 2.46 2.37
    .001 4.95 4.70 4.44 4.17 4.00 3.88 3.74 3.64 3.58 3.42 3.28
    22 .100 1.90 1.86 1.81 1.76 1.73 1.70 1.67 1.65 1.64 1.60 1.57
    .050 2.30 2.23 2.15 2.07 2.02 1.98 1.94 1.91 1.89 1.84 1.79
    .025 2.70 2.60 2.50 2.39 2.32 2.27 2.21 2.17 2.14 2.08 2.01
    .010 3.26 3.12 2.98 2.83 2.73 2.67 2.58 2.53 2.50 2.40 2.32
    .001 4.83 4.58 4.33 4.06 3.89 3.78 3.63 3.54 3.48 3.32 3.17
    23 .100 1.89 1.84 1.80 1.74 1.71 1.69 1.66 1.64 1.62 1.59 1.55
    .050 2.27 2.20 2.13 2.05 2.00 1.96 1.91 1.88 1.86 1.81 1.76
    .025 2.67 2.57 2.47 2.36 2.29 2.24 2.18 2.14 2.11 2.04 1.98
    .010 3.21 3.07 2.93 2.78 2.69 2.62 2.54 2.48 2.45 2.35 2.27
    .001 4.73 4.48 4.23 3.96 3.79 3.68 3.53 3.44 3.38 3.22 3.08

    Table A6 \(F\) critical values (continued)

        Degrees of freedom in the numerator
    Degrees of freedom 
    in the denominator
    \(p\) 1 2 3 4 5 6 7 8 9
    24 .100 2.93 2.54 2.33 2.19 2.10 2.04 1.98 1.94 1.91
    .050 4.26 3.40 3.01 2.78 2.62 2.51 2.42 2.36 2.30
    .025 5.72 4.32 3.72 3.38 3.15 2.99 2.87 2.78 2.70
    .010 7.82 5.61 4.72 4.22 3.90 3.67 3.50 3.36 3.26
    .001 14.03 9.34 7.55 6.59 5.98 5.55 5.23 4.99 4.80
    25 .100 2.92 2.53 2.32 2.18 2.09 2.02 1.97 1.93 1.89
    .050 4.24 3.39 2.99 2.76 2.60 2.49 2.40 2.34 2.28
    .025 5.69 4.29 3.69 3.35 3.13 2.97 2.85 2.75 2.68
    .010 7.77 5.57 4.68 4.18 3.85 3.63 3.46 3.32 3.22
    .001 13.88 9.22 7.45 6.49 5.89 5.46 5.15 4.91 4.71
    26 .100 2.91 2.52 2.31 2.17 2.08 2.01 1.96 1.92 1.88
    .050 4.23 3.37 2.98 2.74 2.59 2.47 2.39 2.32 2.27
    .025 5.66 4.27 3.67 3.33 3.10 2.94 2.82 2.73 2.65
    .010 7.72 5.53 4.64 4.14 3.82 3.59 3.42 3.29 3.18
    .001 13.74 9.12 7.36 6.41 5.80 5.38 5.07 4.83 4.64
    27 .100 2.90 2.51 2.30 2.17 2.07 2.00 1.95 1.91 1.87
    .050 4.21 3.35 2.96 2.73 2.57 2.46 2.37 2.31 2.25
    .025 5.63 4.24 3.65 3.31 3.08 2.92 2.80 2.71 2.63
    .010 7.68 5.49 4.60 4.11 3.78 3.56 3.39 3.26 3.15
    .001 13.61 9.02 7.27 6.33 5.73 5.31 5.00 4.76 4.57
    28 .100 2.89 2.50 2.29 2.16 2.06 2.00 1.94 1.90 1.87
    .050 4.20 3.34 2.95 2.71 2.56 2.45 2.36 2.29 2.24
    .025 5.61 4.22 3.63 3.29 3.06 2.90 2.78 2.69 2.61
    .010 7.64 5.45 4.57 4.07 3.75 3.53 3.36 3.23 3.12
    .001 13.50 8.93 7.19 6.25 5.66 5.24 4.93 4.69 4.50
    29 .100 2.89 2.50 2.28 2.15 2.06 1.99 1.93 1.89 1.86
    .050 4.18 3.33 2.93 2.70 2.55 2.43 2.35 2.28 2.22
    .025 5.59 4.20 3.61 3.27 3.04 2.88 2.76 2.67 2.59
    .010 7.60 5.42 4.54 4.04 3.73 3.50 3.33 3.20 3.09
    .001 13.39 8.85 7.12 6.19 5.59 5.18 4.87 4.64 4.45
    30 .100 2.88 2.49 2.28 2.14 2.05 1.98 1.93 1.88 1.85
    .050 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21
    .025 5.57 4.18 3.59 3.25 3.03 2.87 2.75 2.65 2.57
    .010 7.56 5.39 4.51 4.02 3.70 3.47 3.30 3.17 3.07
    .001 13.29 8.77 7.05 6.12 5.53 5.12 4.82 4.58 4.39
    40 .100 2.84 2.44 2.23 2.09 2.00 1.93 1.87 1.83 1.79
    .050 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.12
    .025 5.42 4.05 3.46 3.13 2.90 2.74 2.62 2.53 2.45
    .010 7.31 5.18 4.31 3.83 3.51 3.29 3.12 2.99 2.89
    .001 12.61 8.25 6.59 5.70 5.13 4.73 4.44 4.21 4.02

    Table A7 \(F\) critical values (continued)

        Degrees of freedom in the numerator
    Degrees of freedom 
    in the denominator
    \(p\) 10 12 15 20 25 30 40 50 60 120 1000
    24 .100 1.88 1.83 1.78 1.73 1.70 1.67 1.64 1.62 1.61 1.57 1.54
    .050 2.25 2.18 2.11 2.03 1.97 1.94 1.89 1.86 1.84 1.79 1.74
    .025 2.64 2.54 2.44 2.33 2.26 2.21 2.15 2.11 2.08 2.01 1.94
    .010 3.17 3.03 2.89 2.74 2.64 2.58 2.49 2.44 2.40 2.31 2.22
    .001 4.64 4.39 4.14 3.87 3.71 3.59 3.45 3.36 3.29 3.14 2.99
    25 .100 1.87 1.82 1.77 1.72 1.68 1.66 1.63 1.61 1.59 1.56 1.52
    .050 2.24 2.16 2.09 2.01 1.96 1.92 1.87 1.84 1.82 1.77 1.72
    .025 2.61 2.51 2.41 2.30 2.23 2.18 2.12 2.08 2.05 1.98 1.91
    .010 3.13 2.99 2.85 2.70 2.60 2.54 2.45 2.40 2.36 2.27 2.18
    .001 4.56 4.31 4.06 3.79 3.63 3.52 3.37 3.28 3.22 3.06 2.91
    26 .100 1.86 1.81 1.76 1.71 1.67 1.65 1.61 1.59 1.58 1.54 1.51
    .050 2.22 2.15 2.07 1.99 1.94 1.90 1.85 1.82 1.80 1.75 1.70
    .025 2.59 2.49 2.39 2.28 2.21 2.16 2.09 2.05 2.03 1.95 1.89
    .010 3.09 2.96 2.81 2.66 2.57 2.50 2.42 2.36 2.33 2.23 2.14
    .001 4.48 4.24 3.99 3.72 3.56 3.44 3.30 3.21 3.15 2.99 2.84
    27 .100 1.85 1.80 1.75 1.70 1.66 1.64 1.60 1.58 1.57 1.53 1.50
    .050 2.20 2.13 2.06 1.97 1.92 1.88 1.84 1.81 1.79 1.73 1.68
    .025 2.57 2.47 2.36 2.25 2.18 2.13 2.07 2.03 2.00 1.93 1.86
    .010 3.06 2.93 2.78 2.63 2.54 2.47 2.38 2.33 2.29 2.20 2.11
    .001 4.41 4.17 3.92 3.66 3.49 3.38 3.23 3.14 3.08 2.92 2.78
    28 .100 1.84 1.79 1.74 1.69 1.65 1.63 1.59 1.57 1.56 1.52 1.48
    .050 2.19 2.12 2.04 1.96 1.91 1.87 1.82 1.79 1.77 1.71 1.66
    .025 2.55 2.45 2.34 2.23 2.16 2.11 2.05 2.01 1.98 1.91 1.84
    .010 3.03 2.90 2.75 2.60 2.51 2.44 2.35 2.30 2.26 2.17 2.08
    .001 4.35 4.11 3.86 3.60 3.43 3.32 3.18 3.09 3.02 2.86 2.72
    29 .100 1.83 1.78 1.73 1.68 1.64 1.62 1.58 1.56 1.55 1.51 1.47
    .050 2.18 2.10 2.03 1.94 1.89 1.85 1.81 1.77 1.75 1.70 1.65
    .025 2.53 2.43 2.32 2.21 2.14 2.09 2.03 1.99 1.96 1.89 1.82
    .010 3.00 2.87 2.73 2.57 2.48 2.41 2.33 2.27 2.23 2.14 2.05
    .001 4.29 4.05 3.80 3.54 3.38 3.27 3.12 3.03 2.97 2.81 2.66
    30 .100 1.82 1.77 1.72 1.67 1.63 1.61 1.57 1.55 1.54 1.50 1.46
    .050 2.16 2.09 2.01 1.93 1.88 1.84 1.79 1.76 1.74 1.68 1.63
    .025 2.51 2.41 2.31 2.20 2.12 2.07 2.01 1.97 1.94 1.87 1.80
    .010 2.98 2.84 2.70 2.55 2.45 2.39 2.30 2.25 2.21 2.11 2.02
    .001 4.24 4.00 3.75 3.49 3.33 3.22 3.07 2.98 2.92 2.76 2.61
    40 .100 1.76 1.71 1.66 1.61 1.57 1.54 1.51 1.48 1.47 1.42 1.38
    .050 2.08 2.00 1.92 1.84 1.78 1.74 1.69 1.66 1.64 1.58 1.52
    .025 2.39 2.29 2.18 2.07 1.99 1.94 1.88 1.83 1.80 1.72 1.65
    .010 2.80 2.66 2.52 2.37 2.27 2.20 2.11 2.06 2.02 1.92 1.82
    .001 3.87 3.64 3.40 3.14 2.98 2.87 2.73 2.64 2.57 2.41 2.25

    Table A8 \(F\) critical values (continued)

        Degrees of freedom in the numerator
    Degrees of freedom 
    in the denominator
    \(p\) 1 2 3 4 5 6 7 8 9
    50 .100 2.81 2.41 2.20 2.06 1.97 1.90 1.84 1.80 1.76
    .050 4.03 3.18 2.79 2.56 2.40 2.29 2.20 2.13 2.07
    .025 5.34 3.97 3.39 3.05 2.83 2.67 2.55 2.46 2.38
    .010 7.17 5.06 4.20 3.72 3.41 3.19 3.02 2.89 2.78
    .001 12.22 7.96 6.34 5.46 4.90 4.51 4.22 4.00 3.82
    60 .100 2.79 2.39 2.18 2.04 1.95 1.87 1.82 1.77 1.74
    .050 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 2.04
    .025 5.29 3.93 3.34 3.01 2.79 2.63 2.51 2.41 2.33
    .010 7.08 4.98 4.13 3.65 3.34 3.12 2.95 2.82 2.72
    .001 11.97 7.77 6.17 5.31 4.76 4.37 4.09 3.86 3.69
    100 .100 2.76 2.36 2.14 2.00 1.91 1.83 1.78 1.73 1.69
    .050 3.94 3.09 2.70 2.46 2.31 2.19 2.10 2.03 1.97
    .025 5.18 3.83 3.25 2.92 2.70 2.54 2.42 2.32 2.24
    .010 6.90 4.82 3.98 3.51 3.21 2.99 2.82 2.69 2.59
    .001 11.50 7.41 5.86 5.02 4.48 4.11 3.83 3.61 3.44
    200 .100 2.73 2.33 2.11 1.97 1.88 1.80 1.75 1.70 1.66
    .050 3.89 3.04 2.65 2.42 2.26 2.14 2.06 1.98 1.93
    .025 5.10 3.76 3.18 2.85 2.63 2.47 2.35 2.26 2.18
    .010 6.76 4.71 3.88 3.41 3.11 2.89 2.73 2.60 2.50
    .001 11.15 7.15 5.63 4.81 4.29 3.92 3.65 3.43 3.26
    1000 .100 2.71 2.31 2.09 1.95 1.85 1.78 1.72 1.68 1.64
    .050 3.85 3.00 2.61 2.38 2.22 2.11 2.02 1.95 1.89
    .025 5.04 3.70 3.13 2.80 2.58 2.42 2.30 2.20 2.13
    .010 6.66 4.63 3.80 3.34 3.04 2.82 2.66 2.53 2.43
    .001 10.89 6.96 5.46 4.65 4.14 3.78 3.51 3.30 3.13

    Table A9 \(F\) critical values (continued)

        Degrees of freedom in the numerator
    Degrees of freedom 
    in the denominator
    \(p\) 10 12 15 20 25 30 40 50 60 120 1000
    50 .100 1.73 1.68 1.63 1.57 1.53 1.50 1.46 1.44 1.42 1.38 1.33
    .050 2.03 1.95 1.87 1.78 1.73 1.69 1.63 1.60 1.58 1.51 1.45
    .025 2.32 2.22 2.11 1.99 1.92 1.87 1.80 1.75 1.72 1.64 1.56
    .010 2.70 2.56 2.42 2.27 2.17 2.10 2.01 1.95 1.91 1.80 1.70
    .001 3.67 3.44 3.20 2.95 2.79 2.68 2.53 2.44 2.38 2.21 2.05
    60 .100 1.71 1.66 1.60 1.54 1.50 1.48 1.44 1.41 1.40 1.35 1.30
    .050 1.99 1.92 1.84 1.75 1.69 1.65 1.59 1.56 1.53 1.47 1.40
    .025 2.27 2.17 2.06 1.94 1.87 1.82 1.74 1.70 1.67 1.58 1.49
    .010 2.63 2.50 2.35 2.20 2.10 2.03 1.94 1.88 1.84 1.73 1.62
    .001 3.54 3.32 3.08 2.83 2.67 2.55 2.41 2.32 2.25 2.08 1.92
    100 .100 1.66 1.61 1.56 1.49 1.45 1.42 1.38 1.35 1.34 1.28 1.22
    .050 1.93 1.85 1.77 1.68 1.62 1.57 1.52 1.48 1.45 1.38 1.30
    .025 2.18 2.08 1.97 1.85 1.77 1.71 1.64 1.59 1.56 1.46 1.36
    .010 2.50 2.37 2.22 2.07 1.97 1.89 1.80 1.74 1.69 1.57 1.45
    .001 3.30 3.07 2.84 2.59 2.43 2.32 2.17 2.08 2.01 1.83 1.64
    200 .100 1.63 1.58 1.52 1.46 1.41 1.38 1.34 1.31 1.29 1.23 1.16
    .050 1.88 1.80 1.72 1.62 1.56 1.52 1.46 1.41 1.39 1.30 1.21
    .025 2.11 2.01 1.90 1.78 1.70 1.64 1.56 1.51 1.47 1.37 1.25
    .010 2.41 2.27 2.13 1.97 1.87 1.79 1.69 1.63 1.58 1.45 1.30
    .001 3.12 2.90 2.67 2.42 2.26 2.15 2.00 1.90 1.83 1.64 1.43
    1000 .100 1.61 1.55 1.49 1.43 1.38 1.35 1.30 1.27 1.25 1.18 1.08
    .050 1.84 1.76 1.68 1.58 1.52 1.47 1.41 1.36 1.33 1.24 1.11
    .025 2.06 1.96 1.85 1.72 1.64 1.58 1.50 1.45 1.41 1.29 1.13
    .010 2.34 2.20 2.06 1.90 1.79 1.72 1.61 1.54 1.50 1.35 1.16
    .001 2.99 2.77 2.54 2.30 2.14 2.02 1.87 1.77 1.69 1.49 1.22

    Table A10 \(F\) critical values (continued)

    Numerical entries represent the probability that a standard normal random variable is between \(0\) and \(z\) where \(z=\frac{x-\mu}{\sigma}\).

    .

    Figure A2

    Standard Normal Probability Distribution: \(Z\) Table

    \(z\) 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
    0.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359
    0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753
    0.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.1141
    0.3 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517
    0.4 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0.1879
    0.5 0.1915 0.1950 0.1985 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0.2224
    0.6 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.2486 0.2517 0.2549
    0.7 0.2580 0.2611 0.2642 0.2673 0.2704 0.2734 0.2764 0.2794 0.2823 0.2852
    0.8 0.2881 0.2910 0.2939 0.2967 0.2995 0.3023 0.3051 0.3078 0.3106 0.3133
    0.9 0.3159 0.3186 0.3212 0.3238 0.3264 0.3289 0.3315 0.3340 0.3365 0.3389
    1.0 0.3413 0.3438 0.3461 0.3485 0.3508 0.3531 0.3554 0.3577 0.3599 0.3621
    1.1 0.3643 0.3665 0.3686 0.3708 0.3729 0.3749 0.3770 0.3790 0.3810 0.3830
    1.2 0.3849 0.3869 0.3888 0.3907 0.3925 0.3944 0.3962 0.3980 0.3997 0.4015
    1.3 0.4032 0.4049 0.4066 0.4082 0.4099 0.4115 0.4131 0.4147 0.4162 0.4177
    1.4 0.4192 0.4207 0.4222 0.4236 0.4251 0.4265 0.4279 0.4292 0.4306 0.4319
    1.5 0.4332 0.4345 0.4357 0.4370 0.4382 0.4394 0.4406 0.4418 0.4429 0.4441
    1.6 0.4452 0.4463 0.4474 0.4484 0.4495 0.4505 0.4515 0.4525 0.4535 0.4545
    1.7 0.4554 0.4564 0.4573 0.4582 0.4591 0.4599 0.4608 0.4616 0.4625 0.4633
    1.8 0.4641 0.4649 0.4656 0.4664 0.4671 0.4678 0.4686 0.4693 0.4699 0.4706
    1.9 0.4713 0.4719 0.4726 0.4732 0.4738 0.4744 0.4750 0.4756 0.4761 0.4767
    2.0 0.4772 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.4817
    2.1 0.4821 0.4826 0.4830 0.4834 0.4838 0.4842 0.4846 0.4850 0.4854 0.4857
    2.2 0.4861 0.4864 0.4868 0.4871 0.4875 0.4878 0.4881 0.4884 0.4887 0.4890
    2.3 0.4893 0.4896 0.4898 0.4901 0.4904 0.4906 0.4909 0.4911 0.4913 0.4916
    2.4 0.4918 0.4920 0.4922 0.4925 0.4927 0.4929 0.4931 0.4932 0.4934 0.4936
    2.5 0.4938 0.4940 0.4941 0.4943 0.4945 0.4946 0.4948 0.4949 0.4951 0.4952
    2.6 0.4953 0.4955 0.4956 0.4957 0.4959 0.4960 0.4961 0.4962 0.4963 0.4964
    2.7 0.4965 0.4966 0.4967 0.4968 0.4969 0.4970 0.4971 0.4972 0.4973 0.4974
    2.8 0.4974 0.4975 0.4976 0.4977 0.4977 0.4978 0.4979 0.4979 0.4980 0.4981
    2.9 0.4981 0.4982 0.4982 0.4983 0.4984 0.4984 0.4985 0.4985 0.4986 0.4986
    3.0 0.4987 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4989 0.4990 0.4990
    3.1 0.4990 0.4991 0.4991 0.4991 0.4992 0.4992 0.4992 0.4992 0.4993 0.4993
    3.2 0.4993 0.4993 0.4994 0.4994 0.4994 0.4994 0.4994 0.4995 0.4995 0.4995
    3.3 0.4995 0.4995 0.4995 0.4996 0.4996 0.4996 0.4996 0.4996 0.4996 0.4997
    3.4 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4998

    Table A11 Standard Normal Distribution

    Student's \(t\) Distribution

    .

    Figure A3 Upper critical values of Student's t Distribution with v Degrees of Freedom

     

    For selected probabilities, a, the table shows the values \(t_{v,a}\) such that \(P(t_v > t_{v,a}) = a\), where \(t_v\) is a Student’s \(t\) random variable with \(v\) degrees of freedom. For example, the probability is .10 that a Student’s \(t\) random variable with 10 degrees of freedom exceeds 1.372.

     
    \(v\) 0.10 0.05 0.025 0.01 0.005 0.001
    1 3.078 6.314 12.706 31.821 63.657 318.313
    2 1.886 2.920 4.303 6.965 9.925 22.327
    3 1.638 2.353 3.182 4.541 5.841 10.215
    4 1.533 2.132 2.776 3.747 4.604 7.173
    5 1.476 2.015 2.571 3.365 4.032 5.893
    6 1.440 1.943 2.447 3.143 3.707 5.208
    7 1.415 1.895 2.365 2.998 3.499 4.782
    8 1.397 1.860 2.306 2.896 3.355 4.499
    9 1.383 1.833 2.262 2.821 3.250 4.296
    10 1.372 1.812 2.228 2.764 3.169 4.143
    11 1.363 1.796 2.201 2.718 3.106 4.024
    12 1.356 1.782 2.179 2.681 3.055 3.929
    13 1.350 1.771 2.160 2.650 3.012 3.852
    14 1.345 1.761 2.145 2.624 2.977 3.787
    15 1.341 1.753 2.131 2.602 2.947 3.733
    16 1.337 1.746 2.120 2.583 2.921 3.686
    17 1.333 1.740 2.110 2.567 2.898 3.646
    18 1.330 1.734 2.101 2.552 2.878 3.610
    19 1.328 1.729 2.093 2.539 2.861 3.579
    20 1.325 1.725 2.086 2.528 2.845 3.552
    21 1.323 1.721 2.080 2.518 2.831 3.527
    22 1.321 1.717 2.074 2.508 2.819 3.505
    23 1.319 1.714 2.069 2.500 2.807 3.485
    24 1.318 1.711 2.064 2.492 2.797 3.467
    25 1.316 1.708 2.060 2.485 2.787 3.450
    26 1.315 1.706* 2.056 2.479 2.779 3.435
    27 1.314 1.703 2.052 2.473 2.771 3.421
    28 1.313 1.701 2.048 2.467 2.763 3.408
    29 1.311 1.699 2.045 2.462 2.756 3.396
    30 1.310 1.697 2.042 2.457 2.750 3.385
    40 1.303 1.684 2.021 2.423 2.704 3.307
    60 1.296 1.671 2.000 2.390 2.660 3.232
    100 1.290 1.660 1.984 2.364 2.626 3.174
    1.282 1.645 1.960 2.326 2.576 3.090

    Table A12 Probability of Exceeding the Critical Value

    .

    Figure A4

    \(\chi^2\) Probability Distribution

    df 0.995 0.990 0.975 0.950 0.900 0.100 0.050 0.025 0.010 0.005
    1 0.000 0.000 0.001 0.004 0.016 2.706 3.841 5.024 6.635 7.879
    2 0.010 0.020 0.051 0.103 0.211 4.605 5.991 7.378 9.210 10.597
    3 0.072 0.115 0.216 0.352 0.584 6.251 7.815 9.348 11.345 12.838
    4 0.207 0.297 0.484 0.711 1.064 7.779 9.488 11.143 13.277 14.860
    5 0.412 0.554 0.831 1.145 1.610 9.236 11.070 12.833 15.086 16.750
    6 0.676 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.812 18.548
    7 0.989 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.475 20.278
    8 1.344 1.646 2.180 2.733 3.490 13.362 15.507 17.535 20.090 21.955
    9 1.735 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666 23.589
    10 2.156 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209 25.188
    11 2.603 3.053 3.816 4.575 5.578 17.275 19.675 21.920 24.725 26.757
    12 3.074 3.571 4.404 5.226 6.304 18.549 21.026 23.337 26.217 28.300
    13 3.565 4.107 5.009 5.892 7.042 19.812 22.362 24.736 27.688 29.819
    14 4.075 4.660 5.629 6.571 7.790 21.064 23.685 26.119 29.141 31.319
    15 4.601 5.229 6.262 7.261 8.547 22.307 24.996 27.488 30.578 32.801
    16 5.142 5.812 6.908 7.962 9.312 23.542 26.296 28.845 32.000 34.267
    17 5.697 6.408 7.564 8.672 10.085 24.769 27.587 30.191 33.409 35.718
    18 6.265 7.015 8.231 9.390 10.865 25.989 28.869 31.526 34.805 37.156
    19 6.844 7.633 8.907 10.117 11.651 27.204 30.144 32.852 36.191 38.582
    20 7.434 8.260 9.591 10.851 12.443 28.412 31.410 34.170 37.566 39.997
    21 8.034 8.897 10.283 11.591 13.240 29.615 32.671 35.479 38.932 41.401
    22 8.643 9.542 10.982 12.338 14.041 30.813 33.924 36.781 40.289 42.796
    23 9.260 10.196 11.689 13.091 14.848 32.007 35.172 38.076 41.638 44.181
    24 9.886 10.856 12.401 13.848 15.659 33.196 36.415 39.364 42.980 45.559
    25 10.520 11.524 13.120 14.611 16.473 34.382 37.652 40.646 44.314 46.928
    26 11.160 12.198 13.844 15.379 17.292 35.563 38.885 41.923 45.642 48.290
    27 11.808 12.879 14.573 16.151 18.114 36.741 40.113 43.195 46.963 49.645
    28 12.461 13.565 15.308 16.928 18.939 37.916 41.337 44.461 48.278 50.993
    29 13.121 14.256 16.047 17.708 19.768 39.087 42.557 45.722 49.588 52.336
    30 13.787 14.953 16.791 18.493 20.599 40.256 43.773 46.979 50.892 53.672
    40 20.707 22.164 24.433 26.509 29.051 51.805 55.758 59.342 63.691 66.766
    50 27.991 29.707 32.357 34.764 37.689 63.167 67.505 71.420 76.154 79.490
    60 35.534 37.485 40.482 43.188 46.459 74.397 79.082 83.298 88.379 91.952
    70 43.275 45.442 48.758 51.739 55.329 85.527 90.531 95.023 100.425 104.215
    80 51.172 53.540 57.153 60.391 64.278 96.578 101.879 106.629 112.329 116.321
    90 59.196 61.754 65.647 69.126 73.291 107.565 113.145 118.136 124.116 128.299
    100 67.328 70.065 74.222 77.929 82.358 118.498 124.342 129.561 135.807 140.169

    Table A13 Area to the Right of the Critical Value of \(\chi^2\)