Statistical Tables F Distribution Table entry for p is the critical value F* with probability p lying to its right. F critical values 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 F critical values (continued) Degrees of freedom in the numerator Degrees of freedom in the denominator p 10 12 15 20 25 30 1 .100 60.19 60.71 61.22 61.74 62.05 62.26 .050 241.88 243.91 245.95 248.01 249.26 250.10 .025 968.63 976.71 984.87 993.10 998.08 1001.4 .010 6055.8 6106.3 6157.3 6208.7 6239.8 6260.6 .001 605621 610668 615764 620908 624017 626099 2 .100 9.39 9.41 9.42 9.44 9.45 9.46 .050 19.40 19.41 19.43 19.45 19.46 19.46 .025 39.40 39.41 39.43 39.45 39.46 39.46 .010 99.40 99.42 99.43 99.45 99.46 99.47 .001 999.40 999.42 999.43 999.45 999.46 999.47 3 .100 5.23 5.22 5.20 5.18 5.17 5.17 .050 8.79 8.74 8.70 8.66 8.63 8.62 .025 14.42 14.34 14.25 14.17 14.12 14.08 .010 27.23 27.05 26.87 26.69 26.58 26.50 .001 129.25 128.32 127.37 126.42 125.84 125.45 4 .100 3.92 3.90 3.87 3.84 3.83 3.82 .050 5.96 5.91 5.86 5.80 5.77 5.75 .025 8.84 8.75 8.66 8.56 8.50 8.46 .010 14.55 14.37 14.20 14.02 13.91 13.84 .001 48.05 47.41 46.76 46.10 45.70 45.43 5 .100 3.30 3.27 3.24 3.21 3.19 3.17 .050 4.74 4.68 4.62 4.56 4.52 4.50 .025 6.62 6.52 6.43 6.33 6.27 6.23 .010 10.05 9.89 9.72 9.55 9.45 9.38 .001 26.92 26.42 25.91 25.39 25.08 24.87 6 .100 2.94 2.90 2.87 2.84 2.81 2.80 .050 4.06 4.00 3.94 3.87 3.83 3.81 .025 5.46 5.37 5.27 5.17 5.11 5.07 .010 7.87 7.72 7.56 7.40 7.30 7.23 .001 18.41 17.99 17.56 17.12 16.85 16.67 7 .100 2.70 2.67 2.63 2.59 2.57 2.56 .050 3.64 3.57 3.51 3.44 3.40 3.38 .025 4.76 4.67 4.57 4.47 4.40 4.36 .010 6.62 6.47 6.31 6.16 6.06 5.99 .001 14.08 13.71 13.32 12.93 12.69 12.53 F critical values (continued) Degrees of freedom in the numerator Degrees of freedom in the denominator p 40 50 60 120 1000 1 .100 62.53 62.69 62.79 63.06 63.30 .050 251.14 251.77 252.20 253.25 254.19 .025 1005.6 1008.1 1009.8 1014.0 1017.7 .010 6286.8 6302.5 6313.0 6339.4 6362.7 .001 628712 630285 631337 633972 636301 2 .100 9.47 9.47 9.47 9.48 9.49 .050 19.47 19.48 19.48 19.49 19.49 .025 39.47 39.48 39.48 39.49 39.50 .010 99.47 99.48 99.48 99.49 99.50 .001 999.47 999.48 999.48 999.49 999.50 3 .100 5.16 5.15 5.15 5.14 5.13 .050 8.59 8.58 8.57 8.55 8.53 .025 14.04 14.01 13.99 13.95 13.91 .010 26.41 26.35 26.32 26.22 26.14 .001 124.96 124.66 124.47 123.97 123.53 4 .100 3.80 3.80 3.79 3.78 3.76 .050 5.72 5.70 5.69 5.66 5.63 .025 8.41 8.38 8.36 8.31 8.26 .010 13.75 13.69 13.65 13.56 13.47 .001 45.09 44.88 44.75 44.40 44.09 5 .100 3.16 3.15 3.14 3.12 3.11 .050 4.46 4.44 4.43 4.40 4.37 .025 6.18 6.14 6.12 6.07 6.02 .010 9.29 9.24 9.20 9.11 9.03 .001 24.60 24.44 24.33 24.06 23.82 6 .100 2.78 2.77 2.76 2.74 2.72 .050 3.77 3.75 3.74 3.70 3.67 .025 5.01 4.98 4.96 4.90 4.86 .010 7.14 7.09 7.06 6.97 6.89 .001 16.44 16.31 16.21 15.98 15.77 7 .100 2.54 2.52 2.51 2.49 2.47 .050 3.34 3.32 3.30 3.27 3.23 .025 4.31 4.28 4.25 4.20 4.15 .010 5.91 5.86 5.82 5.74 5.66 .001 12.33 12.20 12.12 11.91 11.72 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 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 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 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 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 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 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 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 Numerical entries represent the probability that a standard normal random variable is between 0 and z where z = x − μ σ . Standard Normal Probability Distribution: Z Table Standard Normal Distribution 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 Student's t Distribution 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. Probability of Exceeding the Critical Value 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 NIST/SEMATECH e-Handbook of Statistical Methods, http://www.itl.nist.gov/div898/handbook/, September 2011. χ 2 Probability Distribution Area to the Right of the Critical Value of χ 2 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 Data Sets Lap Times The following tables provide lap times from Terri Vogel's log book. Times are recorded in seconds for 2.5-mile laps completed in a series of races and practice runs. Race Lap Times (in seconds) Lap 1 Lap 2 Lap 3 Lap 4 Lap 5 Lap 6 Lap 7 Race 1 135 130 131 132 130 131 133 Race 2 134 131 131 129 128 128 129 Race 3 129 128 127 127 130 127 129 Race 4 125 125 126 125 124 125 125 Race 5 133 132 132 132 131 130 132 Race 6 130 130 130 129 129 130 129 Race 7 132 131 133 131 134 134 131 Race 8 127 128 127 130 128 126 128 Race 9 132 130 127 128 126 127 124 Race 10 135 131 131 132 130 131 130 Race 11 132 131 132 131 130 129 129 Race 12 134 130 130 130 131 130 130 Race 13 128 127 128 128 128 129 128 Race 14 132 131 131 131 132 130 130 Race 15 136 129 129 129 129 129 129 Race 16 129 129 129 128 128 129 129 Race 17 134 131 132 131 132 132 132 Race 18 129 129 130 130 133 133 127 Race 19 130 129 129 129 129 129 128 Race 20 131 128 130 128 129 130 130 Practice Lap Times (in seconds) Lap 1 Lap 2 Lap 3 Lap 4 Lap 5 Lap 6 Lap 7 Practice 1 142 143 180 137 134 134 172 Practice 2 140 135 134 133 128 128 131 Practice 3 130 133 130 128 135 133 133 Practice 4 141 136 137 136 136 136 145 Practice 5 140 138 136 137 135 134 134 Practice 6 142 142 139 138 129 129 127 Practice 7 139 137 135 135 137 134 135 Practice 8 143 136 134 133 134 133 132 Practice 9 135 134 133 133 132 132 133 Practice 10 131 130 128 129 127 128 127 Practice 11 143 139 139 138 138 137 138 Practice 12 132 133 131 129 128 127 126 Practice 13 149 144 144 139 138 138 137 Practice 14 133 132 137 133 134 130 131 Practice 15 138 136 133 133 132 131 131 Stock Prices The following table lists initial public offering (IPO) stock prices for all 1999 stocks that at least doubled in value during the first day of trading. IPO Offer Prices $17.00 $23.00 $14.00 $16.00 $12.00 $26.00 $20.00 $22.00 $14.00 $15.00 $22.00 $18.00 $18.00 $21.00 $21.00 $19.00 $15.00 $21.00 $18.00 $17.00 $15.00 $25.00 $14.00 $30.00 $16.00 $10.00 $20.00 $12.00 $16.00 $17.44 $16.00 $14.00 $15.00 $20.00 $20.00 $16.00 $17.00 $16.00 $15.00 $15.00 $19.00 $48.00 $16.00 $18.00 $9.00 $18.00 $18.00 $20.00 $8.00 $20.00 $17.00 $14.00 $11.00 $16.00 $19.00 $15.00 $21.00 $12.00 $8.00 $16.00 $13.00 $14.00 $15.00 $14.00 $13.41 $28.00 $21.00 $17.00 $28.00 $17.00 $19.00 $16.00 $17.00 $19.00 $18.00 $17.00 $15.00 $14.00 $21.00 $12.00 $18.00 $24.00 $15.00 $23.00 $14.00 $16.00 $12.00 $24.00 $20.00 $14.00 $14.00 $15.00 $14.00 $19.00 $16.00 $38.00 $20.00 $24.00 $16.00 $8.00 $18.00 $17.00 $16.00 $15.00 $7.00 $19.00 $12.00 $8.00 $23.00 $12.00 $18.00 $20.00 $21.00 $34.00 $16.00 $26.00 $14.00 References Data compiled by Jay R. Ritter of University of Florida using data from Securities Data Co. and Bloomberg .

Math


Mathematical Phrases, Symbols, and Formulas English Phrases Written Mathematically When the English says: Interpret this as: X is at least 4. X ≥ 4 The minimum of X is 4. X ≥ 4 X is no less than 4. X ≥ 4 X is greater than or equal to 4. X ≥ 4 X is at most 4. X ≤ 4 The maximum of X is 4. X ≤ 4 X is no more than 4. X ≤ 4 X is less than or equal to 4. X ≤ 4 X does not exceed 4. X ≤ 4 X is greater than 4. X > 4 X is more than 4. X > 4 X exceeds 4. X > 4 X is less than 4. X < 4 There are fewer X than 4. X < 4 X is 4. X = 4 X is equal to 4. X = 4 X is the same as 4. X = 4 X is not 4. X ≠ 4 X is not equal to 4. X ≠ 4 X is not the same as 4. X ≠ 4 X is different than 4. X ≠ 4 Symbols and Their Meanings Symbols and their Meanings Chapter (1st used) Symbol Spoken Meaning Sampling and Data The square root of same Sampling and Data π Pi 3.14159… (a specific number) Descriptive Statistics Q 1 Quartile one the first quartile Descriptive Statistics Q 2 Quartile two the second quartile Descriptive Statistics Q 3 Quartile three the third quartile Descriptive Statistics IQR interquartile range Q 3 – Q 1 = IQR Descriptive Statistics x – x-bar sample mean Descriptive Statistics μ mu population mean Descriptive Statistics s s sample standard deviation Descriptive Statistics s 2 s squared sample variance Descriptive Statistics σ sigma population standard deviation Descriptive Statistics σ 2 sigma squared population variance Descriptive Statistics Σ capital sigma sum Probability Topics { } brackets set notation Probability Topics S S sample space Probability Topics A Event A event A Probability Topics P ( A ) probability of A probability of A occurring Probability Topics P ( A | B ) probability of A given B prob. of A occurring given B has occurred Probability Topics P ( A ∪ B ) prob. of A or B prob. of A or B or both occurring Probability Topics P ( A ∩ B ) prob. of A and B prob. of both A and B occurring (same time) Probability Topics A ′ A-prime, complement of A complement of A, not A Probability Topics P ( A ') prob. of complement of A same Probability Topics G 1 green on first pick same Probability Topics P ( G 1 ) prob. of green on first pick same Discrete Random Variables PDF prob. density function same Discrete Random Variables X X the random variable X Discrete Random Variables X ~ the distribution of X same Discrete Random Variables ≥ greater than or equal to same Discrete Random Variables ≤ less than or equal to same Discrete Random Variables = equal to same Discrete Random Variables ≠ not equal to same Continuous Random Variables f ( x ) f of x function of x Continuous Random Variables pdf prob. density function same Continuous Random Variables U uniform distribution same Continuous Random Variables Exp exponential distribution same Continuous Random Variables f ( x ) = f of x equals same Continuous Random Variables m m decay rate (for exp. dist.) The Normal Distribution N normal distribution same The Normal Distribution z z -score same The Normal Distribution Z standard normal dist. same The Central Limit Theorem X – X -bar the random variable X -bar The Central Limit Theorem μ x – mean of X -bars the average of X -bars The Central Limit Theorem σ x – standard deviation of X -bars same Confidence Intervals CL confidence level same Confidence Intervals CI confidence interval same Confidence Intervals EBM error bound for a mean same Confidence Intervals EBP error bound for a proportion same Confidence Intervals t Student's t -distribution same Confidence Intervals df degrees of freedom same Confidence Intervals t α 2 student t with α /2 area in right tail same Confidence Intervals p ′ p -prime sample proportion of success Confidence Intervals q ′ q -prime sample proportion of failure Hypothesis Testing H 0 H -naught, H -sub 0 null hypothesis Hypothesis Testing H a H-a , H -sub a alternate hypothesis Hypothesis Testing H 1 H -1, H -sub 1 alternate hypothesis Hypothesis Testing α alpha probability of Type I error Hypothesis Testing β beta probability of Type II error Hypothesis Testing X 1 – – X 2 ¯ X 1-bar minus X 2-bar difference in sample means Hypothesis Testing μ 1 − μ 2 mu -1 minus mu -2 difference in population means Hypothesis Testing P ′ 1 − P ′ 2 P 1-prime minus P 2-prime difference in sample proportions Hypothesis Testing p 1 − p 2 p 1 minus p 2 difference in population proportions Chi-Square Distribution Χ 2 Ky -square Chi-square Chi-Square Distribution O Observed Observed frequency Chi-Square Distribution E Expected Expected frequency Linear Regression and Correlation y = a + bx y equals a plus b-x equation of a straight line Linear Regression and Correlation y ^ y -hat estimated value of y Linear Regression and Correlation r "r" same Linear Regression and Correlation ρ rho ("row") population correlation coefficient Linear Regression and Correlation ε error term for a regression line same Linear Regression and Correlation SSE Sum of Squared Errors same F -Distribution and ANOVA F F -ratio F -ratio Formulas Symbols you must know Population Sample N Size n μ Mean x _ σ 2 Variance s 2 σ Standard deviation s p Proportion p ′ Single data set formulae Population Sample μ = E ( x ) = 1 N ∑ i = 1 N ( x i ) Arithmetic mean x – = 1 n ∑ i = 1 n ( x i ) Geometric mean x ~ = ( ∏ i = 1 n X i ) 1 n Q 3 = 3 ( n + 1 ) 4 , Q 1 = ( n + 1 ) 4 Inter-quartile range I Q R = Q 3 − Q 1 Q 3 = 3 ( n + 1 ) 4 , Q 1 = ( n + 1 ) 4 σ 2 = 1 N ∑ i = 1 N ( x i − μ ) 2 Variance s 2 = 1 n ∑ i = 1 n ( x i − x _ ) 2 Single data set formulae Population Sample μ = E ( x ) = 1 N ∑ i = 1 N ( m i · f i ) Arithmetic mean x – = 1 n ∑ i = 1 n ( m i · f i ) Geometric mean x ~ = ( ∏ i = 1 n X i ) 1 n σ 2 = 1 N ∑ i = 1 N ( m i − μ ) 2 · f i Variance s 2 = 1 n ∑ i = 1 n ( m i − x _ ) 2 · f i C V = σ μ · 100 Coefficient of variation C V = s x _ · 100 Basic probability rules P ( A ∩ B ) = P ( A | B ) · P ( B ) Multiplication rule P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ) Addition rule P ( A ∩ B ) = P ( A ) · P ( B ) or P ( A | B ) = P ( A ) Independence test Hypergeometric distribution formulae n C x = ( n x ) = n ! x ! ( n − x ) ! Combinatorial equation P ( x ) = ( A x ) ( N − A n − x ) ( N n ) Probability equation E ( X ) = μ = n p Mean σ 2 = ( N − n N − 1 ) n p ( q ) Variance Binomial distribution formulae P ( x ) = n ! x ! ( n − x ) ! p x ( q ) n − x Probability density function E ( X ) = μ = n p Arithmetic mean σ 2 = n p ( q ) Variance Geometric distribution formulae P ( X = x ) = ( 1 − p ) x − 1 ( p ) Probability when x is the first success. Probability when x is the number of failures before first success P ( X = x ) = ( 1 − p ) x ( p ) μ = 1 p Mean Mean μ = 1 − p p σ 2 = ( 1 − p ) p 2 Variance Variance σ 2 = ( 1 − p ) p 2 Poisson distribution formulae P ( x ) = e − μ μ x x ! Probability equation E ( X ) = μ Mean σ 2 = μ Variance Uniform distribution formulae f ( x ) = 1 b − a for a ≤ x ≤ b PDF E ( X ) = μ = a + b 2 Mean σ 2 = ( b − a ) 2 12 Variance Exponential distribution formulae P ( X ≤ x ) = 1 − e − m x Cumulative probability E ( X ) = μ = 1 m or m = 1 μ Mean and decay factor σ 2 = 1 m 2 = μ 2 Variance The following page of formulae requires the use of the " Z ", " t ", " χ 2 " or " F " tables. Z = x − μ σ Z-transformation for normal distribution Z = x − n p ′ n p ′ ( q ′ ) Normal approximation to the binomial Probability (ignores subscripts) Hypothesis testing Confidence intervals [bracketed symbols equal margin of error] (subscripts denote locations on respective distribution tables) Z c = x ¯ - μ 0 σ n Interval for the population mean when sigma is known x ¯ ± [ Z ( α / 2 ) σ n ] Z c = x ¯ - μ 0 s n Interval for the population mean when sigma is unknown but n > 30 x ¯ ± [ Z ( α / 2 ) s n ] t c = x ¯ - μ 0 s n Interval for the population mean when sigma is unknown but n < 30 x ¯ ± [ t ( n − 1 ) , ( α / 2 ) s n ] Z c = p ′ - p 0 p 0 q 0 n Interval for the population proportion p ′ ± [ Z ( α / 2 ) p ′ q ′ n ] t c = d – - δ 0 s d n Interval for difference between two means with matched pairs d – ± [ t ( n − 1 ) , ( α / 2 ) s d n ] where s d is the deviation of the differences Z c = ( x 1 – - x 2 – ) − δ 0 σ 1 2 n 1 + σ 2 2 n 2 Interval for difference between two means when sigmas are known ( x 1 – - x 2 – ) ± [ Z ( α / 2 ) σ 1 2 n 1 + σ 2 2 n 2 ] t c = ( x ¯ 1 - x ¯ 2 ) - δ 0 ( ( s 1 ) 2 n 1 + ( s 2 ) 2 n 2 ) Interval for difference between two means with equal variances when sigmas are unknown ( x ¯ 1 - x ¯ 2 ) ± [ t d f , ( α / 2 ) ( ( s 1 ) 2 n 1 + ( s 2 ) 2 n 2 ) ] where d f = ( ( s 1 ) 2 n 1 + ( s 2 ) 2 n 2 ) 2 ( 1 n 1 − 1 ) ( ( s 1 ) 2 n 1 ) + ( 1 n 2 − 1 ) ( ( s 2 ) 2 n 2 ) Z c = ( p ′ 1 - p ′ 2 ) − δ 0 p ′ 1 ( q ′ 1 ) n 1 + p ′ 2 ( q ′ 2 ) n 2 Interval for difference between two population proportions ( p ′ 1 - p ′ 2 ) ± [ Z ( α / 2 ) p ′ 1 ( q ′ 1 ) n 1 + p ′ 2 ( q ′ 2 ) n 2 ] χ c 2 = ( n − 1 ) s 2 σ 0 2 Tests for GOF, Independence, and Homogeneity χ c 2 = Σ ( O − E ) 2 E where O = observed values and E = expected values F c = s 1 2 s 2 2 Where s 1 2 is the sample variance which is the larger of the two sample variances The next 3 formulae are for determining sample size with confidence intervals. (note: E represents the margin of error) n = Z ( a 2 ) 2 σ 2 E 2 Use when sigma is known E = x ¯ − μ n = Z ( a 2 ) 2 ( 0.25 ) E 2 Use when p ′ is unknown E = p ′ − p n = Z ( a 2 ) 2 [ p ′ ( q ′ ) ] E 2 Use when p ′ is unknown E = p ′ − p Simple linear regression formulae for y = a + b ( x ) r = Σ [ ( x − x ¯ ) ( y − y ¯ ) ] Σ ( x − x ¯ ) 2 * Σ ( y − y ¯ ) 2 = S x y S x S y = S S R S S T Correlation coefficient b = Σ [ ( x − x ¯ ) ( y − y ¯ ) ] Σ ( x − x ¯ ) 2 = S x y S S x = r y , x ( s y s x ) Coefficient b (slope) a = y ¯ − b ( x ¯ ) y-intercept s e 2 = Σ ( y i − y ^ i ) 2 n − k = Σ i = 1 n e i 2 n − k Estimate of the error variance S b = s e 2 ( x i − x ¯ ) 2 = s e 2 ( n − 1 ) s x 2 Standard error for coefficient b t c = b − β 0 s b Hypothesis test for coefficient β b ± [ t n − 2 , α / 2 S b ] Interval for coefficient β y ^ ± [ t α / 2 * s e ( 1 n + ( x p − x ¯ ) 2 s x ) ] Interval for expected value of y y ^ ± [ t α / 2 * s e ( 1 + 1 n + ( x p − x ¯ ) 2 s x ) ] Prediction interval for an individual y ANOVA formulae S S R = Σ i = 1 n ( y ^ i − y ¯ ) 2 Sum of squares regression S S E = Σ i = 1 n ( y ^ i − y ¯ i ) 2 Sum of squares error S S T = Σ i = 1 n ( y i − y ¯ ) 2 Sum of squares total R 2 = S S R S S T Coefficient of determination The following is the breakdown of a one-way ANOVA table for linear regression. Source of variation Sum of squares Degrees of freedom Mean squares F -ratio Regression S S R 1 or k − 1 M S R = S S R d f R F = M S R M S E Error S S E n − k M S E = S S E d f E Total S S T n − 1

Math


Preface Welcome to Introductory Statistics 2e , an OpenStax resource. This textbook was written to increase student access to high-quality learning materials, maintaining highest standards of academic rigor at little to no cost. The foundation of this textbook is Collaborative Statistics , by Barbara Illowsky and Susan Dean. Additional topics, examples, and innovations in terminology and practical applications have been added, all with a goal of increasing relevance and accessibility for students. About OpenStax OpenStax is part of Rice University, which is a 501(c)(3) nonprofit charitable corporation. Our mission is to make an amazing education accessible for all. Through our partnerships with philanthropic organizations and our alliance with other educational resource companies, we’re breaking down the most common barriers to learning. Because we believe that everyone should and can have access to knowledge. 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Visit your book page on OpenStax.org for more information. Art Attribution In Introductory Statistics 2e , most art contains attribution to its title, creator or rights holder, host platform, and license within the caption. For any art that is openly licensed, non-commercial users or organizations may reuse the art as long as they provide the same attribution to its original source. (Commercial entities should contact OpenStax to discuss reuse rights and permissions.) To maximize readability and content flow, some art does not include attribution in the text. If you reuse art from this text that does not have attribution provided, use the following attribution: Copyright Rice University, OpenStax, under CC BY-NC-SA 4.0 license. Errata All OpenStax textbooks undergo a rigorous review process. However, like any professional-grade textbook, errors sometimes occur. Since our books are web based, we can make updates periodically when deemed pedagogically necessary. If you have a correction to suggest, submit it through the link on your book page on OpenStax.org. Subject matter experts review all errata suggestions. OpenStax is committed to remaining transparent about all updates, so you will also find a list of past errata changes on your book page on OpenStax.org. Format You can access this textbook for free in web view or PDF through OpenStax.org, and in low-cost print and iBooks editions. About Introductory Statistics 2e Introductory Statistics 2e follows scope and sequence requirements of a one-semester introduction to statistics course and is geared toward students majoring in fields other than math or engineering. The text assumes some knowledge of intermediate algebra and focuses on statistics application over theory. Introductory Statistics 2e includes innovative practical applications that make the text relevant and accessible, as well as collaborative exercises, technology integration problems, and statistics labs. Coverage and scope Chapter 1 Sampling and Data Chapter 2 Descriptive Statistics Chapter 3 Probability Topics Chapter 4 Discrete Random Variables Chapter 5 Continuous Random Variables Chapter 6 The Normal Distribution Chapter 7 The Central Limit Theorem Chapter 8 Confidence Intervals Chapter 9 Hypothesis Testing with One Sample Chapter 10 Hypothesis Testing with Two Samples Chapter 11 The Chi-Square Distribution Chapter 12 Linear Regression and Correlation Chapter 13 F Distribution and One-Way ANOVA Alternate sequencing Introductory Statistics 2e was conceived and written to fit a particular topical sequence, but it can be used flexibly to accommodate other course structures. One such potential structure, which fits reasonably well with the textbook content, is provided below. Please consider, however, that the chapters were not written to be completely independent, and that the proposed alternate sequence should be carefully considered for student preparation and textual consistency. Chapter 1 Sampling and Data Chapter 2 Descriptive Statistics Chapter 12 Linear Regression and Correlation Chapter 3 Probability Topics Chapter 4 Discrete Random Variables Chapter 5 Continuous Random Variables Chapter 6 The Normal Distribution Chapter 7 The Central Limit Theorem Chapter 8 Confidence Intervals Chapter 9 Hypothesis Testing with One Sample Chapter 10 Hypothesis Testing with Two Samples Chapter 11 The Chi-Square Distribution Chapter 13 F Distribution and One-Way ANOVA Pedagogical foundation and features Examples are placed strategically throughout the text to show students the step-by-step process of interpreting and solving statistical problems. To keep the text relevant for students, the examples are drawn from a broad spectrum of practical topics, including examples about college life and learning, health and medicine, retail and business, and sports and entertainment. Try It practice problems immediately follow many examples and give students the opportunity to practice as they read the text. They are usually based on practical and familiar topics, like the Examples themselves . Collaborative Exercises provide an in-class scenario for students to work together to explore presented concepts. Using the TI-83, 83+, 84, 84+ Calculator shows students step-by-step instructions to input problems into their calculator. The Technology Icon indicates where the use of a TI calculator or computer software is recommended. Practice, Homework, and Bringing It Together problems give the students problems at various degrees of difficulty while also including real-world scenarios to engage students. Changes to the Second Edition The revision of Introductory Statistics has been undertaken with the support of hundreds of faculty adopters and student users. Most of the edits are focused on currency, relevance, accuracy, and belonging. Dozens of examples and references have been changed in order to align with contemporary contexts and uses. Data and references have been extensively updated, as well. Accuracy checking and errata resolutions resulted in corrections and clarifications, both in the main text and in the answers/solutions. Finally, the second edition includes several hundred edits related to gender, race, ethnicity, age, academic status, and societal issues, designed to present the most informed and inclusive material. OpenStax only undertakes revisions when pedagogically necessary, and we work to ensure that the text maintains its organization in order to ease transition for instructors. Many of the above changes have been made within problems and examples. However, in nearly all cases, the problems retain their original numbering, so that instructors and online homework providers can move to the new edition with minimal impact on their assignment structures. A detailed transition guide is available on the instructor resource page for this textbook. Statistics labs These innovative activities were developed by Barbara Illowsky and Susan Dean in order to offer students the experience of designing, implementing, and interpreting statistical analyses. They are drawn from actual experiments and data-gathering processes and offer a unique hands-on and collaborative experience. The labs provide a foundation for further learning and classroom interaction that will produce a meaningful application of statistics. Statistics Labs appear at the end of each chapter and begin with student learning outcomes, general estimates for time on task, and any global implementation notes. Students are then provided with step-by-step guidance, including sample data tables and calculation prompts. The detailed assistance will help the students successfully apply the concepts in the text and lay the groundwork for future collaborative or individual work. Answers to Questions in the Book Answers to Examples are provided directly under the question. Answers to Try it Questions are provided at the Student Resources page. Answers to an assortment of Practice, Homework, and Bring it Together questions are provided at the end of each chapter for students. All answers to these questions are provided for instructors in the Instructor Answer Guide via the Instructor Resources page. Due to the high variability in responses, answers to Collaborative Exercises and Labs are not provided. Additional resources Student and instructor resources We’ve compiled additional resources for both students and instructors, including Getting Started Guides, an instructor solution manual, and PowerPoint slides. Instructor resources require a verified instructor account, which you can apply for when you log in or create your account on OpenStax.org. Take advantage of these resources to supplement your OpenStax book. Community Hubs OpenStax partners with the Institute for the Study of Knowledge Management in Education (ISKME) to offer Community Hubs on OER Commons – a platform for instructors to share community-created resources that support OpenStax books, free of charge. Through our Community Hubs, instructors can upload their own materials or download resources to use in their own courses, including additional ancillaries, teaching material, multimedia, and relevant course content. We encourage instructors to join the hubs for the subjects most relevant to your teaching and research as an opportunity both to enrich your courses and to engage with other faculty. To reach the Community Hubs, visit www.oercommons.org/hubs/OpenStax . Partner resources OpenStax Partners are our allies in the mission to make high-quality learning materials affordable and accessible to students and instructors everywhere. Their tools integrate seamlessly with our OpenStax titles at a low cost. To access the partner resources for your text, visit your book page on OpenStax.org. About the authors Senior contributing authors Barbara Illowsky, De Anza College Susan Dean, De Anza College Contributing Authors Daniel Birmajer, Nazareth College Bryan Blount, Kentucky Wesleyan College Sheri Boyd, Rollins College Matthew Einsohn, Prescott College Nancy Foreman, West Los Angeles College James Helmreich, Marist College Lynette Kenyon, Collin County Community College Sheldon Lee, Viterbo University Jeff Taub, Maine Maritime Academy Reviewers Birgit Aquilonius, West Valley College Charles Ashbacher, Upper Iowa University, Cedar Rapids Abraham Biggs, Broward Community College Roberta Bloom, De Anza College Ernest Bonat, Portland Community College Sarah Boslaugh, Kennesaw State University David Bosworth, Hutchinson Community College George Bratton, University of Central Arkansas Jing Chang, College of Saint Mary Laurel Chiappetta, University of Pittsburgh Lenore Desilets, De Anza College Ann Flanigan, Kapiolani Community College David French, Tidewater Community College Mo Geraghty, De Anza College Larry Green, Lake Tahoe Community College Michael Greenwich, College of Southern Nevada Inna Grushko, De Anza College Valier Hauber, De Anza College Janice Hector, De Anza College Robert Henderson, Stephen F. Austin State University Mel Jacobsen, Snow College Mary Jo Kane, De Anza College Charles Klein, De Anza College Alexander Kolovos Sara Lenhart, Christopher Newport University Wendy Lightheart, Lane Community College Vladimir Logvenenko, De Anza College Jim Lucas, De Anza College Lisa Markus, De Anza College Miriam Masullo, SUNY Purchase Diane Mathios, De Anza College Robert McDevitt, Germanna Community College Mark Mills, Central College Cindy Moss, Skyline College Nydia Nelson, St. Petersburg College Benjamin Ngwudike, Jackson State University Jonathan Oaks, Macomb Community College Carol Olmstead, De Anza College Adam Pennell, Greensboro College Kathy Plum, De Anza College Lisa Rosenberg, Elon University Sudipta Roy, Kankakee Community College Javier Rueda, De Anza College Yvonne Sandoval, Pima Community College Rupinder Sekhon, De Anza College Travis Short, St. Petersburg College Frank Snow, De Anza College Abdulhamid Sukar, Cameron University Mary Teegarden, San Diego Mesa College John Thomas, College of Lake County Philip J. Verrecchia, York College of Pennsylvania Dennis Walsh, Middle Tennessee State University Cheryl Wartman, University of Prince Edward Island Carol Weideman, St. Petersburg College Andrew Wiesner, Pennsylvania State University

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Introduction We encounter statistics in our daily lives more often than we probably realize and from many different sources, like the news. (credit: modification of work “New office” by Phil Whitehouse/ Flickr, CC BY 2.0) Chapter Objectives By the end of this chapter, the student should be able to: Recognize and differentiate between key terms. Apply various types of sampling methods to data collection. Create and interpret frequency tables. You are probably asking yourself the question, "When and where will I use statistics?" If you read any newspaper, watch television, or use the Internet, you will see statistical information. There are statistics about crime, sports, education, politics, and real estate. Typically, when you read a newspaper article or watch a television news program, you are given sample information. With this information, you may make a decision about the correctness of a statement, claim, or "fact." Statistical methods can help you make the "best educated guess." Since you will undoubtedly be given statistical information at some point in your life, you need to know some techniques for analyzing the information thoughtfully. Think about buying a house or managing a budget. Think about your chosen profession. The fields of economics, business, psychology, education, biology, law, computer science, police science, and early childhood development require at least one course in statistics. Included in this chapter are the basic ideas and words of probability and statistics. You will soon understand that statistics and probability work together. You will also learn how data are gathered and what "good" data can be distinguished from "bad."

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Definitions of Statistics, Probability, and Key Terms The science of statistics deals with the collection, analysis, interpretation, and presentation of data . We see and use data in our everyday lives. In your classroom, try this exercise. Have class members write down the average time (in hours, to the nearest half-hour) they sleep per night. Your instructor will record the data. Then create a simple graph (called a dot plot ) of the data. A dot plot consists of a number line and dots (or points) positioned above the number line. For example, consider the following data: 5 5.5 6 6 6 6.5 6.5 6.5 6.5 7 7 8 8 9 The dot plot for this data would be as follows: Does your dot plot look the same as or different from the example? Why? If you did the same example in an English class with the same number of students, do you think the results would be the same? Why or why not? Where do your data appear to cluster? How might you interpret the clustering? The questions above ask you to analyze and interpret your data. With this example, you have begun your study of statistics. In this course, you will learn how to organize and summarize data. Organizing and summarizing data is called descriptive statistics . Two ways to summarize data are by graphing and by using numbers (for example, finding an average). After you have studied probability and probability distributions, you will use formal methods for drawing conclusions from "good" data. The formal methods are called inferential statistics . Statistical inference uses probability to determine how confident we can be that our conclusions are correct. Effective interpretation of data (inference) is based on good procedures for producing data and thoughtful examination of the data. You will encounter what will seem to be too many mathematical formulas for interpreting data. The goal of statistics is not to perform numerous calculations using the formulas, but to gain an understanding of your data. The calculations can be done using a calculator or a computer. The understanding must come from you. If you can thoroughly grasp the basics of statistics, you can be more confident in the decisions you make in life. Probability Probability is a mathematical tool used to study randomness. It deals with the chance (the likelihood) of an event occurring. For example, if you toss a fair coin four times, the outcomes may not be two heads and two tails. However, if you toss the same coin 4,000 times, the outcomes will be close to half heads and half tails. The expected theoretical probability of heads in any one toss is 1 2 or 0.5. Even though the outcomes of a few repetitions are uncertain, there is a regular pattern of outcomes when there are many repetitions. After reading about the English statistician Karl Pearson who tossed a coin 24,000 times with a result of 12,012 heads, one of the authors tossed a coin 2,000 times. The results were 996 heads. The fraction 996 2000 is equal to 0.498 which is very close to 0.5, the expected probability. The theory of probability began with the study of games of chance such as poker. Predictions take the form of probabilities. To predict the likelihood of an earthquake, of rain, or whether you will get an A in this course, we use probabilities. Doctors use probability to determine the chance of a vaccination causing the disease the vaccination is supposed to prevent. A stockbroker uses probability to determine the rate of return on a client's investments. You might use probability to decide to buy a lottery ticket or not. In your study of statistics, you will use the power of mathematics through probability calculations to analyze and interpret your data. Key Terms In statistics, we generally want to study a population . You can think of a population as a collection of persons, things, or objects under study. To study the population, we select a sample . The idea of sampling is to select a portion (or subset) of the larger population and study that portion (the sample) to gain information about the population. Data are the result of sampling from a population. Because it takes a lot of time and money to examine an entire population, sampling is a very practical technique. If you wished to compute the overall grade point average at your school, it would make sense to select a sample of students who attend the school. The data collected from the sample would be the students' grade point averages. In presidential elections, opinion poll samples of 1,000–2,000 people are taken. The opinion poll is supposed to represent the views of the people in the entire country. Manufacturers of canned carbonated drinks take samples to determine if a 16-ounce can contains 16 ounces of carbonated drink. From the sample data, we can calculate a statistic. A statistic is a number that represents a property of the sample. For example, if we consider one math class to be a sample of the population of all math classes, then the average number of points earned by students in that one math class at the end of the term is an example of a statistic. The statistic is an estimate of a population parameter. A parameter is a numerical characteristic of the whole population that can be estimated by a statistic. Since we considered all math classes to be the population, then the average number of points earned per student over all the math classes is an example of a parameter. One of the main concerns in the field of statistics is how accurately a statistic estimates a parameter. The accuracy really depends on how well the sample represents the population. The sample must contain the characteristics of the population in order to be a representative sample . We are interested in both the sample statistic and the population parameter in inferential statistics. In a later chapter, we will use the sample statistic to test the validity of the established population parameter. A variable , usually notated by capital letters such as X and Y , is a characteristic or measurement that can be determined for each member of a population. Variables may be numerical or categorical . Numerical variables take on values with equal units such as weight in pounds and time in hours. Categorical variables place the person or thing into a category. If we let X equal the number of points earned by one math student at the end of a term, then X is a numerical variable. If we let Y be a person's party affiliation, then some examples of Y include Republican, Democrat, and Independent. Y is a categorical variable. We could do some math with values of X (calculate the average number of points earned, for example), but it makes no sense to do math with values of Y (calculating an average party affiliation makes no sense). Data are the actual values of the variable. They may be numbers or they may be words. Datum is a single value. Two words that come up often in statistics are mean and proportion . If you were to take three exams in your math classes and obtain scores of 86, 75, and 92, you would calculate your mean score by adding the three exam scores and dividing by three (your mean score would be 84.3 to one decimal place). If, in your math class, there are 40 students and 22 are men and 18 are women, then the proportion of men students is 22 40 and the proportion of women students is 18 40 . Mean and proportion are discussed in more detail in later chapters. NOTE The words " mean " and " average " are often used interchangeably. The substitution of one word for the other is common practice. The technical term is "arithmetic mean," and "average" is technically a center location. However, in practice among non-statisticians, "average" is commonly accepted for "arithmetic mean." Determine what the key terms refer to in the following study. We want to know the average (mean) amount of money first year college students spend at ABC College on school supplies that do not include books. We randomly surveyed 100 first year students at the college. Three of those students spent $150, $200, and $225, respectively. The population is all first year students attending ABC College this term. The sample could be all students enrolled in one section of a beginning statistics course at ABC College (although this sample may not represent the entire population). The parameter is the average (mean) amount of money spent (excluding books) by first year college students at ABC College this term. The statistic is the average (mean) amount of money spent (excluding books) by first year college students in the sample. The variable could be the amount of money spent (excluding books) by one first year student. Let X = the amount of money spent (excluding books) by one first year student attending ABC College. The data are the dollar amounts spent by the first year students. Examples of the data are $150, $200, and $225. Try It Determine what the key terms refer to in the following study. We want to know the average (mean) amount of money spent on school uniforms each year by families with children at Knoll Academy. We randomly survey 100 families with children in the school. Three of the families spent $65, $75, and $95, respectively. Determine what the key terms refer to in the following study. A study was conducted at a local college to analyze the average cumulative GPA’s of students who graduated last year. Fill in the letter of the phrase that best describes each of the items below. 1. Population_____ 2. Statistic _____ 3. Parameter _____ 4. Sample _____ 5. Variable _____ 6. Data _____ all students who attended the college last year the cumulative GPA of one student who graduated from the college last year 3.65, 2.80, 1.50, 3.90 a group of students who graduated from the college last year, randomly selected the average cumulative GPA of students who graduated from the college last year all students who graduated from the college last year the average cumulative GPA of students in the study who graduated from the college last year 1. f 2. g 3. e 4. d 5. b 6. c Try It Determine what key terms refer to in the following study. A survey of athletes in a university was conducted to study the heights of athletes, in meters. Fill in the letter of the phrase that best describes each of the items below. Population ____ Statistics ____ Parameter ____ Sample ____ Variable ____ Data _____ the average height of athletes in the university the average height of athletes in the survey all athletes in the university all students in the university the height of one athlete a group of athletes randomly selected 1.82, 1.76, 1.69, 1.93 c b a f e g Determine what the key terms refer to in the following study. As part of a study designed to test the safety of electric automobiles, the National Transportation Safety Board collected and reviewed data about the effects of a crash on test dummies. Here is the criterion they used: Speed at which Cars Crashed Location of “drivers” (i.e. dummies) 35 miles/hour Front Seat Cars with dummies in the front seats were crashed into a wall at a speed of 35 miles per hour. We want to know the proportion of dummies in the driver’s seat that would have had head injuries, if they had been actual drivers. We start with a simple random sample of 75 cars. The population is all cars containing dummies in the front seat. The sample is the 75 cars, selected by a simple random sample. The parameter is the proportion of driver dummies (if they had been real people) who would have suffered head injuries in the population. The statistic is proportion of driver dummies (if they had been real people) who would have suffered head injuries in the sample. The variable X = whether a dummy (if it had been a real person) who would have suffered head injuries. The data are either: yes, had head injury, or no, did not. Try It Determine what the key terms refer to in the following study. A survey is conducted to check the time taken by a mobile for charging of battery from 50% to 100%. The criteria used to collect the data are: Wattage of charger used Type of mobile used 30 W Android We want to know the proportion of Android mobiles that are charged to 100% within 30 minutes. We start with a simple random sample of 200 mobiles. The population is all Android mobiles. The sample is the 200 mobiles, selected by a simple random sample. The parameter is the proportion of Android mobiles that are charged to 100% within 30 minutes in the population. The statistic is the proportion of Android mobiles that are charged to 100% within 30 minutes in the survey. The variable X = the number of Android mobiles that are charged to 100% within 30 minutes. The data are either: yes, charged to 100% within 30 minutes; or no, charged to 100% taking more than 30 minutes. Determine what the key terms refer to in the following study. An insurance company would like to determine the proportion of all medical doctors who have been involved in one or more malpractice lawsuits. The company selects 500 doctors at random from a professional directory and determines the number in the sample who have been involved in a malpractice lawsuit. The population is all medical doctors listed in the professional directory. The parameter is the proportion of medical doctors who have been involved in one or more malpractice suits in the population. The sample is the 500 doctors selected at random from the professional directory. The statistic is the proportion of medical doctors who have been involved in one or more malpractice suits in the sample. The variable X = whether an individual doctor has been involved in a malpractice suit. The data are either: yes, was involved in one or more malpractice lawsuits, or no, was not. Try It Determine what the key terms refer to in the following study. A study is to be conducted by a news agency to find the proportion of all truck drivers that have no points on their license. The agency selects 1000 truck drivers randomly from the directory of truck drivers and determines the number of truck drivers in the sample who have no points on their license. The population is all truck drivers present in the directory. The parameter is the proportion of truck drivers that have no points on their license in the population. The sample is the 1000 truck drivers selected randomly from the directory. The statistics is the proportion of truck drivers that have no points on their license in the sample. The variable X = the number of truck drivers that have no points on their license. The data are either: yes, they have no points on their license; or no, they have points on their license. Do the following exercise collaboratively with up to four people per group. Find a population, a sample, the parameter, the statistic, a variable, and data for the following study: You want to determine the average (mean) number of glasses of milk college students drink per day. Suppose yesterday, in your English class, you asked five students how many glasses of milk they drank the day before. The answers were 1, 0, 1, 3, and 4 glasses of milk. References The Data and Story Library, http://lib.stat.cmu.edu/DASL/Stories/CrashTestDummies.html (accessed May 1, 2013). Chapter Review The mathematical theory of statistics is easier to learn when you know the language. This module presents important terms that will be used throughout the text. Use the following information to answer the next five exercises . Studies are often done by pharmaceutical companies to determine the effectiveness of a treatment program. Suppose that a new drug is currently under study to address a respiratory virus. It is given to patients once the patient exhibits symptoms of the virus. Of interest is the average (mean) length of time in days from the time the patient starts the treatment until the symptoms are alleviated. Two researchers each follow a different set of 40 patients with the respiratory virus from the start of treatment until the symptoms are alleviated. The following data (in days) are collected. Researcher A: 3 4 11 15 16 17 22 44 37 16 14 24 25 15 26 27 33 29 35 44 13 21 22 10 12 8 40 32 26 27 31 34 29 17 8 24 18 47 33 34 Researcher B: 3 14 11 5 16 17 28 41 31 18 14 14 26 25 21 22 31 2 35 44 23 21 21 16 12 18 41 22 16 25 33 34 29 13 18 24 23 42 33 29 Determine what the key terms refer to in the example for Researcher A. population Patients with the virus sample parameter The average length of time (in months) patients with the virus live after treatment. statistic variable X = the length of time (in months) patients with the virus live after treatment HOMEWORK For each of the following eight exercises, identify: a. the population, b. the sample, c. the parameter, d. the statistic, e. the variable, and f. the data. Give examples where appropriate. A fitness center is interested in the mean amount of time a client exercises in the center each week. Ski resorts are interested in the mean age that children take their first ski and snowboard lessons. They need this information to plan their ski classes optimally. all children who take ski or snowboard lessons a group of these children the population mean age of children who take their first snowboard lesson the sample mean age of children who take their first snowboard lesson X = the age of one child who takes their first ski or snowboard lesson values for X , such as 3, 7, and so on A cardiologist is interested in the mean recovery period of their patients who have had heart attacks. Insurance companies are interested in the mean health costs each year of their clients, so that they can determine the costs of health insurance. the clients of the insurance companies a group of the clients the mean health costs of the clients the mean health costs of the sample X = the health costs of one client values for X , such as 34, 9, 82, and so on A politician is interested in the proportion of voters in their district who think the politician doing a good job. A marriage counselor is interested in the proportion of clients they counsel who stay married. all the clients of this counselor a group of clients of this marriage counselor the proportion of all her clients who stay married the proportion of the sample of the counselor’s clients who stay married X = whether a client stayed married yes, no Political pollsters may be interested in the proportion of people who will vote for a particular cause. A marketing company is interested in the proportion of people who will buy a particular product. all people (maybe in a certain geographic area, such as the United States) a group of the people the proportion of all people who will buy the product the proportion of the sample who will buy the product X = whether a person bought the product buy, not buy Use the following information to answer the next three exercises: A Lake Tahoe Community College instructor is interested in the mean number of days Lake Tahoe Community College math students are absent from class during a quarter. What is the population she is interested in? all Lake Tahoe Community College students all Lake Tahoe Community College English students all Lake Tahoe Community College students in her classes all Lake Tahoe Community College math students Consider the following: X = number of days a Lake Tahoe Community College math student is absent In this case, X is an example of a: variable. population. statistic. data. a The instructor’s sample produces a mean number of days absent of 3.5 days. This value is an example of a: parameter. data. statistic. variable. Average also called mean; a number that describes the central tendency of the data Categorical Variable variables that take on values that are names or labels Data a set of observations (a set of possible outcomes); most data can be put into two groups: qualitative (an attribute whose value is indicated by a label) or quantitative (an attribute whose value is indicated by a number). Quantitative data can be separated into two subgroups: discrete and continuous . Data is discrete if it is the result of counting (such as the number of students of a given ethnic group in a class or the number of books on a shelf). Data is continuous if it is the result of measuring (such as distance traveled or weight of luggage) Numerical Variable variables that take on values that are indicated by numbers Parameter a number that is used to represent a population characteristic and that generally cannot be determined easily Population all individuals, objects, or measurements whose properties are being studied Probability a number between zero and one, inclusive, that gives the likelihood that a specific event will occur Proportion the number of successes divided by the total number in the sample Representative Sample a subset of the population that has the same characteristics as the population Sample a subset of the population studied Statistic a numerical characteristic of the sample; a statistic estimates the corresponding population parameter. Variable a characteristic of interest for each person or object in a population

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