2 Contents Contents
Contents
1 Binomial distribution 3
Cumulative distribution function forθ≤0.09 . . . 3
Cumulative distribution function for0.1≤θ≤0.5 . . . 5
2 Poisson distribution 10 Cumulative distribution function . . . 10
3 Normal distribution 14 Cumulative distribution functionΦ(z) . . . 14
InverseΦ−1(p)of the cumulative distribution function (quantiles) . . . . 15
Selected quantilesΦ−1 (p)in high precision . . . 16
4 Student’s t distribution 17 InverseF−1 ν (p)of the cumulative distribution function (quantiles) . . . 17
5
χ
2distribution 19 InverseF−1 ν (p)of the cumulative distribution function (quantiles) . . . 196 F distribution 21 InverseF−1 (ν1,ν2)(p)of the cumulative distribution function (quantiles) . . . 21
7 Non-parametric tests 34 Rejection regions of the Wilcoxon signed-rank test (one sample) . . . 34
Rejection regions of the Mann-Whitney/Wilcoxon two-sample test . . . 34
Rejection regions of the Mann-Whitney/Wilcoxon two-sample test (sizeα: 5% one-sided, 10% two-sided) . . . 35
Rejection regions of the Mann-Whitney/Wilcoxon two-sample test (sizeα: 2.5% one-sided, 5% two-sided) . . . 35
Rejection regions of the Mann-Whitney/Wilcoxon two-sample test (sizeα: 1% one-sided, 2% two-sided) . . . 36
Rejection regions of the Mann-Whitney/Wilcoxon two-sample test (sizeα: 0.5% one-sided, 1% two-sided) . . . 36
8 Correlation 37 Critical values for the linear correlation coefficient . . . 37
9 Fisher’s z-transform 38 Fisher’s z-transform . . . 38
1. Binomial distribution 3
1
Binomial distribution
Cumulative distribution function for
θ
≤
0.09
This section tabulates the cumulative distribution function (c.d.f.) of the binomial distribution, i.e. the distribution of the number of successes innindependent trials of an experiment which leads to a success with probabilityθ. The c.d.f. is
F(x) =P{X ≤x}=
x X
k=0
P{X =k}=
x X
k=0
n k
θk(1−θ)n−k .
The tables only coverθ ≤ 1
2. Forθ > 1
2, the rˆoles of successes and failures need to be reversed, i.e. ifX ∼ Bi(n, θ), and
Y ∼Bi(n,1−θ), thenP{X ≤x}= 1−P{Y ≤n−x−1}.
θ
n x 0.001 0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
1 0 0.9990 0.9950 0.9900 0.9800 0.9700 0.9600 0.9500 0.9400 0.9300 0.9200 0.9100 1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
2 0 0.9980 0.9900 0.9801 0.9604 0.9409 0.9216 0.9025 0.8836 0.8649 0.8464 0.8281 1 1.0000 1.0000 0.9999 0.9996 0.9991 0.9984 0.9975 0.9964 0.9951 0.9936 0.9919 2 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
3 0 0.9970 0.9851 0.9703 0.9412 0.9127 0.8847 0.8574 0.8306 0.8044 0.7787 0.7536 1 1.0000 0.9999 0.9997 0.9988 0.9974 0.9953 0.9928 0.9896 0.9860 0.9818 0.9772 2 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9998 0.9997 0.9995 0.9993 3 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
4 0 0.9960 0.9801 0.9606 0.9224 0.8853 0.8493 0.8145 0.7807 0.7481 0.7164 0.6857 1 1.0000 0.9999 0.9994 0.9977 0.9948 0.9909 0.9860 0.9801 0.9733 0.9656 0.9570 2 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9995 0.9992 0.9987 0.9981 0.9973 3 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 4 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
5 0 0.9950 0.9752 0.9510 0.9039 0.8587 0.8154 0.7738 0.7339 0.6957 0.6591 0.6240 1 1.0000 0.9998 0.9990 0.9962 0.9915 0.9852 0.9774 0.9681 0.9575 0.9456 0.9326 2 1.0000 1.0000 1.0000 0.9999 0.9997 0.9994 0.9988 0.9980 0.9969 0.9955 0.9937 3 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9998 0.9997 4 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
6 0 0.9940 0.9704 0.9415 0.8858 0.8330 0.7828 0.7351 0.6899 0.6470 0.6064 0.5679 1 1.0000 0.9996 0.9985 0.9943 0.9875 0.9784 0.9672 0.9541 0.9392 0.9227 0.9048 2 1.0000 1.0000 1.0000 0.9998 0.9995 0.9988 0.9978 0.9962 0.9942 0.9915 0.9882 3 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9997 0.9995 0.9992 4 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
7 0 0.9930 0.9655 0.9321 0.8681 0.8080 0.7514 0.6983 0.6485 0.6017 0.5578 0.5168 1 1.0000 0.9995 0.9980 0.9921 0.9829 0.9706 0.9556 0.9382 0.9187 0.8974 0.8745 2 1.0000 1.0000 1.0000 0.9997 0.9991 0.9980 0.9962 0.9937 0.9903 0.9860 0.9807 3 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9996 0.9993 0.9988 0.9982 4 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 5 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
8 0 0.9920 0.9607 0.9227 0.8508 0.7837 0.7214 0.6634 0.6096 0.5596 0.5132 0.4703 1 1.0000 0.9993 0.9973 0.9897 0.9777 0.9619 0.9428 0.9208 0.8965 0.8702 0.8423 2 1.0000 1.0000 0.9999 0.9996 0.9987 0.9969 0.9942 0.9904 0.9853 0.9789 0.9711 3 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9996 0.9993 0.9987 0.9978 0.9966 4 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9997 5 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
9 0 0.9910 0.9559 0.9135 0.8337 0.7602 0.6925 0.6302 0.5730 0.5204 0.4722 0.4279 1 1.0000 0.9991 0.9966 0.9869 0.9718 0.9522 0.9288 0.9022 0.8729 0.8417 0.8088 2 1.0000 1.0000 0.9999 0.9994 0.9980 0.9955 0.9916 0.9862 0.9791 0.9702 0.9595 3 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9994 0.9987 0.9977 0.9963 0.9943 4 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9997 0.9995 5 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
4 Cumulative distribution function forθ≤0.09 1. Binomial distribution
n x 0.001 0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
10 2 1.0000 1.0000 0.9999 0.9991 0.9972 0.9938 0.9885 0.9812 0.9717 0.9599 0.9460 3 1.0000 1.0000 1.0000 1.0000 0.9999 0.9996 0.9990 0.9980 0.9964 0.9942 0.9912 4 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9997 0.9994 0.9990 5 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 6 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
11 0 0.9891 0.9464 0.8953 0.8007 0.7153 0.6382 0.5688 0.5063 0.4501 0.3996 0.3544 1 0.9999 0.9987 0.9948 0.9805 0.9587 0.9308 0.8981 0.8618 0.8228 0.7819 0.7399 2 1.0000 1.0000 0.9998 0.9988 0.9963 0.9917 0.9848 0.9752 0.9630 0.9481 0.9305 3 1.0000 1.0000 1.0000 1.0000 0.9998 0.9993 0.9984 0.9970 0.9947 0.9915 0.9871 4 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9995 0.9990 0.9983 5 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 6 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
12 0 0.9881 0.9416 0.8864 0.7847 0.6938 0.6127 0.5404 0.4759 0.4186 0.3677 0.3225 1 0.9999 0.9984 0.9938 0.9769 0.9514 0.9191 0.8816 0.8405 0.7967 0.7513 0.7052 2 1.0000 1.0000 0.9998 0.9985 0.9952 0.9893 0.9804 0.9684 0.9532 0.9348 0.9134 3 1.0000 1.0000 1.0000 0.9999 0.9997 0.9990 0.9978 0.9957 0.9925 0.9880 0.9820 4 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9996 0.9991 0.9984 0.9973 5 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9997 6 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
13 0 0.9871 0.9369 0.8775 0.7690 0.6730 0.5882 0.5133 0.4474 0.3893 0.3383 0.2935 1 0.9999 0.9981 0.9928 0.9730 0.9436 0.9068 0.8646 0.8186 0.7702 0.7206 0.6707 2 1.0000 1.0000 0.9997 0.9980 0.9938 0.9865 0.9755 0.9608 0.9422 0.9201 0.8946 3 1.0000 1.0000 1.0000 0.9999 0.9995 0.9986 0.9969 0.9940 0.9897 0.9837 0.9758 4 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9993 0.9987 0.9976 0.9959 5 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9997 0.9995 6 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 7 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
14 0 0.9861 0.9322 0.8687 0.7536 0.6528 0.5647 0.4877 0.4205 0.3620 0.3112 0.2670 1 0.9999 0.9978 0.9916 0.9690 0.9355 0.8941 0.8470 0.7963 0.7436 0.6900 0.6368 2 1.0000 1.0000 0.9997 0.9975 0.9923 0.9833 0.9699 0.9522 0.9302 0.9042 0.8745 3 1.0000 1.0000 1.0000 0.9999 0.9994 0.9981 0.9958 0.9920 0.9864 0.9786 0.9685 4 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9996 0.9990 0.9980 0.9965 0.9941 5 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9996 0.9992 6 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 7 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
15 0 0.9851 0.9276 0.8601 0.7386 0.6333 0.5421 0.4633 0.3953 0.3367 0.2863 0.2430 1 0.9999 0.9975 0.9904 0.9647 0.9270 0.8809 0.8290 0.7738 0.7168 0.6597 0.6035 2 1.0000 0.9999 0.9996 0.9970 0.9906 0.9797 0.9638 0.9429 0.9171 0.8870 0.8531 3 1.0000 1.0000 1.0000 0.9998 0.9992 0.9976 0.9945 0.9896 0.9825 0.9727 0.9601 4 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9994 0.9986 0.9972 0.9950 0.9918 5 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9997 0.9993 0.9987 6 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 7 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
16 0 0.9841 0.9229 0.8515 0.7238 0.6143 0.5204 0.4401 0.3716 0.3131 0.2634 0.2211 1 0.9999 0.9971 0.9891 0.9601 0.9182 0.8673 0.8108 0.7511 0.6902 0.6299 0.5711 2 1.0000 0.9999 0.9995 0.9963 0.9887 0.9758 0.9571 0.9327 0.9031 0.8689 0.8306 3 1.0000 1.0000 1.0000 0.9998 0.9989 0.9968 0.9930 0.9868 0.9779 0.9658 0.9504 4 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9991 0.9981 0.9962 0.9932 0.9889 5 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9995 0.9990 0.9981 6 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9997 7 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
17 0 0.9831 0.9183 0.8429 0.7093 0.5958 0.4996 0.4181 0.3493 0.2912 0.2423 0.2012 1 0.9999 0.9968 0.9877 0.9554 0.9091 0.8535 0.7922 0.7283 0.6638 0.6005 0.5396 2 1.0000 0.9999 0.9994 0.9956 0.9866 0.9714 0.9497 0.9218 0.8882 0.8497 0.8073 3 1.0000 1.0000 1.0000 0.9997 0.9986 0.9960 0.9912 0.9836 0.9727 0.9581 0.9397 4 1.0000 1.0000 1.0000 1.0000 0.9999 0.9996 0.9988 0.9974 0.9949 0.9911 0.9855 5 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9993 0.9985 0.9973 6 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9996 7 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
1. Binomial distribution Cumulative distribution function for0.1≤θ≤0.5 5
n x 0.001 0.005 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
18 1 0.9998 0.9964 0.9862 0.9505 0.8997 0.8393 0.7735 0.7055 0.6378 0.5719 0.5091 2 1.0000 0.9999 0.9993 0.9948 0.9843 0.9667 0.9419 0.9102 0.8725 0.8298 0.7832 3 1.0000 1.0000 1.0000 0.9996 0.9982 0.9950 0.9891 0.9799 0.9667 0.9494 0.9277 4 1.0000 1.0000 1.0000 1.0000 0.9998 0.9994 0.9985 0.9966 0.9933 0.9884 0.9814 5 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9995 0.9990 0.9979 0.9962 6 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9994 7 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 8 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
19 0 0.9812 0.9092 0.8262 0.6812 0.5606 0.4604 0.3774 0.3086 0.2519 0.2051 0.1666 1 0.9998 0.9960 0.9847 0.9454 0.8900 0.8249 0.7547 0.6829 0.6121 0.5440 0.4798 2 1.0000 0.9999 0.9991 0.9939 0.9817 0.9616 0.9335 0.8979 0.8561 0.8092 0.7585 3 1.0000 1.0000 1.0000 0.9995 0.9978 0.9939 0.9868 0.9757 0.9602 0.9398 0.9147 4 1.0000 1.0000 1.0000 1.0000 0.9998 0.9993 0.9980 0.9956 0.9915 0.9853 0.9765 5 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9994 0.9986 0.9971 0.9949 6 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9996 0.9991 7 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 8 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
20 0 0.9802 0.9046 0.8179 0.6676 0.5438 0.4420 0.3585 0.2901 0.2342 0.1887 0.1516 1 0.9998 0.9955 0.9831 0.9401 0.8802 0.8103 0.7358 0.6605 0.5869 0.5169 0.4516 2 1.0000 0.9999 0.9990 0.9929 0.9790 0.9561 0.9245 0.8850 0.8390 0.7879 0.7334 3 1.0000 1.0000 1.0000 0.9994 0.9973 0.9926 0.9841 0.9710 0.9529 0.9294 0.9007 4 1.0000 1.0000 1.0000 1.0000 0.9997 0.9990 0.9974 0.9944 0.9893 0.9817 0.9710 5 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9991 0.9981 0.9962 0.9932 6 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9994 0.9987 7 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 8 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Cumulative distribution function for
0
.
1
≤
θ
≤
0
.
5
θ
n x 0.1 0.125 0.15 0.2 0.25 0.3 0.3333 0.35 0.4 0.45 0.5
1 0 0.9000 0.8750 0.8500 0.8000 0.7500 0.7000 0.6667 0.6500 0.6000 0.5500 0.5000 1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
2 0 0.8100 0.7656 0.7225 0.6400 0.5625 0.4900 0.4444 0.4225 0.3600 0.3025 0.2500 1 0.9900 0.9844 0.9775 0.9600 0.9375 0.9100 0.8889 0.8775 0.8400 0.7975 0.7500 2 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
3 0 0.7290 0.6699 0.6141 0.5120 0.4219 0.3430 0.2963 0.2746 0.2160 0.1664 0.1250 1 0.9720 0.9570 0.9392 0.8960 0.8438 0.7840 0.7407 0.7183 0.6480 0.5748 0.5000 2 0.9990 0.9980 0.9966 0.9920 0.9844 0.9730 0.9630 0.9571 0.9360 0.9089 0.8750 3 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
4 0 0.6561 0.5862 0.5220 0.4096 0.3164 0.2401 0.1975 0.1785 0.1296 0.0915 0.0625 1 0.9477 0.9211 0.8905 0.8192 0.7383 0.6517 0.5926 0.5630 0.4752 0.3910 0.3125 2 0.9963 0.9929 0.9880 0.9728 0.9492 0.9163 0.8889 0.8735 0.8208 0.7585 0.6875 3 0.9999 0.9998 0.9995 0.9984 0.9961 0.9919 0.9877 0.9850 0.9744 0.9590 0.9375 4 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
5 0 0.5905 0.5129 0.4437 0.3277 0.2373 0.1681 0.1317 0.1160 0.0778 0.0503 0.0312 1 0.9185 0.8793 0.8352 0.7373 0.6328 0.5282 0.4609 0.4284 0.3370 0.2562 0.1875 2 0.9914 0.9839 0.9734 0.9421 0.8965 0.8369 0.7901 0.7648 0.6826 0.5931 0.5000 3 0.9995 0.9989 0.9978 0.9933 0.9844 0.9692 0.9547 0.9460 0.9130 0.8688 0.8125 4 1.0000 1.0000 0.9999 0.9997 0.9990 0.9976 0.9959 0.9947 0.9898 0.9815 0.9688 5 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
6 Cumulative distribution function for0.1≤θ≤0.5 1. Binomial distribution
n x 0.1 0.125 0.15 0.2 0.25 0.3 0.3333 0.35 0.4 0.45 0.5
6 3 0.9987 0.9970 0.9941 0.9830 0.9624 0.9295 0.8999 0.8826 0.8208 0.7447 0.6562 4 0.9999 0.9998 0.9996 0.9984 0.9954 0.9891 0.9822 0.9777 0.9590 0.9308 0.8906 5 1.0000 1.0000 1.0000 0.9999 0.9998 0.9993 0.9986 0.9982 0.9959 0.9917 0.9844 6 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
7 0 0.4783 0.3927 0.3206 0.2097 0.1335 0.0824 0.0585 0.0490 0.0280 0.0152 0.0078 1 0.8503 0.7854 0.7166 0.5767 0.4449 0.3294 0.2634 0.2338 0.1586 0.1024 0.0625 2 0.9743 0.9537 0.9262 0.8520 0.7564 0.6471 0.5706 0.5323 0.4199 0.3164 0.2266 3 0.9973 0.9938 0.9879 0.9667 0.9294 0.8740 0.8267 0.8002 0.7102 0.6083 0.5000 4 0.9998 0.9995 0.9988 0.9953 0.9871 0.9712 0.9547 0.9444 0.9037 0.8471 0.7734 5 1.0000 1.0000 0.9999 0.9996 0.9987 0.9962 0.9931 0.9910 0.9812 0.9643 0.9375 6 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9995 0.9994 0.9984 0.9963 0.9922 7 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
8 0 0.4305 0.3436 0.2725 0.1678 0.1001 0.0576 0.0390 0.0319 0.0168 0.0084 0.0039 1 0.8131 0.7363 0.6572 0.5033 0.3671 0.2553 0.1951 0.1691 0.1064 0.0632 0.0352 2 0.9619 0.9327 0.8948 0.7969 0.6785 0.5518 0.4682 0.4278 0.3154 0.2201 0.1445 3 0.9950 0.9888 0.9786 0.9437 0.8862 0.8059 0.7414 0.7064 0.5941 0.4770 0.3633 4 0.9996 0.9988 0.9971 0.9896 0.9727 0.9420 0.9121 0.8939 0.8263 0.7396 0.6367 5 1.0000 0.9999 0.9998 0.9988 0.9958 0.9887 0.9803 0.9747 0.9502 0.9115 0.8555 6 1.0000 1.0000 1.0000 0.9999 0.9996 0.9987 0.9974 0.9964 0.9915 0.9819 0.9648 7 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9998 0.9993 0.9983 0.9961 8 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
9 0 0.3874 0.3007 0.2316 0.1342 0.0751 0.0404 0.0260 0.0207 0.0101 0.0046 0.0020 1 0.7748 0.6872 0.5995 0.4362 0.3003 0.1960 0.1431 0.1211 0.0705 0.0385 0.0195 2 0.9470 0.9081 0.8591 0.7382 0.6007 0.4628 0.3772 0.3373 0.2318 0.1495 0.0898 3 0.9917 0.9817 0.9661 0.9144 0.8343 0.7297 0.6503 0.6089 0.4826 0.3614 0.2539 4 0.9991 0.9975 0.9944 0.9804 0.9511 0.9012 0.8552 0.8283 0.7334 0.6214 0.5000 5 0.9999 0.9998 0.9994 0.9969 0.9900 0.9747 0.9576 0.9464 0.9006 0.8342 0.7461 6 1.0000 1.0000 1.0000 0.9997 0.9987 0.9957 0.9917 0.9888 0.9750 0.9502 0.9102 7 1.0000 1.0000 1.0000 1.0000 0.9999 0.9996 0.9990 0.9986 0.9962 0.9909 0.9805 8 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9997 0.9992 0.9980 9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
10 0 0.3487 0.2631 0.1969 0.1074 0.0563 0.0282 0.0173 0.0135 0.0060 0.0025 0.0010 1 0.7361 0.6389 0.5443 0.3758 0.2440 0.1493 0.1040 0.0860 0.0464 0.0233 0.0107 2 0.9298 0.8805 0.8202 0.6778 0.5256 0.3828 0.2991 0.2616 0.1673 0.0996 0.0547 3 0.9872 0.9725 0.9500 0.8791 0.7759 0.6496 0.5593 0.5138 0.3823 0.2660 0.1719 4 0.9984 0.9955 0.9901 0.9672 0.9219 0.8497 0.7869 0.7515 0.6331 0.5044 0.3770 5 0.9999 0.9995 0.9986 0.9936 0.9803 0.9527 0.9234 0.9051 0.8338 0.7384 0.6230 6 1.0000 1.0000 0.9999 0.9991 0.9965 0.9894 0.9803 0.9740 0.9452 0.8980 0.8281 7 1.0000 1.0000 1.0000 0.9999 0.9996 0.9984 0.9966 0.9952 0.9877 0.9726 0.9453 8 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9996 0.9995 0.9983 0.9955 0.9893 9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9990
10 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
11 0 0.3138 0.2302 0.1673 0.0859 0.0422 0.0198 0.0116 0.0088 0.0036 0.0014 0.0005 1 0.6974 0.5919 0.4922 0.3221 0.1971 0.1130 0.0751 0.0606 0.0302 0.0139 0.0059 2 0.9104 0.8503 0.7788 0.6174 0.4552 0.3127 0.2341 0.2001 0.1189 0.0652 0.0327 3 0.9815 0.9610 0.9306 0.8389 0.7133 0.5696 0.4726 0.4256 0.2963 0.1911 0.1133 4 0.9972 0.9927 0.9841 0.9496 0.8854 0.7897 0.7110 0.6683 0.5328 0.3971 0.2744 5 0.9997 0.9990 0.9973 0.9883 0.9657 0.9218 0.8779 0.8513 0.7535 0.6331 0.5000 6 1.0000 0.9999 0.9997 0.9980 0.9924 0.9784 0.9614 0.9499 0.9006 0.8262 0.7256 7 1.0000 1.0000 1.0000 0.9998 0.9988 0.9957 0.9912 0.9878 0.9707 0.9390 0.8867 8 1.0000 1.0000 1.0000 1.0000 0.9999 0.9994 0.9986 0.9980 0.9941 0.9852 0.9673 9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9993 0.9978 0.9941
10 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9995 11 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
1. Binomial distribution Cumulative distribution function for0.1≤θ≤0.5 7
n x 0.1 0.125 0.15 0.2 0.25 0.3 0.3333 0.35 0.4 0.45 0.5
12 7 1.0000 1.0000 0.9999 0.9994 0.9972 0.9905 0.9812 0.9745 0.9427 0.8883 0.8062 8 1.0000 1.0000 1.0000 0.9999 0.9996 0.9983 0.9961 0.9944 0.9847 0.9644 0.9270 9 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9995 0.9992 0.9972 0.9921 0.9807
10 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9989 0.9968 11 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 12 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
13 0 0.2542 0.1762 0.1209 0.0550 0.0238 0.0097 0.0051 0.0037 0.0013 0.0004 0.0001 1 0.6213 0.5035 0.3983 0.2336 0.1267 0.0637 0.0385 0.0296 0.0126 0.0049 0.0017 2 0.8661 0.7841 0.6920 0.5017 0.3326 0.2025 0.1387 0.1132 0.0579 0.0269 0.0112 3 0.9658 0.9310 0.8820 0.7473 0.5843 0.4206 0.3224 0.2783 0.1686 0.0929 0.0461 4 0.9935 0.9835 0.9658 0.9009 0.7940 0.6543 0.5520 0.5005 0.3530 0.2279 0.1334 5 0.9991 0.9970 0.9925 0.9700 0.9198 0.8346 0.7587 0.7159 0.5744 0.4268 0.2905 6 0.9999 0.9996 0.9987 0.9930 0.9757 0.9376 0.8965 0.8705 0.7712 0.6437 0.5000 7 1.0000 1.0000 0.9998 0.9988 0.9944 0.9818 0.9653 0.9538 0.9023 0.8212 0.7095 8 1.0000 1.0000 1.0000 0.9998 0.9990 0.9960 0.9912 0.9874 0.9679 0.9302 0.8666 9 1.0000 1.0000 1.0000 1.0000 0.9999 0.9993 0.9984 0.9975 0.9922 0.9797 0.9539
10 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9997 0.9987 0.9959 0.9888 11 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9995 0.9983 12 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 13 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
14 0 0.2288 0.1542 0.1028 0.0440 0.0178 0.0068 0.0034 0.0024 0.0008 0.0002 0.0001 1 0.5846 0.4626 0.3567 0.1979 0.1010 0.0475 0.0274 0.0205 0.0081 0.0029 0.0009 2 0.8416 0.7490 0.6479 0.4481 0.2811 0.1608 0.1053 0.0839 0.0398 0.0170 0.0065 3 0.9559 0.9127 0.8535 0.6982 0.5213 0.3552 0.2612 0.2205 0.1243 0.0632 0.0287 4 0.9908 0.9770 0.9533 0.8702 0.7415 0.5842 0.4755 0.4227 0.2793 0.1672 0.0898 5 0.9985 0.9953 0.9885 0.9561 0.8883 0.7805 0.6898 0.6405 0.4859 0.3373 0.2120 6 0.9998 0.9993 0.9978 0.9884 0.9617 0.9067 0.8505 0.8164 0.6925 0.5461 0.3953 7 1.0000 0.9999 0.9997 0.9976 0.9897 0.9685 0.9424 0.9247 0.8499 0.7414 0.6047 8 1.0000 1.0000 1.0000 0.9996 0.9978 0.9917 0.9826 0.9757 0.9417 0.8811 0.7880 9 1.0000 1.0000 1.0000 1.0000 0.9997 0.9983 0.9960 0.9940 0.9825 0.9574 0.9102
10 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9993 0.9989 0.9961 0.9886 0.9713 11 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9994 0.9978 0.9935 12 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9991 13 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 14 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
15 0 0.2059 0.1349 0.0874 0.0352 0.0134 0.0047 0.0023 0.0016 0.0005 0.0001 0.0000 1 0.5490 0.4241 0.3186 0.1671 0.0802 0.0353 0.0194 0.0142 0.0052 0.0017 0.0005 2 0.8159 0.7132 0.6042 0.3980 0.2361 0.1268 0.0794 0.0617 0.0271 0.0107 0.0037 3 0.9444 0.8922 0.8227 0.6482 0.4613 0.2969 0.2092 0.1727 0.0905 0.0424 0.0176 4 0.9873 0.9689 0.9383 0.8358 0.6865 0.5155 0.4041 0.3519 0.2173 0.1204 0.0592 5 0.9978 0.9930 0.9832 0.9389 0.8516 0.7216 0.6184 0.5643 0.4032 0.2608 0.1509 6 0.9997 0.9988 0.9964 0.9819 0.9434 0.8689 0.7970 0.7548 0.6098 0.4522 0.3036 7 1.0000 0.9998 0.9994 0.9958 0.9827 0.9500 0.9118 0.8868 0.7869 0.6535 0.5000 8 1.0000 1.0000 0.9999 0.9992 0.9958 0.9848 0.9692 0.9578 0.9050 0.8182 0.6964 9 1.0000 1.0000 1.0000 0.9999 0.9992 0.9963 0.9915 0.9876 0.9662 0.9231 0.8491
10 1.0000 1.0000 1.0000 1.0000 0.9999 0.9993 0.9982 0.9972 0.9907 0.9745 0.9408 11 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9995 0.9981 0.9937 0.9824 12 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9989 0.9963 13 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9995 14 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
16 0 0.1853 0.1181 0.0743 0.0281 0.0100 0.0033 0.0015 0.0010 0.0003 0.0001 0.0000 1 0.5147 0.3879 0.2839 0.1407 0.0635 0.0261 0.0137 0.0098 0.0033 0.0010 0.0003 2 0.7892 0.6771 0.5614 0.3518 0.1971 0.0994 0.0594 0.0451 0.0183 0.0066 0.0021 3 0.9316 0.8698 0.7899 0.5981 0.4050 0.2459 0.1659 0.1339 0.0651 0.0281 0.0106 4 0.9830 0.9593 0.9209 0.7982 0.6302 0.4499 0.3391 0.2892 0.1666 0.0853 0.0384 5 0.9967 0.9900 0.9765 0.9183 0.8103 0.6598 0.5469 0.4900 0.3288 0.1976 0.1051 6 0.9995 0.9981 0.9944 0.9733 0.9204 0.8247 0.7374 0.6881 0.5272 0.3660 0.2272 7 0.9999 0.9997 0.9989 0.9930 0.9729 0.9256 0.8735 0.8406 0.7161 0.5629 0.4018 8 1.0000 1.0000 0.9998 0.9985 0.9925 0.9743 0.9500 0.9329 0.8577 0.7441 0.5982 9 1.0000 1.0000 1.0000 0.9998 0.9984 0.9929 0.9841 0.9771 0.9417 0.8759 0.7728
8 Cumulative distribution function for0.1≤θ≤0.5 1. Binomial distribution
n x 0.1 0.125 0.15 0.2 0.25 0.3 0.3333 0.35 0.4 0.45 0.5
16 10 1.0000 1.0000 1.0000 1.0000 0.9997 0.9984 0.9960 0.9938 0.9809 0.9514 0.8949 11 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997 0.9992 0.9987 0.9951 0.9851 0.9616 12 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9991 0.9965 0.9894 13 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9994 0.9979 14 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
17 0 0.1668 0.1033 0.0631 0.0225 0.0075 0.0023 0.0010 0.0007 0.0002 0.0000 0.0000 1 0.4818 0.3542 0.2525 0.1182 0.0501 0.0193 0.0096 0.0067 0.0021 0.0006 0.0001 2 0.7618 0.6409 0.5198 0.3096 0.1637 0.0774 0.0442 0.0327 0.0123 0.0041 0.0012 3 0.9174 0.8457 0.7556 0.5489 0.3530 0.2019 0.1304 0.1028 0.0464 0.0184 0.0064 4 0.9779 0.9482 0.9013 0.7582 0.5739 0.3887 0.2814 0.2348 0.1260 0.0596 0.0245 5 0.9953 0.9862 0.9681 0.8943 0.7653 0.5968 0.4777 0.4197 0.2639 0.1471 0.0717 6 0.9992 0.9971 0.9917 0.9623 0.8929 0.7752 0.6739 0.6188 0.4478 0.2902 0.1662 7 0.9999 0.9995 0.9983 0.9891 0.9598 0.8954 0.8281 0.7872 0.6405 0.4743 0.3145 8 1.0000 0.9999 0.9997 0.9974 0.9876 0.9597 0.9245 0.9006 0.8011 0.6626 0.5000 9 1.0000 1.0000 1.0000 0.9995 0.9969 0.9873 0.9727 0.9617 0.9081 0.8166 0.6855
10 1.0000 1.0000 1.0000 0.9999 0.9994 0.9968 0.9920 0.9880 0.9652 0.9174 0.8338 11 1.0000 1.0000 1.0000 1.0000 0.9999 0.9993 0.9981 0.9970 0.9894 0.9699 0.9283 12 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9994 0.9975 0.9914 0.9755 13 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9995 0.9981 0.9936 14 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9988 15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 16 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
18 0 0.1501 0.0904 0.0536 0.0180 0.0056 0.0016 0.0007 0.0004 0.0001 0.0000 0.0000 1 0.4503 0.3228 0.2241 0.0991 0.0395 0.0142 0.0068 0.0046 0.0013 0.0003 0.0001 2 0.7338 0.6051 0.4797 0.2713 0.1353 0.0600 0.0326 0.0236 0.0082 0.0025 0.0007 3 0.9018 0.8201 0.7202 0.5010 0.3057 0.1646 0.1017 0.0783 0.0328 0.0120 0.0038 4 0.9718 0.9354 0.8794 0.7164 0.5187 0.3327 0.2311 0.1886 0.0942 0.0411 0.0154 5 0.9936 0.9814 0.9581 0.8671 0.7175 0.5344 0.4122 0.3550 0.2088 0.1077 0.0481 6 0.9988 0.9957 0.9882 0.9487 0.8610 0.7217 0.6085 0.5491 0.3743 0.2258 0.1189 7 0.9998 0.9992 0.9973 0.9837 0.9431 0.8593 0.7767 0.7283 0.5634 0.3915 0.2403 8 1.0000 0.9999 0.9995 0.9957 0.9807 0.9404 0.8924 0.8609 0.7368 0.5778 0.4073 9 1.0000 1.0000 0.9999 0.9991 0.9946 0.9790 0.9567 0.9403 0.8653 0.7473 0.5927
10 1.0000 1.0000 1.0000 0.9998 0.9988 0.9939 0.9856 0.9788 0.9424 0.8720 0.7597 11 1.0000 1.0000 1.0000 1.0000 0.9998 0.9986 0.9961 0.9938 0.9797 0.9463 0.8811 12 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997 0.9991 0.9986 0.9942 0.9817 0.9519 13 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9987 0.9951 0.9846 14 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9990 0.9962 15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9993 16 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 17 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
19 0 0.1351 0.0791 0.0456 0.0144 0.0042 0.0011 0.0005 0.0003 0.0001 0.0000 0.0000 1 0.4203 0.2938 0.1985 0.0829 0.0310 0.0104 0.0047 0.0031 0.0008 0.0002 0.0000 2 0.7054 0.5698 0.4413 0.2369 0.1113 0.0462 0.0240 0.0170 0.0055 0.0015 0.0004 3 0.8850 0.7933 0.6841 0.4551 0.2631 0.1332 0.0787 0.0591 0.0230 0.0077 0.0022 4 0.9648 0.9209 0.8556 0.6733 0.4654 0.2822 0.1879 0.1500 0.0696 0.0280 0.0096 5 0.9914 0.9757 0.9463 0.8369 0.6678 0.4739 0.3519 0.2968 0.1629 0.0777 0.0318 6 0.9983 0.9939 0.9837 0.9324 0.8251 0.6655 0.5431 0.4812 0.3081 0.1727 0.0835 7 0.9997 0.9988 0.9959 0.9767 0.9225 0.8180 0.7207 0.6656 0.4878 0.3169 0.1796 8 1.0000 0.9998 0.9992 0.9933 0.9713 0.9161 0.8538 0.8145 0.6675 0.4940 0.3238 9 1.0000 1.0000 0.9999 0.9984 0.9911 0.9674 0.9352 0.9125 0.8139 0.6710 0.5000
10 1.0000 1.0000 1.0000 0.9997 0.9977 0.9895 0.9759 0.9653 0.9115 0.8159 0.6762 11 1.0000 1.0000 1.0000 1.0000 0.9995 0.9972 0.9926 0.9886 0.9648 0.9129 0.8204 12 1.0000 1.0000 1.0000 1.0000 0.9999 0.9994 0.9981 0.9969 0.9884 0.9658 0.9165 13 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9996 0.9993 0.9969 0.9891 0.9682 14 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9994 0.9972 0.9904 15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9995 0.9978 16 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9996 17 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
1. Binomial distribution Cumulative distribution function for0.1≤θ≤0.5 9
n x 0.1 0.125 0.15 0.2 0.25 0.3 0.3333 0.35 0.4 0.45 0.5
20 2 0.6769 0.5353 0.4049 0.2061 0.0913 0.0355 0.0176 0.0121 0.0036 0.0009 0.0002 3 0.8670 0.7653 0.6477 0.4114 0.2252 0.1071 0.0604 0.0444 0.0160 0.0049 0.0013 4 0.9568 0.9050 0.8298 0.6296 0.4148 0.2375 0.1515 0.1182 0.0510 0.0189 0.0059 5 0.9887 0.9688 0.9327 0.8042 0.6172 0.4164 0.2972 0.2454 0.1256 0.0553 0.0207 6 0.9976 0.9916 0.9781 0.9133 0.7858 0.6080 0.4793 0.4166 0.2500 0.1299 0.0577 7 0.9996 0.9981 0.9941 0.9679 0.8982 0.7723 0.6615 0.6010 0.4159 0.2520 0.1316 8 0.9999 0.9997 0.9987 0.9900 0.9591 0.8867 0.8095 0.7624 0.5956 0.4143 0.2517 9 1.0000 0.9999 0.9998 0.9974 0.9861 0.9520 0.9081 0.8782 0.7553 0.5914 0.4119
10 Cumulative distribution function 2. Poisson distribution
2
Poisson distribution
Cumulative distribution function
This section tabulates the cumulative distribution function (c.d.f.) of the Poisson distribution with expected value (“rate”)λ, which is
F(x) =P{X ≤x}=
x X
k=0
P{X =k}=
x X
k=0
exp(−λ)λ
k
k!.
λ
x 0.02 0.04 0.06 0.08 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0 0.9802 0.9608 0.9418 0.9231 0.9048 0.8607 0.8187 0.7788 0.7408 0.7047 0.6703 0.6376 0.6065 1 0.9998 0.9992 0.9983 0.9970 0.9953 0.9898 0.9825 0.9735 0.9631 0.9513 0.9384 0.9246 0.9098 2 1.0000 1.0000 1.0000 0.9999 0.9998 0.9995 0.9989 0.9978 0.9964 0.9945 0.9921 0.9891 0.9856 3 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9997 0.9995 0.9992 0.9988 0.9982 4 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9998 5 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
λ
x 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.1 1.2 1.3
0 0.5769 0.5488 0.5220 0.4966 0.4724 0.4493 0.4274 0.4066 0.3867 0.3679 0.3329 0.3012 0.2725 1 0.8943 0.8781 0.8614 0.8442 0.8266 0.8088 0.7907 0.7725 0.7541 0.7358 0.6990 0.6626 0.6268 2 0.9815 0.9769 0.9717 0.9659 0.9595 0.9526 0.9451 0.9371 0.9287 0.9197 0.9004 0.8795 0.8571 3 0.9975 0.9966 0.9956 0.9942 0.9927 0.9909 0.9889 0.9865 0.9839 0.9810 0.9743 0.9662 0.9569 4 0.9997 0.9996 0.9994 0.9992 0.9989 0.9986 0.9982 0.9977 0.9971 0.9963 0.9946 0.9923 0.9893 5 1.0000 1.0000 0.9999 0.9999 0.9999 0.9998 0.9997 0.9997 0.9995 0.9994 0.9990 0.9985 0.9978 6 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9999 0.9997 0.9996 7 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 8 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
λ
x 1.4 1.5 1.6 1.7 1.8 1.9 2 2.2 2.4 2.6 2.8 3 3.2
0 0.2466 0.2231 0.2019 0.1827 0.1653 0.1496 0.1353 0.1108 0.0907 0.0743 0.0608 0.0498 0.0408 1 0.5918 0.5578 0.5249 0.4932 0.4628 0.4337 0.4060 0.3546 0.3084 0.2674 0.2311 0.1991 0.1712 2 0.8335 0.8088 0.7834 0.7572 0.7306 0.7037 0.6767 0.6227 0.5697 0.5184 0.4695 0.4232 0.3799 3 0.9463 0.9344 0.9212 0.9068 0.8913 0.8747 0.8571 0.8194 0.7787 0.7360 0.6919 0.6472 0.6025 4 0.9857 0.9814 0.9763 0.9704 0.9636 0.9559 0.9473 0.9275 0.9041 0.8774 0.8477 0.8153 0.7806 5 0.9968 0.9955 0.9940 0.9920 0.9896 0.9868 0.9834 0.9751 0.9643 0.9510 0.9349 0.9161 0.8946 6 0.9994 0.9991 0.9987 0.9981 0.9974 0.9966 0.9955 0.9925 0.9884 0.9828 0.9756 0.9665 0.9554 7 0.9999 0.9998 0.9997 0.9996 0.9994 0.9992 0.9989 0.9980 0.9967 0.9947 0.9919 0.9881 0.9832 8 1.0000 1.0000 1.0000 0.9999 0.9999 0.9998 0.9998 0.9995 0.9991 0.9985 0.9976 0.9962 0.9943 9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9996 0.9993 0.9989 0.9982
10 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9997 0.9995 11 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 12 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
λ
x 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8
2. Poisson distribution Cumulative distribution function 11
x 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8
8 0.9917 0.9883 0.9840 0.9786 0.9721 0.9642 0.9549 0.9442 0.9319 0.9181 0.9027 0.8857 0.8672 9 0.9973 0.9960 0.9942 0.9919 0.9889 0.9851 0.9805 0.9749 0.9682 0.9603 0.9512 0.9409 0.9292
10 0.9992 0.9987 0.9981 0.9972 0.9959 0.9943 0.9922 0.9896 0.9863 0.9823 0.9775 0.9718 0.9651 11 0.9998 0.9996 0.9994 0.9991 0.9986 0.9980 0.9971 0.9960 0.9945 0.9927 0.9904 0.9875 0.9841 12 0.9999 0.9999 0.9998 0.9997 0.9996 0.9993 0.9990 0.9986 0.9980 0.9972 0.9962 0.9949 0.9932 13 1.0000 1.0000 1.0000 0.9999 0.9999 0.9998 0.9997 0.9995 0.9993 0.9990 0.9986 0.9980 0.9973 14 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9999 0.9998 0.9997 0.9995 0.9993 0.9990 15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9998 0.9998 0.9996 16 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9999 17 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
λ
x 6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8 8.5 9
0 0.0025 0.0020 0.0017 0.0014 0.0011 0.0009 0.0007 0.0006 0.0005 0.0004 0.0003 0.0002 0.0001 1 0.0174 0.0146 0.0123 0.0103 0.0087 0.0073 0.0061 0.0051 0.0043 0.0036 0.0030 0.0019 0.0012 2 0.0620 0.0536 0.0463 0.0400 0.0344 0.0296 0.0255 0.0219 0.0188 0.0161 0.0138 0.0093 0.0062 3 0.1512 0.1342 0.1189 0.1052 0.0928 0.0818 0.0719 0.0632 0.0554 0.0485 0.0424 0.0301 0.0212 4 0.2851 0.2592 0.2351 0.2127 0.1920 0.1730 0.1555 0.1395 0.1249 0.1117 0.0996 0.0744 0.0550 5 0.4457 0.4141 0.3837 0.3547 0.3270 0.3007 0.2759 0.2526 0.2307 0.2103 0.1912 0.1496 0.1157 6 0.6063 0.5742 0.5423 0.5108 0.4799 0.4497 0.4204 0.3920 0.3646 0.3384 0.3134 0.2562 0.2068 7 0.7440 0.7160 0.6873 0.6581 0.6285 0.5987 0.5689 0.5393 0.5100 0.4812 0.4530 0.3856 0.3239 8 0.8472 0.8259 0.8033 0.7796 0.7548 0.7291 0.7027 0.6757 0.6482 0.6204 0.5925 0.5231 0.4557 9 0.9161 0.9016 0.8858 0.8686 0.8502 0.8305 0.8096 0.7877 0.7649 0.7411 0.7166 0.6530 0.5874
10 0.9574 0.9486 0.9386 0.9274 0.9151 0.9015 0.8867 0.8707 0.8535 0.8352 0.8159 0.7634 0.7060 11 0.9799 0.9750 0.9693 0.9627 0.9552 0.9467 0.9371 0.9265 0.9148 0.9020 0.8881 0.8487 0.8030 12 0.9912 0.9887 0.9857 0.9821 0.9779 0.9730 0.9673 0.9609 0.9536 0.9454 0.9362 0.9091 0.8758 13 0.9964 0.9952 0.9937 0.9920 0.9898 0.9872 0.9841 0.9805 0.9762 0.9714 0.9658 0.9486 0.9261 14 0.9986 0.9981 0.9974 0.9966 0.9956 0.9943 0.9927 0.9908 0.9886 0.9859 0.9827 0.9726 0.9585 15 0.9995 0.9993 0.9990 0.9986 0.9982 0.9976 0.9969 0.9959 0.9948 0.9934 0.9918 0.9862 0.9780 16 0.9998 0.9997 0.9996 0.9995 0.9993 0.9990 0.9987 0.9983 0.9978 0.9971 0.9963 0.9934 0.9889 17 0.9999 0.9999 0.9999 0.9998 0.9997 0.9996 0.9995 0.9993 0.9991 0.9988 0.9984 0.9970 0.9947 18 1.0000 1.0000 1.0000 0.9999 0.9999 0.9999 0.9998 0.9997 0.9996 0.9995 0.9993 0.9987 0.9976 19 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9999 0.9998 0.9997 0.9995 0.9989
20 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9998 0.9996 21 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 22 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 23 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
λ
x 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 15.5
0 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1 0.0008 0.0005 0.0003 0.0002 0.0001 0.0001 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 2 0.0042 0.0028 0.0018 0.0012 0.0008 0.0005 0.0003 0.0002 0.0001 0.0001 0.0001 0.0000 0.0000 3 0.0149 0.0103 0.0071 0.0049 0.0034 0.0023 0.0016 0.0011 0.0007 0.0005 0.0003 0.0002 0.0001 4 0.0403 0.0293 0.0211 0.0151 0.0107 0.0076 0.0053 0.0037 0.0026 0.0018 0.0012 0.0009 0.0006 5 0.0885 0.0671 0.0504 0.0375 0.0277 0.0203 0.0148 0.0107 0.0077 0.0055 0.0039 0.0028 0.0020 6 0.1649 0.1301 0.1016 0.0786 0.0603 0.0458 0.0346 0.0259 0.0193 0.0142 0.0105 0.0076 0.0055 7 0.2687 0.2202 0.1785 0.1432 0.1137 0.0895 0.0698 0.0540 0.0415 0.0316 0.0239 0.0180 0.0135 8 0.3918 0.3328 0.2794 0.2320 0.1906 0.1550 0.1249 0.0998 0.0790 0.0621 0.0484 0.0374 0.0288 9 0.5218 0.4579 0.3971 0.3405 0.2888 0.2424 0.2014 0.1658 0.1353 0.1094 0.0878 0.0699 0.0552
12 Cumulative distribution function 2. Poisson distribution
x 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 15.5
19 0.9980 0.9965 0.9942 0.9907 0.9857 0.9787 0.9694 0.9573 0.9421 0.9235 0.9012 0.8752 0.8455
20 0.9991 0.9984 0.9972 0.9953 0.9925 0.9884 0.9827 0.9750 0.9649 0.9521 0.9362 0.9170 0.8944 21 0.9996 0.9993 0.9987 0.9977 0.9962 0.9939 0.9906 0.9859 0.9796 0.9712 0.9604 0.9469 0.9304 22 0.9999 0.9997 0.9994 0.9990 0.9982 0.9970 0.9951 0.9924 0.9885 0.9833 0.9763 0.9673 0.9558 23 0.9999 0.9999 0.9998 0.9995 0.9992 0.9985 0.9975 0.9960 0.9938 0.9907 0.9863 0.9805 0.9730 24 1.0000 1.0000 0.9999 0.9998 0.9996 0.9993 0.9988 0.9980 0.9968 0.9950 0.9924 0.9888 0.9840 25 1.0000 1.0000 1.0000 0.9999 0.9998 0.9997 0.9994 0.9990 0.9984 0.9974 0.9959 0.9938 0.9909 26 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9997 0.9995 0.9992 0.9987 0.9979 0.9967 0.9950 27 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9998 0.9996 0.9994 0.9989 0.9983 0.9973 28 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9997 0.9995 0.9991 0.9986 29 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9998 0.9996 0.9993
30 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9998 0.9997 31 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 32 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 33 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
λ
x 16 17 18 19 20 21 22 23 24 25 30 35 40
2 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 4 0.0004 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 5 0.0014 0.0007 0.0003 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 6 0.0040 0.0021 0.0010 0.0005 0.0003 0.0001 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 7 0.0100 0.0054 0.0029 0.0015 0.0008 0.0004 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 8 0.0220 0.0126 0.0071 0.0039 0.0021 0.0011 0.0006 0.0003 0.0002 0.0001 0.0000 0.0000 0.0000 9 0.0433 0.0261 0.0154 0.0089 0.0050 0.0028 0.0015 0.0008 0.0004 0.0002 0.0000 0.0000 0.0000
10 0.0774 0.0491 0.0304 0.0183 0.0108 0.0063 0.0035 0.0020 0.0011 0.0006 0.0000 0.0000 0.0000 11 0.1270 0.0847 0.0549 0.0347 0.0214 0.0129 0.0076 0.0044 0.0025 0.0014 0.0001 0.0000 0.0000 12 0.1931 0.1350 0.0917 0.0606 0.0390 0.0245 0.0151 0.0091 0.0054 0.0031 0.0002 0.0000 0.0000 13 0.2745 0.2009 0.1426 0.0984 0.0661 0.0434 0.0278 0.0174 0.0107 0.0065 0.0004 0.0000 0.0000 14 0.3675 0.2808 0.2081 0.1497 0.1049 0.0716 0.0477 0.0311 0.0198 0.0124 0.0009 0.0000 0.0000 15 0.4667 0.3715 0.2867 0.2148 0.1565 0.1111 0.0769 0.0520 0.0344 0.0223 0.0019 0.0001 0.0000 16 0.5660 0.4677 0.3751 0.2920 0.2211 0.1629 0.1170 0.0821 0.0563 0.0377 0.0039 0.0003 0.0000 17 0.6593 0.5640 0.4686 0.3784 0.2970 0.2270 0.1690 0.1228 0.0871 0.0605 0.0073 0.0006 0.0000 18 0.7423 0.6550 0.5622 0.4695 0.3814 0.3017 0.2325 0.1748 0.1283 0.0920 0.0129 0.0012 0.0001 19 0.8122 0.7363 0.6509 0.5606 0.4703 0.3843 0.3060 0.2377 0.1803 0.1336 0.0219 0.0023 0.0002
20 0.8682 0.8055 0.7307 0.6472 0.5591 0.4710 0.3869 0.3101 0.2426 0.1855 0.0353 0.0043 0.0004 21 0.9108 0.8615 0.7991 0.7255 0.6437 0.5577 0.4716 0.3894 0.3139 0.2473 0.0544 0.0076 0.0007 22 0.9418 0.9047 0.8551 0.7931 0.7206 0.6405 0.5564 0.4723 0.3917 0.3175 0.0806 0.0128 0.0014 23 0.9633 0.9367 0.8989 0.8490 0.7875 0.7160 0.6374 0.5551 0.4728 0.3939 0.1146 0.0208 0.0026 24 0.9777 0.9594 0.9317 0.8933 0.8432 0.7822 0.7117 0.6346 0.5540 0.4734 0.1572 0.0324 0.0045 25 0.9869 0.9748 0.9554 0.9269 0.8878 0.8377 0.7771 0.7077 0.6319 0.5529 0.2084 0.0486 0.0076 26 0.9925 0.9848 0.9718 0.9514 0.9221 0.8826 0.8324 0.7723 0.7038 0.6294 0.2673 0.0705 0.0123 27 0.9959 0.9912 0.9827 0.9687 0.9475 0.9175 0.8775 0.8274 0.7677 0.7002 0.3329 0.0988 0.0193 28 0.9978 0.9950 0.9897 0.9805 0.9657 0.9436 0.9129 0.8726 0.8225 0.7634 0.4031 0.1343 0.0294 29 0.9989 0.9973 0.9941 0.9882 0.9782 0.9626 0.9398 0.9085 0.8679 0.8179 0.4757 0.1770 0.0432
30 0.9994 0.9986 0.9967 0.9930 0.9865 0.9758 0.9595 0.9360 0.9042 0.8633 0.5484 0.2269 0.0617 31 0.9997 0.9993 0.9982 0.9960 0.9919 0.9848 0.9735 0.9564 0.9322 0.8999 0.6186 0.2833 0.0855 32 0.9999 0.9996 0.9990 0.9978 0.9953 0.9907 0.9831 0.9711 0.9533 0.9285 0.6845 0.3449 0.1153 33 0.9999 0.9998 0.9995 0.9988 0.9973 0.9945 0.9895 0.9813 0.9686 0.9502 0.7444 0.4102 0.1514 34 1.0000 0.9999 0.9998 0.9994 0.9985 0.9968 0.9936 0.9882 0.9794 0.9662 0.7973 0.4775 0.1939 35 1.0000 1.0000 0.9999 0.9997 0.9992 0.9982 0.9962 0.9927 0.9868 0.9775 0.8426 0.5448 0.2424 36 1.0000 1.0000 0.9999 0.9998 0.9996 0.9990 0.9978 0.9956 0.9918 0.9854 0.8804 0.6102 0.2963 37 1.0000 1.0000 1.0000 0.9999 0.9998 0.9995 0.9988 0.9974 0.9950 0.9908 0.9110 0.6721 0.3547 38 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9993 0.9985 0.9970 0.9943 0.9352 0.7291 0.4160 39 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9996 0.9992 0.9983 0.9966 0.9537 0.7802 0.4790
2. Poisson distribution Cumulative distribution function 13
x 16 17 18 19 20 21 22 23 24 25 30 35 40
44 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9937 0.9415 0.7657 45 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9960 0.9575 0.8097 46 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9975 0.9697 0.8479 47 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9985 0.9788 0.8804 48 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9991 0.9854 0.9075 49 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9995 0.9902 0.9297
50 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997 0.9935 0.9474 51 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9957 0.9613 52 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9973 0.9719 53 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9983 0.9800 54 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9989 0.9860 55 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9993 0.9903 56 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9996 0.9934 57 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9956 58 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9971 59 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9981
14 Cumulative distribution functionΦ(z) 3. Normal distribution
3
Normal distribution
Cumulative distribution function
Φ(z)
The table below contains the probability
Φ(z) =P{Z≤z}=
Z z
−∞
φ(u)du=
Z z
−∞ 1 √
2πexp
−u
2
2
du
for a normally distributed random variableZwith expected value0and variance1. The table only contains the probabilities forz≥0. Forz <0, the probabilities can be obtained using the identityΦ(z) = 1−Φ(−z).
φ(u)
z
u
Φ(z)
z
z .0 .1 .2 .3 .4 .5 .6 .7 .8 .9
0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517
0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441
1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545
1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633
1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857
2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890
2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916
2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952
2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964
2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974
2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981
2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990
3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993
3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995
3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997
3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998
3.5 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998
3.6 0.9998 0.9998 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999
3.7 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999
3.8 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999
3. Normal distribution InverseΦ−1(p)of the cumulative distribution function (quantiles) 15
Inverse
Φ
−1(p)
of the cumulative distribution function (quantiles)
The table below contains the quantiles of the standard normal distribution. For0< p <1 the quantile is the value of z for whichP{Z ≤ z} = p, whereZ ∼ N(0,1). Thus z= Φ−1(p).
The table only contains the quantiles forp≥ 12. Forp < 12 quantiles can be obtained by exploiting the symmetry of the normal distribution:Φ−1
(p) =−Φ−1 (1−p).
φ(z)
Φ−1 (p)
z
p
p
p 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.50 0.0000 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 0.0175 0.0201 0.0226
0.51 0.0251 0.0276 0.0301 0.0326 0.0351 0.0376 0.0401 0.0426 0.0451 0.0476
0.52 0.0502 0.0527 0.0552 0.0577 0.0602 0.0627 0.0652 0.0677 0.0702 0.0728
0.53 0.0753 0.0778 0.0803 0.0828 0.0853 0.0878 0.0904 0.0929 0.0954 0.0979
0.54 0.1004 0.1030 0.1055 0.1080 0.1105 0.1130 0.1156 0.1181 0.1206 0.1231
0.55 0.1257 0.1282 0.1307 0.1332 0.1358 0.1383 0.1408 0.1434 0.1459 0.1484
0.56 0.1510 0.1535 0.1560 0.1586 0.1611 0.1637 0.1662 0.1687 0.1713 0.1738
0.57 0.1764 0.1789 0.1815 0.1840 0.1866 0.1891 0.1917 0.1942 0.1968 0.1993
0.58 0.2019 0.2045 0.2070 0.2096 0.2121 0.2147 0.2173 0.2198 0.2224 0.2250
0.59 0.2275 0.2301 0.2327 0.2353 0.2378 0.2404 0.2430 0.2456 0.2482 0.2508
0.60 0.2533 0.2559 0.2585 0.2611 0.2637 0.2663 0.2689 0.2715 0.2741 0.2767
0.61 0.2793 0.2819 0.2845 0.2871 0.2898 0.2924 0.2950 0.2976 0.3002 0.3029
0.62 0.3055 0.3081 0.3107 0.3134 0.3160 0.3186 0.3213 0.3239 0.3266 0.3292
0.63 0.3319 0.3345 0.3372 0.3398 0.3425 0.3451 0.3478 0.3505 0.3531 0.3558
0.64 0.3585 0.3611 0.3638 0.3665 0.3692 0.3719 0.3745 0.3772 0.3799 0.3826
0.65 0.3853 0.3880 0.3907 0.3934 0.3961 0.3989 0.4016 0.4043 0.4070 0.4097
0.66 0.4125 0.4152 0.4179 0.4207 0.4234 0.4261 0.4289 0.4316 0.4344 0.4372
0.67 0.4399 0.4427 0.4454 0.4482 0.4510 0.4538 0.4565 0.4593 0.4621 0.4649
0.68 0.4677 0.4705 0.4733 0.4761 0.4789 0.4817 0.4845 0.4874 0.4902 0.4930
0.69 0.4959 0.4987 0.5015 0.5044 0.5072 0.5101 0.5129 0.5158 0.5187 0.5215
0.70 0.5244 0.5273 0.5302 0.5330 0.5359 0.5388 0.5417 0.5446 0.5476 0.5505
0.71 0.5534 0.5563 0.5592 0.5622 0.5651 0.5681 0.5710 0.5740 0.5769 0.5799
0.72 0.5828 0.5858 0.5888 0.5918 0.5948 0.5978 0.6008 0.6038 0.6068 0.6098
0.73 0.6128 0.6158 0.6189 0.6219 0.6250 0.6280 0.6311 0.6341 0.6372 0.6403
0.74 0.6433 0.6464 0.6495 0.6526 0.6557 0.6588 0.6620 0.6651 0.6682 0.6713
0.75 0.6745 0.6776 0.6808 0.6840 0.6871 0.6903 0.6935 0.6967 0.6999 0.7031
0.76 0.7063 0.7095 0.7128 0.7160 0.7192 0.7225 0.7257 0.7290 0.7323 0.7356
0.77 0.7388 0.7421 0.7454 0.7488 0.7521 0.7554 0.7588 0.7621 0.7655 0.7688
0.78 0.7722 0.7756 0.7790 0.7824 0.7858 0.7892 0.7926 0.7961 0.7995 0.8030
0.79 0.8064 0.8099 0.8134 0.8169 0.8204 0.8239 0.8274 0.8310 0.8345 0.8381
0.80 0.8416 0.8452 0.8488 0.8524 0.8560 0.8596 0.8633 0.8669 0.8705 0.8742
0.81 0.8779 0.8816 0.8853 0.8890 0.8927 0.8965 0.9002 0.9040 0.9078 0.9116
0.82 0.9154 0.9192 0.9230 0.9269 0.9307 0.9346 0.9385 0.9424 0.9463 0.9502
0.83 0.9542 0.9581 0.9621 0.9661 0.9701 0.9741 0.9782 0.9822 0.9863 0.9904
0.84 0.9945 0.9986 1.0027 1.0069 1.0110 1.0152 1.0194 1.0237 1.0279 1.0322
0.85 1.0364 1.0407 1.0450 1.0494 1.0537 1.0581 1.0625 1.0669 1.0714 1.0758
0.86 1.0803 1.0848 1.0893 1.0939 1.0985 1.1031 1.1077 1.1123 1.1170 1.1217
0.87 1.1264 1.1311 1.1359 1.1407 1.1455 1.1503 1.1552 1.1601 1.1650 1.1700
0.88 1.1750 1.1800 1.1850 1.1901 1.1952 1.2004 1.2055 1.2107 1.2160 1.2212
0.89 1.2265 1.2319 1.2372 1.2426 1.2481 1.2536 1.2591 1.2646 1.2702 1.2759
0.90 1.2816 1.2873 1.2930 1.2988 1.3047 1.3106 1.3165 1.3225 1.3285 1.3346
0.91 1.3408 1.3469 1.3532 1.3595 1.3658 1.3722 1.3787 1.3852 1.3917 1.3984
0.92 1.4051 1.4118 1.4187 1.4255 1.4325 1.4395 1.4466 1.4538 1.4611 1.4684
0.93 1.4758 1.4833 1.4909 1.4985 1.5063 1.5141 1.5220 1.5301 1.5382 1.5464
0.94 1.5548 1.5632 1.5718 1.5805 1.5893 1.5982 1.6072 1.6164 1.6258 1.6352
0.95 1.6449 1.6546 1.6646 1.6747 1.6849 1.6954 1.7060 1.7169 1.7279 1.7392
0.96 1.7507 1.7624 1.7744 1.7866 1.7991 1.8119 1.8250 1.8384 1.8522 1.8663
0.97 1.8808 1.8957 1.9110 1.9268 1.9431 1.9600 1.9774 1.9954 2.0141 2.0335
0.98 2.0537 2.0749 2.0969 2.1201 2.1444 2.1701 2.1973 2.2262 2.2571 2.2904
16 Selected quantilesΦ−1(p)in high precision 3. Normal distribution
Selected quantiles
Φ
−1(p)
in high precision
p Φ−1(p)
0.9 1.2815515655 0.95 1.6448536270 0.975 1.9599639845 0.99 2.3263478740 0.995 2.5758293035 0.9975 2.8070337683
p Φ−1(p)
0.999 3.0902323062 0.9995 3.2905267315 0.99975 3.4807564043 0.9999 3.7190164855 0.99995 3.8905918864 0.999975 4.0556269811
p Φ−1(p)
4. Student’s t distribution 17
4
Student’s t distribution
Inverse
F
−1ν
(p)
of the cumulative distribution function (quantiles)
The table below contains the quantiles of Student’s t distribution withν degrees of free-dom. For0 < p < 1 the quantile is the value ofxfor whichP{X ≤ x} = p, where X ∼t(ν). Thusx=F−1
ν (p).
The table only contains the quantiles forp≥ 12. Forp < 12 quantiles can be obtained by exploiting the symmetry of the t distribution:F−1
ν (p) =−F
−1
ν (1−p).
fν(x)
F−1
ν (p)
x
p
p
ν 0.6 0.7 0.75 0.8 0.85 0.9 0.95 0.975 0.99 0.995 0.999 0.9995
1 0.3249 0.7265 1.0000 1.3764 1.9626 3.0777 6.3138 12.706 31.821 63.657 318.31 636.62 2 0.2887 0.6172 0.8165 1.0607 1.3862 1.8856 2.9200 4.3027 6.9646 9.9248 22.327 31.599 3 0.2767 0.5844 0.7649 0.9785 1.2498 1.6377 2.3534 3.1824 4.5407 5.8409 10.215 12.924 4 0.2707 0.5686 0.7407 0.9410 1.1896 1.5332 2.1318 2.7764 3.7469 4.6041 7.1732 8.6103 5 0.2672 0.5594 0.7267 0.9195 1.1558 1.4759 2.0150 2.5706 3.3649 4.0321 5.8934 6.8688 6 0.2648 0.5534 0.7176 0.9057 1.1342 1.4398 1.9432 2.4469 3.1427 3.7074 5.2076 5.9588 7 0.2632 0.5491 0.7111 0.8960 1.1192 1.4149 1.8946 2.3646 2.9980 3.4995 4.7853 5.4079 8 0.2619 0.5459 0.7064 0.8889 1.1081 1.3968 1.8595 2.3060 2.8965 3.3554 4.5008 5.0413 9 0.2610 0.5435 0.7027 0.8834 1.0997 1.3830 1.8331 2.2622 2.8214 3.2498 4.2968 4.7809
10 0.2602 0.5415 0.6998 0.8791 1.0931 1.3722 1.8125 2.2281 2.7638 3.1693 4.1437 4.5869 11 0.2596 0.5399 0.6974 0.8755 1.0877 1.3634 1.7959 2.2010 2.7181 3.1058 4.0247 4.4370 12 0.2590 0.5386 0.6955 0.8726 1.0832 1.3562 1.7823 2.1788 2.6810 3.0545 3.9296 4.3178 13 0.2586 0.5375 0.6938 0.8702 1.0795 1.3502 1.7709 2.1604 2.6503 3.0123 3.8520 4.2208 14 0.2582 0.5366 0.6924 0.8681 1.0763 1.3450 1.7613 2.1448 2.6245 2.9768 3.7874 4.1405 15 0.2579 0.5357 0.6912 0.8662 1.0735 1.3406 1.7531 2.1314 2.6025 2.9467 3.7328 4.0728 16 0.2576 0.5350 0.6901 0.8647 1.0711 1.3368 1.7459 2.1199 2.5835 2.9208 3.6862 4.0150 17 0.2573 0.5344 0.6892 0.8633 1.0690 1.3334 1.7396 2.1098 2.5669 2.8982 3.6458 3.9651 18 0.2571 0.5338 0.6884 0.8620 1.0672 1.3304 1.7341 2.1009 2.5524 2.8784 3.6105 3.9216 19 0.2569 0.5333 0.6876 0.8610 1.0655 1.3277 1.7291 2.0930 2.5395 2.8609 3.5794 3.8834
20 0.2567 0.5329 0.6870 0.8600 1.0640 1.3253 1.7247 2.0860 2.5280 2.8453 3.5518 3.8495 21 0.2566 0.5325 0.6864 0.8591 1.0627 1.3232 1.7207 2.0796 2.5176 2.8314 3.5272 3.8193 22 0.2564 0.5321 0.6858 0.8583 1.0614 1.3212 1.7171 2.0739 2.5083 2.8188 3.5050 3.7921 23 0.2563 0.5317 0.6853 0.8575 1.0603 1.3195 1.7139 2.0687 2.4999 2.8073 3.4850 3.7676 24 0.2562 0.5314 0.6848 0.8569 1.0593 1.3178 1.7109 2.0639 2.4922 2.7969 3.4668 3.7454 25 0.2561 0.5312 0.6844 0.8562 1.0584 1.3163 1.7081 2.0595 2.4851 2.7874 3.4502 3.7251 26 0.2560 0.5309 0.6840 0.8557 1.0575 1.3150 1.7056 2.0555 2.4786 2.7787 3.4350 3.7066 27 0.2559 0.5306 0.6837 0.8551 1.0567 1.3137 1.7033 2.0518 2.4727 2.7707 3.4210 3.6896 28 0.2558 0.5304 0.6834 0.8546 1.0560 1.3125 1.7011 2.0484 2.4671 2.7633 3.4082 3.6739 29 0.2557 0.5302 0.6830 0.8542 1.0553 1.3114 1.6991 2.0452 2.4620 2.7564 3.3962 3.6594
30 0.2556 0.5300 0.6828 0.8538 1.0547 1.3104 1.6973 2.0423 2.4573 2.7500 3.3852 3.6460 31 0.2555 0.5298 0.6825 0.8534 1.0541 1.3095 1.6955 2.0395 2.4528 2.7440 3.3749 3.6335 32 0.2555 0.5297 0.6822 0.8530 1.0535 1.3086 1.6939 2.0369 2.4487 2.7385 3.3653 3.6218 33 0.2554 0.5295 0.6820 0.8526 1.0530 1.3077 1.6924 2.0345 2.4448 2.7333 3.3563 3.6109 34 0.2553 0.5294 0.6818 0.8523 1.0525 1.3070 1.6909 2.0322 2.4411 2.7284 3.3479 3.6007 35 0.2553 0.5292 0.6816 0.8520 1.0520 1.3062 1.6896 2.0301 2.4377 2.7238 3.3400 3.5911 36 0.2552 0.5291 0.6814 0.8517 1.0516 1.3055 1.6883 2.0281 2.4345 2.7195 3.3326 3.5821 37 0.2552 0.5289 0.6812 0.8514 1.0512 1.3049 1.6871 2.0262 2.4314 2.7154 3.3256 3.5737 38 0.2551 0.5288 0.6810 0.8512 1.0508 1.3042 1.6860 2.0244 2.4286 2.7116 3.3190 3.5657 39 0.2551 0.5287 0.6808 0.8509 1.0504 1.3036 1.6849 2.0227 2.4258 2.7079 3.3128 3.5581
40 0.2550 0.5286 0.6807 0.8507 1.0500 1.3031 1.6839 2.0211 2.4233 2.7045 3.3069 3.5510 41 0.2550 0.5285 0.6805 0.8505 1.0497 1.3025 1.6829 2.0195 2.4208 2.7012 3.3013 3.5442 42 0.2550 0.5284 0.6804 0.8503 1.0494 1.3020 1.6820 2.0181 2.4185 2.6981 3.2960 3.5377 43 0.2549 0.5283 0.6802 0.8501 1.0491 1.3016 1.6811 2.0167 2.4163 2.6951 3.2909 3.5316 44 0.2549 0.5282 0.6801 0.8499 1.0488 1.3011 1.6802 2.0154 2.4141 2.6923 3.2861 3.5258 45 0.2549 0.5281 0.6800 0.8497 1.0485 1.3006 1.6794 2.0141 2.4121 2.6896 3.2815 3.5203 46 0.2548 0.5281 0.6799 0.8495 1.0483 1.3002 1.6787 2.0129 2.4102 2.6870 3.2771 3.5150 47 0.2548 0.5280 0.6797 0.8493 1.0480 1.2998 1.6779 2.0117 2.4083 2.6846 3.2729 3.5099 48 0.2548 0.5279 0.6796 0.8492 1.0478 1.2994 1.6772 2.0106 2.4066 2.6822 3.2689 3.5051 49 0.2547 0.5278 0.6795 0.8490 1.0475 1.2991 1.6766 2.0096 2.4049 2.6800 3.2651 3.5004
18 InverseF−1
ν (p)of the cumulative distribution function (quantiles) 4. Student’s t distribution
ν 0.6 0.7 0.75 0.8 0.85 0.9 0.95 0.975 0.99 0.995 0.999 0.9995
50 0.2547 0.5278 0.6794 0.8489 1.0473 1.2987 1.6759 2.0086 2.4033 2.6778 3.2614 3.4960 51 0.2547 0.5277 0.6793 0.8487 1.0471 1.2984 1.6753 2.0076 2.4017 2.6757 3.2579 3.4918 52 0.2546 0.5276 0.6792 0.8486 1.0469 1.2980 1.6747 2.0066 2.4002 2.6737 3.2545 3.4877 53 0.2546 0.5276 0.6791 0.8485 1.0467 1.2977 1.6741 2.0057 2.3988 2.6718 3.2513 3.4838 54 0.2546 0.5275 0.6791 0.8483 1.0465 1.2974 1.6736 2.0049 2.3974 2.6700 3.2481 3.4800 55 0.2546 0.5275 0.6790 0.8482 1.0463 1.2971 1.6730 2.0040 2.3961 2.6682 3.2451 3.4764 56 0.2546 0.5274 0.6789 0.8481 1.0461 1.2969 1.6725 2.0032 2.3948 2.6665 3.2423 3.4729 57 0.2545 0.5273 0.6788 0.8480 1.0459 1.2966 1.6720 2.0025 2.3936 2.6649 3.2395 3.4696 58 0.2545 0.5273 0.6787 0.8479 1.0458 1.2963 1.6716 2.0017 2.3924 2.6633 3.2368 3.4663 59 0.2545 0.5272 0.6787 0.8478 1.0456 1.2961 1.6711 2.0010 2.3912 2.6618 3.2342 3.4632
60 0.2545 0.5272 0.6786 0.8477 1.0455 1.2958 1.6706 2.0003 2.3901 2.6603 3.2317 3.4602 61 0.2545 0.5272 0.6785 0.8476 1.0453 1.2956 1.6702 1.9996 2.3890 2.6589 3.2293 3.4573 62 0.2544 0.5271 0.6785 0.8475 1.0452 1.2954 1.6698 1.9990 2.3880 2.6575 3.2270 3.4545 63 0.2544 0.5271 0.6784 0.8474 1.0450 1.2951 1.6694 1.9983 2.3870 2.6561 3.2247 3.4518 64 0.2544 0.5270 0.6783 0.8473 1.0449 1.2949 1.6690 1.9977 2.3860 2.6549 3.2225 3.4491 65 0.2544 0.5270 0.6783 0.8472 1.0448 1.2947 1.6686 1.9971 2.3851 2.6536 3.2204 3.4466 66 0.2544 0.5269 0.6782 0.8471 1.0446 1.2945 1.6683 1.9966 2.3842 2.6524 3.2184 3.4441 67 0.2544 0.5269 0.6782 0.8470 1.0445 1.2943 1.6679 1.9960 2.3833 2.6512 3.2164 3.4417 68 0.2543 0.5269 0.6781 0.8469 1.0444 1.2941 1.6676 1.9955 2.3824 2.6501 3.2145 3.4394 69 0.2543 0.5268 0.6781 0.8469 1.0443 1.2939 1.6672 1.9949 2.3816 2.6490 3.2126 3.4372
70 0.2543 0.5268 0.6780 0.8468 1.0442 1.2938 1.6669 1.9944 2.3808 2.6479 3.2108 3.4350 71 0.2543 0.5268 0.6780 0.8467 1.0441 1.2936 1.6666 1.9939 2.3800 2.6469 3.2090 3.4329 72 0.2543 0.5267 0.6779 0.8466 1.0440 1.2934 1.6663 1.9935 2.3793 2.6459 3.2073 3.4308 73 0.2543 0.5267 0.6779 0.8466 1.0438 1.2933 1.6660 1.9930 2.3785 2.6449 3.2057 3.4289 74 0.2543 0.5267 0.6778 0.8465 1.0437 1.2931 1.6657 1.9925 2.3778 2.6439 3.2041 3.4269 75 0.2542 0.5266 0.6778 0.8464 1.0436 1.2929 1.6654 1.9921 2.3771 2.6430 3.2025 3.4250 76 0.2542 0.5266 0.6777 0.8464 1.0436 1.2928 1.6652 1.9917 2.3764 2.6421 3.2010 3.4232 77 0.2542 0.5266 0.6777 0.8463 1.0435 1.2926 1.6649 1.9913 2.3758 2.6412 3.1995 3.4214 78 0.2542 0.5266 0.6776 0.8463 1.0434 1.2925 1.6646 1.9908 2.3751 2.6403 3.1980 3.4197 79 0.2542 0.5265 0.6776 0.8462 1.0433 1.2924 1.6644 1.9905 2.3745 2.6395 3.1966 3.4180
80 0.2542 0.5265 0.6776 0.8461 1.0432 1.2922 1.6641 1.9901 2.3739 2.6387 3.1953 3.4163 81 0.2542 0.5265 0.6775 0.8461 1.0431 1.2921 1.6639 1.9897 2.3733 2.6379 3.1939 3.4147 82 0.2542 0.5264 0.6775 0.8460 1.0430 1.2920 1.6636 1.9893 2.3727 2.6371 3.1926 3.4132 83 0.2542 0.5264 0.6775 0.8460 1.0429 1.2918 1.6634 1.9890 2.3721 2.6364 3.1913 3.4116 84 0.2542 0.5264 0.6774 0.8459 1.0429 1.2917 1.6632 1.9886 2.3716 2.6356 3.1901 3.4102 85 0.2541 0.5264 0.6774 0.8459 1.0428 1.2916 1.6630 1.9883 2.3710 2.6349 3.1889 3.4087 86 0.2541 0.5263 0.6774 0.8458 1.0427 1.2915 1.6628 1.9879 2.3705 2.6342 3.1877 3.4073 87 0.2541 0.5263 0.6773 0.8458 1.0426 1.2914 1.6626 1.9876 2.3700 2.6335 3.1866 3.4059 88 0.2541 0.5263 0.6773 0.8457 1.0426 1.2912 1.6624 1.9873 2.3695 2.6329 3.1854 3.4045 89 0.2541 0.5263 0.6773 0.8457 1.0425 1.2911 1.6622 1.9870 2.3690 2.6322 3.1843 3.4032
90 0.2541 0.5263 0.6772 0.8456 1.0424 1.2910 1.6620 1.9867 2.3685 2.6316 3.1833 3.4019 91 0.2541 0.5262 0.6772 0.8456 1.0424 1.2909 1.6618 1.9864 2.3680 2.6309 3.1822 3.4007 92 0.2541 0.5262 0.6772 0.8455 1.0423 1.2908 1.6616 1.9861 2.3676 2.6303 3.1812 3.3994 93 0.2541 0.5262 0.6771 0.8455 1.0422 1.2907 1.6614 1.9858 2.3671 2.6297 3.1802 3.3982 94 0.2541 0.5262 0.6771 0.8455 1.0422 1.2906 1.6612 1.9855 2.3667 2.6291 3.1792 3.3971 95 0.2541 0.5262 0.6771 0.8454 1.0421 1.2905 1.6611 1.9853 2.3662 2.6286 3.1782 3.3959 96 0.2541 0.5261 0.6771 0.8454 1.0421 1.2904 1.6609 1.9850 2.3658 2.6280 3.1773 3.3948 97 0.2540 0.5261 0.6770 0.8453 1.0420 1.2903 1.6607 1.9847 2.3654 2.6275 3.1764 3.3937 98 0.2540 0.5261 0.6770 0.8453 1.0419 1.2902 1.6606 1.9845 2.3650 2.6269 3.1755 3.3926 99 0.2540 0.5261 0.6770 0.8453 1.0419 1.2902 1.6604 1.9842 2.3646 2.6264 3.1746 3.3915
5.χ2distribution 19
5
χ
2distribution
Inverse
F
−1ν
(p)
of the cumulative distribution function (quantiles)
The table below contains the quantiles of theχ2(chi-squared) distribution withνdegrees of freedom. For0< p <1the quantile is the value ofxfor whichP{X≤x}=p, where X ∼χ2(ν). Thusx=F−1
ν (p).
fν(x)
F−1
ν (p)
x
p
p
ν 0.005 0.01 0.025 0.05 0.1 0.5 0.9 0.95 0.975 0.99 0.995 0.999
1 0.0000 0.0002 0.0010 0.0039 0.0158 0.4549 2.7055 3.8415 5.0239 6.6349 7.8794 10.828 2 0.0100 0.0201 0.0506 0.1026 0.2107 1.3863 4.6052 5.9915 7.3778 9.2103 10.597 13.816 3 0.0717 0.1148 0.2158 0.3518 0.5844 2.3660 6.2514 7.8147 9.3484 11.345 12.838 16.266 4 0.2070 0.2971 0.4844 0.7107 1.0636 3.3567 7.7794 9.4877 11.143 13.277 14.860 18.467 5 0.4117 0.5543 0.8312 1.1455 1.6103 4.3515 9.2364 11.070 12.833 15.086 16.750 20.515 6 0.6757 0.8721 1.2373 1.6354 2.2041 5.3481 10.645 12.592 14.449 16.812 18.548 22.458 7 0.9893 1.2390 1.6899 2.1673 2.8331 6.3458 12.017 14.067 16.013 18.475 20.278 24.322 8 1.3444 1.6465 2.1797 2.7326 3.4895 7.3441 13.362 15.507 17.535 20.090 21.955 26.124 9 1.7349 2.0879 2.7004 3.3251 4.1682 8.3428 14.684 16.919 19.023 21.666 23.589 27.877
10 2.1559 2.5582 3.2470 3.9403 4.8652 9.3418 15.987 18.307 20.483 23.209 25.188 29.588 11 2.6032 3.0535 3.8157 4.5748 5.5778 10.341 17.275 19.675 21.920 24.725 26.757 31.264 12 3.0738 3.5706 4.4038 5.2260 6.3038 11.340 18.549 21.026 23.337 26.217 28.300 32.909 13 3.5650 4.1069 5.0088 5.8919 7.0415 12.340 19.812 22.362 24.736 27.688 29.819 34.528 14 4.0747 4.6604 5.6287 6.5706 7.7895 13.339 21.064 23.685 26.119 29.141 31.319 36.123 15 4.6009 5.2293 6.2621 7.2609 8.5468 14.339 22.307 24.996 27.488 30.578 32.801 37.697 16 5.1422 5.8122 6.9077 7.9616 9.3122 15.338 23.542 26.296 28.845 32.000 34.267 39.252 17 5.6972 6.4078 7.5642 8.6718 10.085 16.338 24.769 27.587 30.191 33.409 35.718 40.790 18 6.2648 7.0149 8.2307 9.3905 10.865 17.338 25.989 28.869 31.526 34.805 37.156 42.312 19 6.8440 7.6327 8.9065 10.117 11.651 18.338 27.204 30.144 32.852 36.191 38.582 43.820
20 7.4338 8.2604 9.5908 10.851 12.443 19.337 28.412 31.410 34.170 37.566 39.997 45.315 21 8.0337 8.8972 10.283 11.591 13.240 20.337 29.615 32.671 35.479 38.932 41.401 46.797 22 8.6427 9.5425 10.982 12.338 14.041 21.337 30.813 33.924 36.781 40.289 42.796 48.268 23 9.2604 10.196 11.689 13.091 14.848 22.337 32.007 35.172 38.076 41.638 44.181 49.728 24 9.8862 10.856 12.401 13.848 15.659 23.337 33.196 36.415 39.364 42.980 45.559 51.179 25 10.520 11.524 13.120 14.611 16.473 24.337 34.382 37.652 40.646 44.314 46.928 52.620 26 11.160 12.198 13.844 15.379 17.292 25.336 35.563 38.885 41.923 45.642 48.290 54.052 27 11.808 12.879 14.573 16.151 18.114 26.336 36.741 40.113 43.195 46.963 49.645 55.476 28 12.461 13.565 15.308 16.928 18.939 27.336 37.916 41.337 44.461 48.278 50.993 56.892 29 13.121 14.256 16.047 17.708 19.768 28.336 39.087 42.557 45.722 49.588 52.336 58.301
30 13.787 14.953 16.791 18.493 20.599 29.336 40.256 43.773 46.979 50.892 53.672 59.703 31 14.458 15.655 17.539 19.281 21.434 30.336 41.422 44.985 48.232 52.191 55.003 61.098 32 15.134 16.362 18.291 20.072 22.271 31.336 42.585 46.194 49.480 53.486 56.328 62.487 33 15.815 17.074 19.047 20.867 23.110 32.336 43.745 47.400 50.725 54.776 57.648 63.870 34 16.501 17.789 19.806 21.664 23.952 33.336 44.903 48.602 51.966 56.061 58.964 65.247 35 17.192 18.509 20.569 22.465 24.797 34.336 46.059 49.802 53.203 57.342 60.275 66.619 36 17.887 19.233 21.336 23.269 25.643 35.336 47.212 50.998 54.437 58.619 61.581 67.985 37 18.586 19.960 22.106 24.075 26.492 36.336 48.363 52.192 55.668 59.893 62.883 69.346 38 19.289 20.691 22.878 24.884 27.343 37.335 49.513 53.384 56.896 61.162 64.181 70.703 39 19.996 21.426 23.654 25.695 28.196 38.335 50.660 54.572 58.120 62.428 65.476 72.055
20 InverseF−1
ν (p)of the cumulative distribution function (quantiles) 5.χ2distribution
ν 0.005 0.01 0.025 0.05 0.1 0.5 0.9 0.95 0.975 0.99 0.995 0.999
49 27.249 28.941 31.555 33.930 36.818 48.335 62.038 66.339 70.222 74.919 78.231 85.351
50 27.991 29.707 32.357 34.764 37.689 49.335 63.167 67.505 71.420 76.154 79.490 86.661 51 28.735 30.475 33.162 35.600 38.560 50.335 64.295 68.669 72.616 77.386 80.747 87.968 52 29.481 31.246 33.968 36.437 39.433 51.335 65.422 69.832 73.810 78.616 82.001 89.272 53 30.230 32.018 34.776 37.276 40.308 52.335 66.548 70.993 75.002 79.843 83.253 90.573 54 30.981 32.793 35.586 38.116 41.183 53.335 67.673 72.153 76.192 81.069 84.502 91.872 55 31.735 33.570 36.398 38.958 42.060 54.335 68.796 73.311 77.380 82.292 85.749 93.168 56 32.490 34.350 37.212 39.801 42.937 55.335 69.919 74.468 78.567 83.513 86.994 94.461 57 33.248 35.131 38.027 40.646 43.816 56.335 71.040 75.624 79.752 84.733 88.236 95.751 58 34.008 35.913 38.844 41.492 44.696 57.335 72.160 76.778 80.936 85.950 89.477 97.039 59 34.770 36.698 39.662 42.339 45.577 58.335 73.279 77.931 82.117 87.166 90.715 98.324
60 35.534 37.485 40.482 43.188 46.459 59.335 74.397 79.082 83.298 88.379 91.952 99.607 61 36.301 38.273 41.303 44.038 47.342 60.335 75.514 80.232 84.476 89.591 93.186 100.89 62 37.068 39.063 42.126 44.889 48.226 61.335 76.630 81.381 85.654 90.802 94.419 102.17 63 37.838 39.855 42.950 45.741 49.111 62.335 77.745 82.529 86.830 92.010 95.649 103.44 64 38.610 40.649 43.776 46.595 49.996 63.335 78.860 83.675 88.004 93.217 96.878 104.72 65 39.383 41.444 44.603 47.450 50.883 64.335 79.973 84.821 89.177 94.422 98.105 105.99 66 40.158 42.240 45.431 48.305 51.770 65.335 81.085 85.965 90.349 95.626 99.330 107.26 67 40.935 43.038 46.261 49.162 52.659 66.335 82.197 87.108 91.519 96.828 100.55 108.53 68 41.713 43.838 47.092 50.020 53.548 67.335 83.308 88.250 92.689 98.028 101.78 109.79 69 42.494 44.639 47.924 50.879 54.438 68.334 84.418 89.391 93.856 99.228 103.00 111.06
70 43.275 45.442 48.758 51.739 55.329 69.334 85.527 90.531 95.023 100.43 104.21 112.32 71 44.058 46.246 49.592 52.600 56.221 70.334 86.635 91.670 96.189 101.62 105.43 113.58 72 44.843 47.051 50.428 53.462 57.113 71.334 87.743 92.808 97.353 102.82 106.65 114.84 73 45.629 47.858 51.265 54.325 58.006 72.334 88.850 93.945 98.516 104.01 107.86 116.09 74 46.417 48.666 52.103 55.189 58.900 73.334 89.956 95.081 99.678 105.20 109.07 117.35 75 47.206 49.475 52.942 56.054 59.795 74.334 91.061 96.217 100.84 106.39 110.29 118.60 76 47.997 50.286 53.782 56.920 60.690 75.334 92.166 97.351 102.00 107.58 111.50 119.85 77 48.788 51.097 54.623 57.786 61.586 76.334 93.270 98.484 103.16 108.77 112.70 121.10 78 49.582 51.910 55.466 58.654 62.483 77.334 94.374 99.617 104.32 109.96 113.91 122.35 79 50.376 52.725 56.309 59.522 63.380 78.334 95.476 100.75 105.47 111.14 115.12 123.59
80 51.172 53.540 57.153 60.391 64.278 79.334 96.578 101.88 106.63 112.33 116.32 124.84 81 51.969 54.357 57.998 61.261 65.176 80.334 97.680 103.01 107.78 113.51 117.52 126.08 82 52.767 55.174 58.845 62.132 66.076 81.334 98.780 104.14 108.94 114.69 118.73 127.32 83 53.567 55.993 59.692 63.004 66.976 82.334 99.880 105.27 110.09 115.88 119.93 128.56 84 54.368 56.813 60.540 63.876 67.876 83.334 100.98 106.39 111.24 117.06 121.13 129.80 85 55.170 57.634 61.389 64.749 68.777 84.334 102.08 107.52 112.39 118.24 122.32 131.04 86 55.973 58.456 62.239 65.623 69.679 85.334 103.18 108.65 113.54 119.41 123.52 132.28 87 56.777 59.279 63.089 66.498 70.581 86.334 104.28 109.77 114.69 120.59 124.72 133.51 88 57.582 60.103 63.941 67.373 71.484 87.334 105.37 110.90 115.84 121.77 125.91 134.75 89 58.389 60.928 64.793 68.249 72.387 88.334 106.47 112.02 116.99 122.94 127.11 135.98
90 59.196 61.754 65.647 69.126 73.291 89.334 107.57 113.15 118.14 124.12 128.30 137.21 91 60.005 62.581 66.501 70.003 74.196 90.334 108.66 114.27 119.28 125.29 129.49 138.44 92 60.815 63.409 67.356 70.882 75.100 91.334 109.76 115.39 120.43 126.46 130.68 139.67 93 61.625 64.238 68.211 71.760 76.006 92.334 110.85 116.51 121.57 127.63 131.87 140.89 94 62.437 65.068 69.068 72.640 76.912 93.334 111.94 117.63 122.72 128.80 133.06 142.12 95 63.250 65.898 69.925 73.520 77.818 94.334 113.04 118.75 123.86 129.97 134.25 143.34 96 64.063 66.730 70.783 74.401 78.725 95.334 114.13 119.87 125.00 131.14 135.43 144.57 97 64.878 67.562 71.642 75.282 79.633 96.334 115.22 120.99 126.14 132.31 136.62 145.79 98 65.694 68.396 72.501 76.164 80.541 97.334 116.32 122.11 127.28 133.48 137.80 147.01 99 66.510 69.230 73.361 77.046 81.449 98.334 117.41 123.23 128.42 134.64 138.99 148.23
6. F distribution InverseF−1
(ν1,ν2)(p)of the cumulative distribution function (quantiles) 21
6
F distribution
Inverse
F
(−ν11,ν2)
(p)
of the cumulative distribution function (quantiles)
The table below contains the quantiles of the F distribution with ν1 andν2 degrees of
22 InverseF−1
(ν1,ν2)(p)of the cumulative distribution function (quantiles) 6. F distribution
6. F distribution InverseF−1
(ν1,ν2)(p)of the cumulative distribution function (quantiles) 23
24 InverseF−1
(ν1,ν2)(p)of the cumulative distribution function (quantiles) 6. F distribution
6. F distribution InverseF−1
(ν1,ν2)(p)of the cumulative distribution function (quantiles) 25
26 InverseF−1
(ν1,ν2)(p)of the cumulative distribution function (quantiles) 6. F distribution
6. F distribution InverseF−1
(ν1,ν2)(p)of the cumulative distribution function (quantiles) 27
28 InverseF−1
(ν1,ν2)(p)of the cumulative distribution function (quantiles) 6. F distribution
6. F distribution InverseF−1
(ν1,ν2)(p)of the cumulative distribution function (quantiles) 29
30 InverseF−1
(ν1,ν2)(p)of the cumulative distribution function (quantiles) 6. F distribution
6. F distribution InverseF−1
(ν1,ν2)(p)of the cumulative distribution function (quantiles) 31
32 InverseF−1
(ν1,ν2)(p)of the cumulative distribution function (quantiles) 6. F distribution