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Normal Probability Table and Chi-Squared Distribution Critical Values, Schemes and Mind Maps of Statistics

Cukurova UniversityStatistics

Normal probability table (z-scores) and critical values of the chi-squared distribution. The normal probability table lists the probability of finding a z-score below a given value, while the chi-squared distribution critical values indicate the chi-square value required to achieve a specific level of significance. These tables are essential in statistical analysis, particularly in hypothesis testing and confidence interval estimation.

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Table A.3 Normal Probability Table 735
0z
Area
Table A.3 Areas under the Normal Curve
z.00 .01 .02 .03 .04 .05 .06 .07 .08 .09
−3.40.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002
−3.30.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003
−3.20.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005
−3.10.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007
−3.00.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010
−2.90.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014
−2.80.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019
−2.70.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026
−2.60.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036
−2.50.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048
−2.40.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064
−2.30.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084
−2.20.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110
−2.10.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143
−2.00.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183
−1.90.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233
−1.80.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294
−1.70.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367
−1.60.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455
−1.50.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559
−1.40.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681
−1.30.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823
−1.20.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985
−1.10.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170
−1.00.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379
−0.90.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611
−0.80.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867
−0.70.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148
−0.60.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451
−0.50.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776
−0.40.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121
−0.30.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483
−0.20.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859
−0.10.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247
−0.00.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641
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Download Normal Probability Table and Chi-Squared Distribution Critical Values and more Schemes and Mind Maps Statistics in PDF only on Docsity!

0 z

Area

736 Appendix A Statistical Tables and Proofs

738 Appendix A Statistical Tables and Proofs

Table A.4 (continued) Critical Values of the t-Distribution

α

Table A.5 Critical Values of the Chi-Squared Distribution^ α

0 f

α Table A.6 Critical Values of the F-Distribution α

Reproduced from Table 18 of Biometrika Tables for Statisticians, Vol. I, by permission of E.S.

742 Appendix A Statistical Tables and Proofs

Table A.6 (continued) Critical Values of the F-Distribution

744 Appendix A Statistical Tables and Proofs

Table A.6 (continued) Critical Values of the F-Distribution

Tolerance Factors for Normal Distributions Two-Sided Intervals One-Sided Intervals

.^05

.^01 .^05 .^01 1 −

α

α

α

α Adapted from C. Eisenhart, M. W. Hastay, and W. A. Wallis, Techniques of Statistical Analysis , Chapter 2, McGraw-

  • Table A.3 Normal Probability Table - z .00 .01 .02 .03 .04 .05 .06 .07 .08. Table A.3 Areas under the Normal Curve
    • − 3 4 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.
    • − 3 3 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.
    • − 3 2 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.
    • − 3 1 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.
    • − 3 0 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.
    • − 2 9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.
    • − 2 8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.
    • − 2 7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.
    • − 2 6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.
    • − 2 5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.
    • − 2 4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.
    • − 2 3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.
    • − 2 2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.
    • − 2 1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.
    • − 2 0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.
    • − 1 9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.
    • − 1 8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.
    • − 1 7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.
    • − 1 6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.
    • − 1 5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.
    • − 1 4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.
    • − 1 3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.
    • − 1 2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.
    • − 1 1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.
    • − 1 0 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.
    • − 0 9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.
    • − 0 8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.
    • − 0 7 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.
    • − 0 6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.
    • − 0 5 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.
    • − 0 4 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.
    • − 0 3 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.
    • − 0 2 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.
    • − 0 1 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.
    • − 0 0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.
    • z .00 .01 .02 .03 .04 .05 .06 .07 .08. Table A.3 (continued) Areas under the Normal Curve
  • 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.
  • 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.
  • 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.
  • 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.
  • 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.
  • 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.
  • 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.
  • 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.
  • 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.
  • 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.
  • 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.
  • 1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.
  • 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.
  • 1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.
  • 1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.
  • 1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.
  • 1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.
  • 1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.
  • 1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.
  • 1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.
  • 2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.
  • 2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.
  • 2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.
  • 2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.
  • 2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.
  • 2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.
  • 2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.
  • 2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.
  • 2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.
  • 2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.
  • 3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.
  • 3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.
  • 3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.
  • 3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.
  • 3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.
  • Table A.4 Student t-Distribution Probability Table - v 0.40 0.30 0.20 0.15 0.10 0.05 0. α - 1 0.325 0.727 1.376 1.963 3.078 6.314 12. - 2 0.289 0.617 1.061 1.386 1.886 2.920 4. - 3 0.277 0.584 0.978 1.250 1.638 2.353 3. - 4 0.271 0.569 0.941 1.190 1.533 2.132 2. - 5 0.267 0.559 0.920 1.156 1.476 2.015 2. - 6 0.265 0.553 0.906 1.134 1.440 1.943 2. - 7 0.263 0.549 0.896 1.119 1.415 1.895 2. - 8 0.262 0.546 0.889 1.108 1.397 1.860 2. - 9 0.261 0.543 0.883 1.100 1.383 1.833 2. - 10 0.260 0.542 0.879 1.093 1.372 1.812 2. - 11 0.260 0.540 0.876 1.088 1.363 1.796 2. - 12 0.259 0.539 0.873 1.083 1.356 1.782 2. - 13 0.259 0.538 0.870 1.079 1.350 1.771 2. - 14 0.258 0.537 0.868 1.076 1.345 1.761 2. - 15 0.258 0.536 0.866 1.074 1.341 1.753 2. - 16 0.258 0.535 0.865 1.071 1.337 1.746 2. - 17 0.257 0.534 0.863 1.069 1.333 1.740 2. - 18 0.257 0.534 0.862 1.067 1.330 1.734 2. - 19 0.257 0.533 0.861 1.066 1.328 1.729 2. - 20 0.257 0.533 0.860 1.064 1.325 1.725 2. - 21 0.257 0.532 0.859 1.063 1.323 1.721 2. - 22 0.256 0.532 0.858 1.061 1.321 1.717 2. - 23 0.256 0.532 0.858 1.060 1.319 1.714 2. - 24 0.256 0.531 0.857 1.059 1.318 1.711 2. - 25 0.256 0.531 0.856 1.058 1.316 1.708 2. - 26 0.256 0.531 0.856 1.058 1.315 1.706 2. - 27 0.256 0.531 0.855 1.057 1.314 1.703 2. - 28 0.256 0.530 0.855 1.056 1.313 1.701 2. - 29 0.256 0.530 0.854 1.055 1.311 1.699 2. - 30 0.256 0.530 0.854 1.055 1.310 1.697 2. - 40 0.255 0.529 0.851 1.050 1.303 1.684 2. - 60 0.254 0.527 0.848 1.045 1.296 1.671 2.
    • 120 0.254 0.526 0.845 1.041 1.289 1.658 1.
      • ∞ 0.253 0.524 0.842 1.036 1.282 1.645 1.
      • v 0.02 0.015 0.01 0.0075 0.005 0.0025 0. α
        • 1 15.894 21.205 31.821 42.433 63.656 127.321 636.
        • 2 4.849 5.643 6.965 8.073 9.925 14.089 31.
        • 3 3.482 3.896 4.541 5.047 5.841 7.453 12.
        • 4 2.999 3.298 3.747 4.088 4.604 5.598 8.
        • 5 2.757 3.003 3.365 3.634 4.032 4.773 6.
        • 6 2.612 2.829 3.143 3.372 3.707 4.317 5.
        • 7 2.517 2.715 2.998 3.203 3.499 4.029 5.
        • 8 2.449 2.634 2.896 3.085 3.355 3.833 5.
        • 9 2.398 2.574 2.821 2.998 3.250 3.690 4.
    • 10 2.359 2.527 2.764 2.932 3.169 3.581 4.
    • 11 2.328 2.491 2.718 2.879 3.106 3.497 4.
    • 12 2.303 2.461 2.681 2.836 3.055 3.428 4.
    • 13 2.282 2.436 2.650 2.801 3.012 3.372 4.
    • 14 2.264 2.415 2.624 2.771 2.977 3.326 4.
    • 15 2.249 2.397 2.602 2.746 2.947 3.286 4.
    • 16 2.235 2.382 2.583 2.724 2.921 3.252 4.
    • 17 2.224 2.368 2.567 2.706 2.898 3.222 3.
    • 18 2.214 2.356 2.552 2.689 2.878 3.197 3.
    • 19 2.205 2.346 2.539 2.674 2.861 3.174 3.
    • 20 2.197 2.336 2.528 2.661 2.845 3.153 3.
    • 21 2.189 2.328 2.518 2.649 2.831 3.135 3.
    • 22 2.183 2.320 2.508 2.639 2.819 3.119 3.
    • 23 2.177 2.313 2.500 2.629 2.807 3.104 3.
    • 24 2.172 2.307 2.492 2.620 2.797 3.091 3.
    • 25 2.167 2.301 2.485 2.612 2.787 3.078 3.
    • 26 2.162 2.296 2.479 2.605 2.779 3.067 3.
    • 27 2.158 2.291 2.473 2.598 2.771 3.057 3.
    • 28 2.154 2.286 2.467 2.592 2.763 3.047 3.
    • 29 2.150 2.282 2.462 2.586 2.756 3.038 3.
    • 30 2.147 2.278 2.457 2.581 2.750 3.030 3.
    • 40 2.123 2.250 2.423 2.542 2.704 2.971 3.
    • 60 2.099 2.223 2.390 2.504 2.660 2.915 3.
  • 120 2.076 2.196 2.358 2.468 2.617 2.860 3.
    • ∞ 2.054 2.170 2.326 2.432 2.576 2.807 3.
  • Table A.5 Chi-Squared Distribution Probability Table - 0 χ - v 0.995 0.99 0.98 0.975 0.95 0.90 0.80 0.75 0.70 0. α - 1 0.0^4393 0.0^3157 0.0^3628 0.0^3982 0.00393 0.0158 0.0642 0.102 0.148 0. - 2 0.0100 0.0201 0.0404 0.0506 0.103 0.211 0.446 0.575 0.713 1. - 3 0.0717 0.115 0.185 0.216 0.352 0.584 1.005 1.213 1.424 2. - 4 0.207 0.297 0.429 0.484 0.711 1.064 1.649 1.923 2.195 3. - 5 0.412 0.554 0.752 0.831 1.145 1.610 2.343 2.675 3.000 4. - 6 0.676 0.872 1.134 1.237 1.635 2.204 3.070 3.455 3.828 5. - 7 0.989 1.239 1.564 1.690 2.167 2.833 3.822 4.255 4.671 6. - 8 1.344 1.647 2.032 2.180 2.733 3.490 4.594 5.071 5.527 7. - 9 1.735 2.088 2.532 2.700 3.325 4.168 5.380 5.899 6.393 8.
    • 10 2.156 2.558 3.059 3.247 3.940 4.865 6.179 6.737 7.267 9.
    • 11 2.603 3.053 3.609 3.816 4.575 5.578 6.989 7.584 8.148 10.
    • 12 3.074 3.571 4.178 4.404 5.226 6.304 7.807 8.438 9.034 11.
    • 13 3.565 4.107 4.765 5.009 5.892 7.041 8.634 9.299 9.926 12.
    • 14 4.075 4.660 5.368 5.629 6.571 7.790 9.467 10.165 10.821 13.
    • 15 4.601 5.229 5.985 6.262 7.261 8.547 10.307 11.037 11.721 14.
    • 16 5.142 5.812 6.614 6.908 7.962 9.312 11.152 11.912 12.624 15.
    • 17 5.697 6.408 7.255 7.564 8.672 10.085 12.002 12.792 13.531 16.
    • 18 6.265 7.015 7.906 8.231 9.390 10.865 12.857 13.675 14.440 17.
    • 19 6.844 7.633 8.567 8.907 10.117 11.651 13.716 14.562 15.352 18.
    • 20 7.434 8.260 9.237 9.591 10.851 12.443 14.578 15.452 16.266 19.
    • 21 8.034 8.897 9.915 10.283 11.591 13.240 15.445 16.344 17.182 20.
    • 22 8.643 9.542 10.600 10.982 12.338 14.041 16.314 17.240 18.101 21.
    • 23 9.260 10.196 11.293 11.689 13.091 14.848 17.187 18.137 19.021 22.
    • 24 9.886 10.856 11.992 12.401 13.848 15.659 18.062 19.037 19.943 23.
    • 25 10.520 11.524 12.697 13.120 14.611 16.473 18.940 19.939 20.867 24.
    • 26 11.160 12.198 13.409 13.844 15.379 17.292 19.820 20.843 21.792 25.
    • 27 11.808 12.878 14.125 14.573 16.151 18.114 20.703 21.749 22.719 26.
    • 28 12.461 13.565 14.847 15.308 16.928 18.939 21.588 22.657 23.647 27.
    • 29 13.121 14.256 15.574 16.047 17.708 19.768 22.475 23.567 24.577 28.
    • 30 13.787 14.953 16.306 16.791 18.493 20.599 23.364 24.478 25.508 29.
    • 40 20.707 22.164 23.838 24.433 26.509 29.051 32.345 33.66 34.872 39.
    • 50 27.991 29.707 31.664 32.357 34.764 37.689 41.449 42.942 44.313 49.
    • 60 35.534 37.485 39.699 40.482 43.188 46.459 50.641 52.294 53.809 59.
    • v 0.30 0.25 0.20 0.10 0.05 0.025 0.02 0.01 0.005 0. α
      • 1 1.074 1.323 1.642 2.706 3.841 5.024 5.412 6.635 7.879 10.
      • 2 2.408 2.773 3.219 4.605 5.991 7.378 7.824 9.210 10.597 13.
      • 3 3.665 4.108 4.642 6.251 7.815 9.348 9.837 11.345 12.838 16.
      • 4 4.878 5.385 5.989 7.779 9.488 11.143 11.668 13.277 14.860 18.
      • 5 6.064 6.626 7.289 9.236 11.070 12.832 13.388 15.086 16.750 20.
      • 6 7.231 7.841 8.558 10.645 12.592 14.449 15.033 16.812 18.548 22.
      • 7 8.383 9.037 9.803 12.017 14.067 16.013 16.622 18.475 20.278 24.
      • 8 9.524 10.219 11.030 13.362 15.507 17.535 18.168 20.090 21.955 26.
      • 9 10.656 11.389 12.242 14.684 16.919 19.023 19.679 21.666 23.589 27.
  • 10 11.781 12.549 13.442 15.987 18.307 20.483 21.161 23.209 25.188 29.
  • 11 12.899 13.701 14.631 17.275 19.675 21.920 22.618 24.725 26.757 31.
  • 12 14.011 14.845 15.812 18.549 21.026 23.337 24.054 26.217 28.300 32.
  • 13 15.119 15.984 16.985 19.812 22.362 24.736 25.471 27.688 29.819 34.
  • 14 16.222 17.117 18.151 21.064 23.685 26.119 26.873 29.141 31.319 36.
  • 15 17.322 18.245 19.311 22.307 24.996 27.488 28.259 30.578 32.801 37.
  • 16 18.418 19.369 20.465 23.542 26.296 28.845 29.633 32.000 34.267 39.
  • 17 19.511 20.489 21.615 24.769 27.587 30.191 30.995 33.409 35.718 40.
  • 18 20.601 21.605 22.760 25.989 28.869 31.526 32.346 34.805 37.156 42.
  • 19 21.689 22.718 23.900 27.204 30.144 32.852 33.687 36.191 38.582 43.
  • 20 22.775 23.828 25.038 28.412 31.410 34.170 35.020 37.566 39.997 45.
  • 21 23.858 24.935 26.171 29.615 32.671 35.479 36.343 38.932 41.401 46.
  • 22 24.939 26.039 27.301 30.813 33.924 36.781 37.659 40.289 42.796 48.
  • 23 26.018 27.141 28.429 32.007 35.172 38.076 38.968 41.638 44.181 49.
  • 24 27.096 28.241 29.553 33.196 36.415 39.364 40.270 42.980 45.558 51.
  • 25 28.172 29.339 30.675 34.382 37.652 40.646 41.566 44.314 46.928 52.
  • 26 29.246 30.435 31.795 35.563 38.885 41.923 42.856 45.642 48.290 54.
  • 27 30.319 31.528 32.912 36.741 40.113 43.195 44.140 46.963 49.645 55.
  • 28 31.391 32.620 34.027 37.916 41.337 44.461 45.419 48.278 50.994 56.
  • 29 32.461 33.711 35.139 39.087 42.557 45.722 46.693 49.588 52.335 58.
  • 30 33.530 34.800 36.250 40.256 43.773 46.979 47.962 50.892 53.672 59.
  • 40 44.165 45.616 47.269 51.805 55.758 59.342 60.436 63.691 66.766 73.
  • 50 54.723 56.334 58.164 63.167 67.505 71.420 72.613 76.154 79.490 86.
  • 60 65.226 66.981 68.972 74.397 79.082 83.298 84.58 88.379 91.952 99.
  • Table A.6 F-Distribution Probability Table - v f 0 05 (v 1 , v 2 )
    • v
      • 1 161.45 199.50 215.71 224.58 230.16 233.99 236.77 238.88 240.
      • 2 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.
      • 3 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.
      • 4 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.
      • 5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.
      • 6 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.
      • 7 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.
      • 8 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.
      • 9 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.
    • 10 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.
    • 11 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.
    • 12 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.
    • 13 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.
    • 14 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.
    • 15 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.
    • 16 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.
    • 17 4.45 3.59 3.20 2.96 2.81 2.70 2.61 2.55 2.
    • 18 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.
    • 19 4.38 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.
    • 20 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.
    • 21 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.
    • 22 4.30 3.44 3.05 2.82 2.66 2.55 2.46 2.40 2.
    • 23 4.28 3.42 3.03 2.80 2.64 2.53 2.44 2.37 2.
    • 24 4.26 3.40 3.01 2.78 2.62 2.51 2.42 2.36 2.
    • 25 4.24 3.39 2.99 2.76 2.60 2.49 2.40 2.34 2.
    • 26 4.23 3.37 2.98 2.74 2.59 2.47 2.39 2.32 2.
    • 27 4.21 3.35 2.96 2.73 2.57 2.46 2.37 2.31 2.
    • 28 4.20 3.34 2.95 2.71 2.56 2.45 2.36 2.29 2.
    • 29 4.18 3.33 2.93 2.70 2.55 2.43 2.35 2.28 2.
    • 30 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.
    • 40 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.
    • 60 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 2.
  • 120 3.92 3.07 2.68 2.45 2.29 2.18 2.09 2.02 1.
    • ∞ 3.84 3.00 2.60 2.37 2.21 2.10 2.01 1.94 1. - v f 0 05 (v 1 , v 2 )
      • 1 241.88 243.91 245.95 248.01 249.05 250.10 251.14 252.20 253.25 254. v 2 10 12 15 20 24 30 40 60 120 ∞
      • 2 19.40 19.41 19.43 19.45 19.45 19.46 19.47 19.48 19.49 19.
      • 3 8.79 8.74 8.70 8.66 8.64 8.62 8.59 8.57 8.55 8.
      • 4 5.96 5.91 5.86 5.80 5.77 5.75 5.72 5.69 5.66 5.
      • 5 4.74 4.68 4.62 4.56 4.53 4.50 4.46 4.43 4.40 4.
      • 6 4.06 4.00 3.94 3.87 3.84 3.81 3.77 3.74 3.70 3.
      • 7 3.64 3.57 3.51 3.44 3.41 3.38 3.34 3.30 3.27 3.
      • 8 3.35 3.28 3.22 3.15 3.12 3.08 3.04 3.01 2.97 2.
      • 9 3.14 3.07 3.01 2.94 2.90 2.86 2.83 2.79 2.75 2.
    • 10 2.98 2.91 2.85 2.77 2.74 2.70 2.66 2.62 2.58 2.
    • 11 2.85 2.79 2.72 2.65 2.61 2.57 2.53 2.49 2.45 2.
    • 12 2.75 2.69 2.62 2.54 2.51 2.47 2.43 2.38 2.34 2.
    • 13 2.67 2.60 2.53 2.46 2.42 2.38 2.34 2.30 2.25 2.
    • 14 2.60 2.53 2.46 2.39 2.35 2.31 2.27 2.22 2.18 2.
    • 15 2.54 2.48 2.40 2.33 2.29 2.25 2.20 2.16 2.11 2.
    • 16 2.49 2.42 2.35 2.28 2.24 2.19 2.15 2.11 2.06 2.
    • 17 2.45 2.38 2.31 2.23 2.19 2.15 2.10 2.06 2.01 1.
    • 18 2.41 2.34 2.27 2.19 2.15 2.11 2.06 2.02 1.97 1.
    • 19 2.38 2.31 2.23 2.16 2.11 2.07 2.03 1.98 1.93 1.
    • 20 2.35 2.28 2.20 2.12 2.08 2.04 1.99 1.95 1.90 1.
    • 21 2.32 2.25 2.18 2.10 2.05 2.01 1.96 1.92 1.87 1.
    • 22 2.30 2.23 2.15 2.07 2.03 1.98 1.94 1.89 1.84 1.
    • 23 2.27 2.20 2.13 2.05 2.01 1.96 1.91 1.86 1.81 1.
    • 24 2.25 2.18 2.11 2.03 1.98 1.94 1.89 1.84 1.79 1.
    • 25 2.24 2.16 2.09 2.01 1.96 1.92 1.87 1.82 1.77 1.
    • 26 2.22 2.15 2.07 1.99 1.95 1.90 1.85 1.80 1.75 1.
    • 27 2.20 2.13 2.06 1.97 1.93 1.88 1.84 1.79 1.73 1.
    • 28 2.19 2.12 2.04 1.96 1.91 1.87 1.82 1.77 1.71 1.
    • 29 2.18 2.10 2.03 1.94 1.90 1.85 1.81 1.75 1.70 1.
    • 30 2.16 2.09 2.01 1.93 1.89 1.84 1.79 1.74 1.68 1.
    • 40 2.08 2.00 1.92 1.84 1.79 1.74 1.69 1.64 1.58 1.
    • 60 1.99 1.92 1.84 1.75 1.70 1.65 1.59 1.53 1.47 1.
  • 120 1.91 1.83 1.75 1.66 1.61 1.55 1.50 1.43 1.35 1.
    • ∞ 1.83 1.75 1.67 1.57 1.52 1.46 1.39 1.32 1.22 1.
  • Table A.6 F-Distribution Probability Table - v f 0 01 (v 1 , v 2 ) - v - 1 4052.18 4999.50 5403.35 5624.58 5763.65 5858.99 5928.36 5981.07 6022. - 2 98.50 99.00 99.17 99.25 99.30 99.33 99.36 99.37 99. - 3 34.12 30.82 29.46 28.71 28.24 27.91 27.67 27.49 27. - 4 21.20 18.00 16.69 15.98 15.52 15.21 14.98 14.80 14. - 5 16.26 13.27 12.06 11.39 10.97 10.67 10.46 10.29 10. - 6 13.75 10.92 9.78 9.15 8.75 8.47 8.26 8.10 7. - 7 12.25 9.55 8.45 7.85 7.46 7.19 6.99 6.84 6. - 8 11.26 8.65 7.59 7.01 6.63 6.37 6.18 6.03 5. - 9 10.56 8.02 6.99 6.42 6.06 5.80 5.61 5.47 5. - 10 10.04 7.56 6.55 5.99 5.64 5.39 5.20 5.06 4. - 11 9.65 7.21 6.22 5.67 5.32 5.07 4.89 4.74 4. - 12 9.33 6.93 5.95 5.41 5.06 4.82 4.64 4.50 4. - 13 9.07 6.70 5.74 5.21 4.86 4.62 4.44 4.30 4. - 14 8.86 6.51 5.56 5.04 4.69 4.46 4.28 4.14 4. - 15 8.68 6.36 5.42 4.89 4.56 4.32 4.14 4.00 3. - 16 8.53 6.23 5.29 4.77 4.44 4.20 4.03 3.89 3. - 17 8.40 6.11 5.18 4.67 4.34 4.10 3.93 3.79 3. - 18 8.29 6.01 5.09 4.58 4.25 4.01 3.84 3.71 3. - 19 8.18 5.93 5.01 4.50 4.17 3.94 3.77 3.63 3. - 20 8.10 5.85 4.94 4.43 4.10 3.87 3.70 3.56 3. - 21 8.02 5.78 4.87 4.37 4.04 3.81 3.64 3.51 3. - 22 7.95 5.72 4.82 4.31 3.99 3.76 3.59 3.45 3. - 23 7.88 5.66 4.76 4.26 3.94 3.71 3.54 3.41 3. - 24 7.82 5.61 4.72 4.22 3.90 3.67 3.50 3.36 3. - 25 7.77 5.57 4.68 4.18 3.85 3.63 3.46 3.32 3. - 26 7.72 5.53 4.64 4.14 3.82 3.59 3.42 3.29 3. - 27 7.68 5.49 4.60 4.11 3.78 3.56 3.39 3.26 3. - 28 7.64 5.45 4.57 4.07 3.75 3.53 3.36 3.23 3. - 29 7.60 5.42 4.54 4.04 3.73 3.50 3.33 3.20 3. - 30 7.56 5.39 4.51 4.02 3.70 3.47 3.30 3.17 3. - 40 7.31 5.18 4.31 3.83 3.51 3.29 3.12 2.99 2. - 60 7.08 4.98 4.13 3.65 3.34 3.12 2.95 2.82 2.
    • 120 6.85 4.79 3.95 3.48 3.17 2.96 2.79 2.66 2.
      • ∞ 6.63 4.61 3.78 3.32 3.02 2.80 2.64 2.51 2.
        • v f 0 01 (v 1 , v 2 )
      • 1 6055.85 6106.32 6157.28 6208.73 6234.63 6260.65 6286.78 6313.03 6339.39 6365. v 2 10 12 15 20 24 30 40 60 120 ∞
      • 2 99.40 99.42 99.43 99.45 99.46 99.47 99.47 99.48 99.49 99.
      • 3 27.23 27.05 26.87 26.69 26.60 26.50 26.41 26.32 26.22 26.
      • 4 14.55 14.37 14.20 14.02 13.93 13.84 13.75 13.65 13.56 13.
      • 5 10.05 9.89 9.72 9.55 9.47 9.38 9.29 9.20 9.11 9.
      • 6 7.87 7.72 7.56 7.40 7.31 7.23 7.14 7.06 6.97 6.
      • 7 6.62 6.47 6.31 6.16 6.07 5.99 5.91 5.82 5.74 5.
      • 8 5.81 5.67 5.52 5.36 5.28 5.20 5.12 5.03 4.95 4.
      • 9 5.26 5.11 4.96 4.81 4.73 4.65 4.57 4.48 4.40 4.
    • 10 4.85 4.71 4.56 4.41 4.33 4.25 4.17 4.08 4.00 3.
    • 11 4.54 4.40 4.25 4.10 4.02 3.94 3.86 3.78 3.69 3.
    • 12 4.30 4.16 4.01 3.86 3.78 3.70 3.62 3.54 3.45 3.
    • 13 4.10 3.96 3.82 3.66 3.59 3.51 3.43 3.34 3.25 3.
    • 14 3.94 3.80 3.66 3.51 3.43 3.35 3.27 3.18 3.09 3.
    • 15 3.80 3.67 3.52 3.37 3.29 3.21 3.13 3.05 2.96 2.
    • 16 3.69 3.55 3.41 3.26 3.18 3.10 3.02 2.93 2.84 2.
    • 17 3.59 3.46 3.31 3.16 3.08 3.00 2.92 2.83 2.75 2.
    • 18 3.51 3.37 3.23 3.08 3.00 2.92 2.84 2.75 2.66 2.
    • 19 3.43 3.30 3.15 3.00 2.92 2.84 2.76 2.67 2.58 2.
    • 20 3.37 3.23 3.09 2.94 2.86 2.78 2.69 2.61 2.52 2.
    • 21 3.31 3.17 3.03 2.88 2.80 2.72 2.64 2.55 2.46 2.
    • 22 3.26 3.12 2.98 2.83 2.75 2.67 2.58 2.50 2.40 2.
    • 23 3.21 3.07 2.93 2.78 2.70 2.62 2.54 2.45 2.35 2.
    • 24 3.17 3.03 2.89 2.74 2.66 2.58 2.49 2.40 2.31 2.
    • 25 3.13 2.99 2.85 2.70 2.62 2.54 2.45 2.36 2.27 2.
    • 26 3.09 2.96 2.81 2.66 2.58 2.50 2.42 2.33 2.23 2.
    • 27 3.06 2.93 2.78 2.63 2.55 2.47 2.38 2.29 2.20 2.
    • 28 3.03 2.90 2.75 2.60 2.52 2.44 2.35 2.26 2.17 2.
    • 29 3.00 2.87 2.73 2.57 2.49 2.41 2.33 2.23 2.14 2.
    • 30 2.98 2.84 2.70 2.55 2.47 2.39 2.30 2.21 2.11 2.
    • 40 2.80 2.66 2.52 2.37 2.29 2.20 2.11 2.02 1.92 1.
    • 60 2.63 2.50 2.35 2.20 2.12 2.03 1.94 1.84 1.73 1.
  • 120 2.47 2.34 2.19 2.03 1.95 1.86 1.76 1.66 1.53 1.
    • ∞ 2.32 2.18 2.04 1.88 1.79 1.70 1.59 1.47 1.32 1.
  • Table A.7 Tolerance Factors for Normal Distributions
    • Table A.
      • = γ
      • = γ
      • = γ
      • = γ
          1. n
                    • 3. - 4. - 5. - 4. - 5. - 7. - 2. - 3. - 4. - 3. - 4. - 6. - 3. - 3. - 4. - 4. - 4. - 6. - 2. - 3. - 4. - 3. - 4. - 5. - 2. - 3. - 4. - 3. - 4. - 5. - 2. - 3. - 4. - 3. - 3. - 5. - 2. - 3. - 4. - 3. - 4. - 5. - 2. - 2. - 3. - 3. - 3. - 5. - 2. - 3. - 4. - 3. - 4. - 5. - 2. - 2. - 3. - 2. - 3. - 4. - 2. - 3. - 4. - 3. - 3. - 5. - 2. - 2. - 3. - 2. - 3. - 4. - 2. - 3. - 4. - 3. - 3. - 4. - 2. - 2. - 3. - 2. - 3. - 4. - 2. - 3. - 3. - 3. - 3. - 4. - 2. - 2. - 3. - 2. - 1. - 4. - 2. - 2. - 3. - 2. - 3. - 4. - 2. - 2. - 3. - 2. - 3. - 4. - 2. - 2. - 3. - 2. - 3. - 4. - 2. - 2. - 3. - 2. - 3. - 4. - 2. - 2. - 3. - 2. - 3. - 4. - 2. - 2. - 3. - 2. - 2. - 4. - 2. - 2. - 3. - 2. - 3. - 4. - 1. - 2. - 3. - 2. - 2. - 3. - 2. - 2. - 3. - 2. - 3. - 4. - 1. - 2. - 3. - 2. - 2. - 3. - 2. - 2. - 3. - 2. - 3. - 4. - 1. - 2. - 3. - 2. - 2. - 1. - 2. - 2. - 3. - 2. - 2. - 3. - 1. - 2. - 3. - 2. - 2. - 3. - 2. - 2. - 3. - 2. - 2. - 3. - 1. - 2. - 3. - 2. - 2. - 3. - 2. - 2. - 3. - 2. - 2. - 3. - 1. - 2. - 2. - 1. - 2. - 3. - 2. - 2. - 3. - 2. - 2. - 3. - 1. - 2. - 2. - 1. - 2. - 3. - 2. - 2. - 3. - 2. - 2. - 3. - 1. - 2. - 2. - 1. - 2. - 3. - 1. - 2. - 3. - 2. - 2. - 3. - 1. - 2. - 2. - 1. - 2. - 3. - 1. - 2. - 3. - 2. - 2. - 3. - 1. - 2. - 2. - 1. - 2. - 3. - 1. - 2. - 3. - 2. - 2. - 3. - 1. - 1. - 2. - 1. - 2. - 2. - 1. - 2. - 2. - 2. - 2. - 3. - 1. - 1. - 2. - 1. - 2. - 2. - 1. - 2. - 2. - 1. - 2. - 3. - 1. - 1. - 2. - 1. - 2. - 2. - 1. - 2. - 2. - 1. - 2. - 3. - 1. - 1. - 2. - 1. - 2. - 2. - 1. - 2. - 2. - 1. - 2. - 2. - 1. - 1. - 2. - 1. - 1. - 2. - 1. - 2. - 2. - 1. - 2. - 2. - 1. - 1. - 2. - 1. - 1. - 2. - 1. - 2. - 2. - 1. - 2. - 2. - 1. - 1. - 2. - 1. - 1. - 2. - 1. - 2. - 2. - 1. - 2. - 2. - 1. - 1. - 2. - 1. - 1. - 2. - 1. ∞ - 1. - 2. - 1. - 1. - 2. - 1. - 1. - 2. - 1. - 1. - 2.