Page:Biometrika - Volume 6, Issue 1.djvu/18

Mean value of standard deviations; calculated	2.186	±.023
Mean value of„ standard„ deviations;„ observed	2.179
Difference=	−.007

Standard deviation of standard deviations:—

	Calculated	.9224	±.016
	Observed	.9802
Difference=+		.0578

Comparison of fit. Theoretical Equation: $y={\frac {16\times 750}{{\sqrt {2\pi }}\sigma ^{3}}}x^{2}e^{-{\frac {2x^{2}}{\sigma ^{2}}}}$.

Scale in terms
of standard
deviation of
population 0 to .1 .1 to .2 .2 to .3 .3 to .4 .4 to .5 .5 to .6 .6. to .7 .7 to .8 .8 to .9 .9. to 1.0 1.0 to 1.1 1.1 to 1.2 1.2 to 1.3 1.3 to 1.4 1.4 to 1.5 1.5 to 1.6 1.6 to 1.7 greater
than 1.7 Calculated
frequency 11/2 101/2 27 451/2 641/2 781/2 87 88 811/2 71 58 45 33 23 15 91/2 51/2 7 Observed
frequency 2 14 271/2 51 641/2 91 941/2 681/2 651/2 73 481/2 401/2 421/2 20 221/2 12 5 71/2 Difference +1/2 +31/2 +1/2 +51/2 — +121/2 +71/2 −191/2 −16 +2 −91/2 −41/2 +91/2 −3 +71/2 +21/2 −1/2 +1/2

whence $\chi ^{2}=21.80$, $P=.19$.

Calculated	value of standard deviation	1	(±.017)
Observed	value of„ standard„ deviation„		.982
	Difference	=	−.018

Comparison of Fit. Theoretical Equation: $y={\frac {2}{\pi }}{\textrm {cos}}^{4}\theta$, $z={\textrm {tan}}\theta$.

Scale of $z$ less than −3.05 −3.05 to −2.05 −2.05 to −1.55 −1.55 to −1.05 −1.05 to −.75 −.75 to −.45 −.45 to −.15 −.15 to +.15 +.15 to +.45 +.45 to +.75 +.75 to +1.05 +1.05 to +1.55 +1.55 to +2.05 +2.05 to +3.05 more than +3.05 Calculated
frequency 5 91/2 131/2 341/2 441/2 781/2 119 141 119 781/2 441/2 341/2 131/2 91/2 5 Observed
frequency 4 151/2 18 331/2 44 75 122 138 1201/2 71 461/2 36 11 9 6 Difference −1 +6 +41/2 −1 −1/2 −31/2 +3 −3 +11/2 −71/2 +2 +11/2 −21/2 −1/2 +1

whence $\chi ^{2}=7.39$, $P=.92$.

A very close fit.

We see then that if the distribution is approximately normal our theory gives us a satisfactory measure of the certainty to be derived from a small sample in both the cases we have tested; but we have an indication that a fine grouping is