Page:Biometrika - Volume 6, Issue 1.djvu/19

of advantage. If the distribution is not normal, the mean and the standard deviation of a sample will be positively correlated, so that although both will have greater variability, yet they will tend to counteract each other, a mean deviating largely from the general mean tending to be divided by a larger standard deviation. Consequently I believe that the tables at the end of the present paper may be used in estimating the degree of certainty arrived at by the mean of a few experiments, in the case of most laboratory or biological work where the distributions are as a rule of a ‘cocked hat’ type and so sufficiently nearly normal.

Section VII. Tables of ${\frac {n-2}{n-3}}{\frac {n-4}{n-5}}{\textrm {...}}{\binom {{\frac {3}{2}}\cdot {\frac {1}{2}}n\;{\textrm {odd}}}{{\frac {2}{1}}\cdot {\frac {1}{\pi }}n\;{\textrm {even}}}}\int _{-{\frac {\pi }{2}}}^{{\textrm {tan}}^{-1}z}{\textrm {cos}}^{n-2}\theta d\theta$ for values of $n$ from $4$ to $10$ inclusive.

Together with ${\frac {\sqrt {7}}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-{\frac {7x^{2}}{2}}}dx$ for comparison when $n=10$.

$z\left(={\frac {x}{s}}\right)$ $n=4$ $n=5$ $n=6$ $n=7$ $n=8$ $n=9$ $n=10$ For comparison
$\left({\frac {\sqrt {7}}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-{\frac {7x^{2}}{2}}}dx\right)$ ​ 0.1 .5633 .5745 .5841 .5928 .6006 .60787 .61462 .60411 0.2 .6241 .6458 .6634 .6798 .6936 .70705 .71846 .70159 0.3 .6804 .7096 .7340 .7549 .7733 .78961 .80423 .78641 0.4 .7309 .7657 .7939 .8175 .8376 .85465 .86970 .85520 0.5 .7749 .8131 .8428 .8667 .8863 .90251 .91609 .90691 0.6 .8125 .8518 .8813 .9040 .9218 .93600 .94732 .94375 0.7 .8440 .8830 .9109 .9314 .9468 .95851 .96747 .96799 0.8 .8701 .9076 .9332 .9512 .9640 .97328 .98007 .98253 0.9 .8915 .9269 .9498 .9652 .9756 .98279 .98780 .99137 1.0 .9092 .9419 .9622 .9751 .9834 .98890 .99252 .99820 ​ ​ 1.1 .9236 .9537 .9714 .9821 .9887 .99280 .99539 .99926 1.2 .9354 .9628 .9782 .9870 .9922 .99528 .99713 .99971 1.3 .9451 .9700 .9832 .9905 .9946 .99688 .99819 .99986 1.4 .9531 .9756 .9870 .9930 .9962 .99791 .99885 .99989 1.5 .9598 .9800 .9899 .9948 .9973 .99859 .99926 .99999 1.6 .9653 .9836 .9920 .9961 .9981 .99903 .99951 1.7 .9699 .9864 .9937 .9970 .9986 .99933 .99968 1.8 .9737 .9886 .9950 .9977 .9990 .99953 .99978 1.9 .9770 .9904 .9959 .9983 .9992 .99967 .99985 2.0 .9797 .9919 .9967 .9986 .9994 .99976 .99990 ​ ​ 2.1 .9821 .9931 .9973 .9989 .9996 .99983 .99993 2.2 .9841 .9941 .9978 .9992 .9997 .99987 .99995 2.3 .9858 .9950 .9982 .9993 .9998 .99991 .99996 2.4 .9873 .9957 .9985 .9995 .9998 .99993 .99997 2.5 .9886 .9963 .9987 .9996 .9998 .99995 .99998 2.6 .9898 .9967 .9989 .9996 .9999 .99996 .99999 2.7 .9908 .9972 .9991 .9997 .9999 .99997 .99999 2.8 .9916 .9975 .9992 .9998 .9999 .99998 .99999 2.9 .9924 .9978 .9993 .9998 .9999 .99998 .99999 3.0 .9931 .9981 .9994 .9998 — .99999 — — ​