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159 Critical point. For example: x 05 leaves 5% probability in the tail. 3 x2 Critical Points d.f. | X225 | | X205 | Xi)25 | X2oio | X2005 | 001 | | 1.32 | 2.71 | 3.84 | 5.02 | 6.63 | 7.88 | 10.8 | | 2.77 | 4.61 | 5.99 | 7.38 | 9.21 | 10.6 | 13.8 | | 4.11 | 6.25 | 7.81 | 9.35 | 11.3 | 12.8 | 16.3 | | 5.39 | 7.78 | 9.49 | 11.1 | 13.3 | 14.9 | 18.5 | | 6.63 | 9.24 | 11.1 | 12.8 | 15.1 | 16.7 | 20.5 | | 7.84 | 10.6 | 12.6 | 14.4 | 16.8 | 18.5 | 22.5 | | 9.04 | 12.0 | 14.1 | 16.0 | 18.5 | 20.3 | 24.3 | | 10.2 | 13.4 | 15.5 | 17.5 | 20.1 | 22.0 | 26.1 | | 11.4 | 14.7 | 16.9 | 19.0 | 21.7 | 23.6 | 27.9 | | 12.5 | 16.0 | 18.3 | 20.5 | 23.2 | 25.2 | 29.6 | | 13.7 | 17.3 | 19.7 | 21.9 | 24.7 | 26.8 | 31.3 | | 14.8 | 18.5 | 21.0 | 23.3 | 26.2 | 28.3 | 32.9 | | 16.0 | 19.8 | 22.4 | 24.7 | 27.7 | 29.8 | 34.5 | | 17.1 | 21.1 | 23.7 | 26.1 | 29.1 | 31.3 | 36.1 | | 18.2 | 22.3 | 25.0 | 27.5 | 30.6 | 32.8 | 37.7 | | 19.4 | 23.5 | 26.3 | 28.8 | 32.0 | 34.3 | 39.3 | | 20.5 | 24.8 | 27.6 | 30.2 | 33.4 | 35.7 | 40.8 | | 21.6 | 26.0 | 28.9 | 31.5 | 34.8 | 37.2 | 42.3 | | 22.7 | 27.2 | 30.1 | 32.9 | 36.2 | 38.6 | 32.8 | | 23.8 | 28.4 | 31.4 | 34.2 | 37.6 | 40.0 | 45.3 | | 24.9 | 29.6 | 32.7 | 35.5 | 38.9 | 41.4 | 46.8 | | 26.0 | 30.8 | 33.9 | 36.8 | 40.3 | 42.8 | 48.3 | | 27.1 | 32.0 | 35.2 | 38.1 | 41.6 | 44.2 | 49.7 | | 28.2 | 33.2 | 36.4 | 39.4 | 32.0 | 45.6 | 51.2 | | 29.3 | 34.4 | 37.7 | 40.6 | 44.3 | 46.9 | 52.6 | | 30.4 | 35.6 | 38.9 | 41.9 | 45.6 | 48.3 | 54.1 | | 31.5 | 36.7 | 40.1 | 43.2 | 47.0 | 49.6 | 55.5 | | 32.6 | 37.9 | 41.3 | 44.5 | 48.3 | 51.0 | 56.9 | | 33.7 | 39.1 | 42.6 | 45.7 | 49.6 | 52.3 | 58.3 | | 34.8 | 40.3 | 43.8 | 47.0 | 50.9 | 53.7 | 59.7 | | 45.6 | 51.8 | 55.8 | 59.3 | 63.7 | 66.8 | 73.4 | | 56.3 | 63.2 | 67.5 | 71.4 | 76.2 | 79.5 | 86.7 | | 67.0 | 74.4 | 79.1 | 83.3 | 88.4 | 92.0 | 99.6 | | 77.6 | 85.5 | 90.5 | 95.0 | | | | | 88.1 | 96.6 | | | | | | | 98.6 | | | | | | | | | | | | | | |
Critical point. For example: F.05 leaves 5% probability in the tail. 4 F Critical Points DEGREES OF FREEDOM FOR NUMERATOR 10 20 40 1 F25 5.83 7.50 8.20 8.58 8.82 8.98 9.19 9.32 9.58 9.71 9.85 F,o 39.9 49.5 53.6 55.8 57.2 58.2 59.4 60.2 61.7 62.5 63.3 Fm 161 200 216 225 230 234 239 242 248 251 254 2 F,5 2.57 3.00 3.15 3.23 3.28 3.31 3.35 3.38 3.43 3.45 3.48 F10 8.53 9.00 9.16 9.24 9.29 9.33 9.37 9.39 9.44 9.47 9.49 Fos 18.5 19.0 19.2 19.2 19.3 19.3 19.4 19.4 19.4 19.5 19.5 Fm 98.5 99.0 99.2 99.2 99.3 99.3 99.4 99.4 99.4 99.5 99.5 (- -Tool Fmi 998 999 999 999 999 999 999 999 999 999 999 < 3 F,5 2.02 2.28 2.36 2.39 2.41 2.42 2.44 2.44 2.46 2.47 2.47 Fl0 5.54 5.46 5.39 5.34 5.31 5.28 5.25 5.23 5.18 5.16 5.13 2 F05 10.1 9.55 9.28 9.12 9.10 8.94 8.85 8.79 8.66 8.59 8.53 Fm 34.1 30.8 29.5 28.7 28.2 27.9 27.5 27.2 26.7 26.4 26.1 Fm 167 149 141 137 135 133 131 129 126 125 124 Q S Q u. W W <& w Q | F.25 | 1.81 | 2.00 | 2.05 | 2.06 | 2.07 | 2.08 | 2.08 | 2.08 | 2.08 | 2.08 | 2.08 | | | 4.54 | 4.32 | 4.19 | 4.11 | 4.05 | 4.01 | 3.95 | 3.92 | 3.84 | 3.80 | 3.76 | | F.05 | 7.71 | 6.94 | 6.59 | 6.39 | 6.26 | 6.16 | 6.04 | 5.96 | 5.80 | 5.72 | 5.63 | | | 21.2 | 18.0 | 16.7 | 16.0 | 15.5 | 15.2 | 14.8 | 14.5 | 14.0 | 13.7 | 13.5 | | F.oo\ | 74.1 | 61.3 | 56.2 | 53.4 | 51.7 | 50.5 | 49.0 | 48.1 | 46.1 | 45.1 | 44.1 | | F.25 | 1.69 | 1.85 | 1.88 | 1.89 | 1.89 | 1.89 | 1.89 | 1.89 | 1.88 | 1.88 | 1.87 | | | 4.06 | 3.78 | 3.62 | 3.52 | 3.45 | 3.40 | 3.34 | 3.30 | 3.21 | 3.16 | 3.10 | | F.05 | 6.61 | 5.79 | 5.41 | 5.19 | 5.05 | 4.95 | 4.82 | 4.74 | 4.56 | 4.46 | 4.36 | | F.oi | 16.3 | 13.3 | 12.1 | 11.4 | 11.0 | 10.7 | 10.3 | 10.1 | 9.55 | 9.29 | 9.02 | | F 001 | 47.2 | 37.1 | 33.2 | 31.1 | 29.8 | 28.8 | 27.6 | 26.9 | 25.4 | 24.6 | 23.8 | | F.25 | 1.62 | 1.76 | 1.78 | 1.79 | 1.79 | 1.78 | 1.77 | 1.77 | 1.76 | 1.75 | 1.74 | | | 3.78 | 3.46 | 3.29 | 3.18 | 3.11 | 3.05 | 2.98 | 2.94 | 2.84 | 2.78 | 2.72 | | F.05 | 5.99 | 5.14 | 4.76 | 4.53 | 4.39 | 4.28 | 4.15 | 4.06 | 3.87 | 3.77 | 3.67 | | | 13.7 | 10.9 | 9.78 | 9.15 | 8.75 | 8.47 | 8.10 | 7.87 | 7.40 | 7.14 | 6.88 | | | 35.5 | 27.0 | 23.7 | 21.9 | 20.8 | 20.0 | 19.0 | 18.4 | 17.1 | 16.4 | 15.8 | | F.25 | 1.57 | 1.70 | 1.72 | 1.72 | 1.71 | 1.71 | 1.70 | 1.69 | 1.67 | 1.66 | 1.65 | | F.io | 3.59 | 3.26 | 3.07 | 2.96 | 2.88 | 2.83 | 2.75 | 2.70 | 2.59 | 2.54 | 2.47 | | F.05 | 5.59 | 4.74 | 4.35 | 4.12 | 3.97 | 3.87 | 3.73 | 3.64 | 3.44 | 3.34 | 3.23 | | | 12.2 | 9.55 | 8.45 | 7.85 | 7.46 | 7.19 | 6.84 | 6.62 | 6.16 | 5.91 | 5.65 | | F.oo] | 29.3 | 21.7 | 18.8 | 17.2 | 16.2 | 15.5 | 14.6 | 14.1 | 12.9 | 12.3 | 11.7 | | | 1.54 | 1.66 | 1.69 | 1.66 | 1.66 | 1.65 | 1.64 | 1.63 | 1.61 | 1.59 | 1.58 | | | 3.46 | 3.11 | 2.92 | 2.81 | 2.73 | 2.67 | 2.59 | 2.54 | 2.42 | 2.36 | 2.29 | | F.05 | 5.32 | 4.46 | 4.07 | 3.84 | 3.69 | 3.58 | 3.44 | 3.35 | 3.15 | 3.04 | 2.93 | | | 11.3 | 8.65 | 7.59 | 7.01 | 6.63 | 6.37 | 6.03 | 5.81 | 5.36 | 5.12 | 4.86 | | F.ooi | 25.4 | 18.5 | 15.8 | 14.4 | 13.5 | 12.9 | 12.0 | 11.5 | 10.5 | 9.92 | 9.33 | | F.25 | 1.51 | 1.62 | 1.63 | 1.63 | 1.62 | 1.61 | 1.60 | 1.59 | 1.56 | 1.55 | 1.53 | | | 3.36 | 3.01 | 2.81 | 2.69 | 2.61 | 2.55 | 2.47 | 2.42 | 2.30 | 2.23 | 2.16 |
DEGREES OF FREEDOM FOR NUMERATOR 10 20 1.01 -001 10 F25 F,o F.o5 F.oi F.ooi 12 F25 F.w FM FM Fn Pi H $ F.ooi 2 16 F.25 § £> U. 01 S 20 F25 Q f10 F.05 Fm F.ooi .001 14 F.25 Q uu uu a: u! 30 F25 F J .05 Fm F.ooi W 40 F25 D f10 F.05 F.01 F.ooi 60 f,5 F,o F0> F0, Fooi 120 F,< F.m F05 F.oi F.ooi 00 F.25 F.05 F.oi F.ooi 5.12 4.26 3.86 3.63 3.48 10.6 8.02 6.99 6.42 6.06 22.9 16.4 13.9 12.6 11.7 1.49 1.60 1.60 1.59 1.59 3.28 2.92 2.73 2.61 2.52 4.96 4.10 3.71 3.48 3.33 10.0 7.56 6.55 5.99 5.64 21.0 14.9 12.6 11.3 10.5 1.56 1.56 1.56 1.55 1.54 3.18 2.81 2.61 2.48 2.39 4.75 3.89 3.49 3.26 3.11 9.33 6.93 5.95 5.41 5.06 18.6 13.0 10.8 9.63 8.89 1.44 1.53 1.53 1.52 1.51 3.10 2.73 2.52 2.39 2.31 4.60 3.74 3.34 3.11 2.96 8.86 5.51 5.56 5.04 4.69 17.1 11.8 9.73 8.62 7.92 1.42 1.51 1.51 1.50 1.48 3.05 2.67 2.46 2.33 2.24 4.49 3.63 3.24 3.01 2.85 8.53 6.23 5.29 4.77 4.44 16.1 11.0 9.00 7.94 7.27 1.40 1.49 1.48 1.46 1.45 2.97 2.59 2.38 2.25 2.16 4.35 3.49 3.10 2.87 2.71 8.10 5.85 4.94 4.43 4.10 14.8 9.95 8.10 7.10 6.46 1.38 1.45 1.44 1.42 1.41 2.88 2.49 2.28 2.14 2.05 4.17 3.32 2.92 2.69 2.53 7.56 5.39 4.51 4.02 3.70 13.3 8.77 7.05 6.12 5.53 1.36 1.44 1.42 1.40 1.39 2.84 2.44 2.23 2.09 2.00 4.08 3.23 2.84 2.61 2.45 7.31 5.18 4.31 3.83 3.51 12.6 8.25 6.60 5.70 5.13 1.35 1.42 1.41 1.38 1.37 2.79 2.39 2.18 2.04 1.95 4.00 3.15 2.76 2.53 2.37 7.08 4.98 4.13 3.65 3.34 12.0 7.76 6.17 5.31 4.76 1.34 1.40 1.39 1.37 1.35 2.75 2.35 2.13 1.99 1.90 3.92 3.07 2.68 2.45 2.29 6.85 4.79 3.95 3.48 3.17 11.4 7.32 5.79 4.95 4.42 1.32 1.39 1.37 1.35 1.33 2.71 2.30 2.08 1.94 1.85 3.84 3.00 2.60 2.37 2.21 6.63 4.61 3.78 3.32 3.02 10.8 6.91 5.42 4.62 4.10 3.37 3.23 3.14 2.94 2.83 2.71 5.80 5.47 5.26 4.81 4.57 4.31 11.1 10.4 9.89 8.90 8.37 7.81 1.58 1.56 1.55 1.52 1.51 1.48 2.46 2.38 2.32 2.20 2.13 2.06 3.22 3.07 2.98 2.77 2.66 2.54 5.39 5.06 4.85 4.41 4.17 3.91 9.92 9.20 8.75 7.80 7.30 6.76 1.53 1.51 1.50 1.47 1.45 1.42 2.33 2.24 2.19 2.06 1.99 1.90 3.00 2.85 2.75 2.54 2.43 2.30 4.82 4.50 4.30 3.86 3.62 3.36 8.38 7.71 7.29 6.40 5.93 5.42 1.50 1.48 1.46 1.43 1.41 1.38 2.24 2.15 2.10 1.96 1.89 1.80 2.85 2.70 2.60 2.39 2.27 2.13 4.46 4.14 3.94 3.51 3.27 3.00 7.43 6.80 6.40 5.56 5.10 4.60 1.48 1.46 1.45 1.40 1.37 1.34 2.18 2.09 2.03 1.89 1.81 1.72 2.74 2.59 2.49 2.28 2.15 2.01 4.20 3.89 3.69 3.26 3.02 2.75 6.81 6.19 5.81 4.99 4.54 4.06 1.44 1.42 1.40 1.36 1.33 1.29 2.09 2.00 1.94 1.79 1.71 1.61 2.60 2.45 2.35 2.12 1.99 1.84 3.87 3.56 3.37 2.94 2.69 2.42 6.02 5.44 5.08 4.29 3.86 3.38 1.39 1.37 1.35 1.30 1.27 1.23 1.98 1.88 1.82 1.67 1.57 1.46 2.42 2.27 2.16 1.93 1.79 1.62 3.47 3.17 2.98 2.55 2.30 2.01 5.12 4.58 4.24 3.49 3.07 2.59 1.37 1.35 1.33 1.28 1.24 1.19 1.93 1.83 1.76 1.61 1.51 1.38 2.34 2.18 2.08 1.84 1.69 1.51 3.29 2.99 2.80 2.37 2.11 1.80 4.73 4.21 3.87 3.15 2.73 2.23 1.35 1.32 1.30 1.25 1.21 1.15 1.87 1.77 1.71 1.54 1.44 1.29 2.25 2.10 1.99 1.75 1.59 1.39 3.12 2.82 2.63 2.20 1.94 1.60 4.37 3.87 3.54 2.83 2.41 1.89 1.33 1.30 1.28 1.22 1.18 1.10 1.82 1.72 1.65 1.48 1.37 1.19 2.17 2.02 1.91 1.66 1.50 1.25 2.96 2.66 2.47 2.03 1.76 1.38 4.04 3.55 3.24 2.53 2.11 1.54 1.31 1.28 1.25 1.19 1.14 1.00 1.77 1.67 1.60 1.42 1.30 1.00 2.10 1.94 1.83 1.57 1.39 1.00 2.80 2.51 2.32 1.88 1.59 1.00 3.74 3.27 2.96 2.27 1.84 1.00
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