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