Page:Biometrika - Volume 6, Issue 1.djvu/16

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 9 141/2 111/2 33 431/2 701/2 1191/2 1511/2 122 671/2 49 261/2 16 10 6 Difference +4 +5 −2 −11/2 −1 −8 +1/2 +101/2 +3 −11 +41/2 −8 +21/2 +1/2 +1

whence $\chi ^{2}=12.44$, $P=.56$.

This is very satisfactory, especially when we consider that as a rule observations are tested against curves fitted from the mean and one or more other moments of the observations, so that considerable correspondence is only to be expected; while this curve is exposed to the full errors of random sampling, its constants having been calculated quite apart from the observations.

Diagram III. Comparison of Calculated Standard Deviation Frequency Curve with 750 actual Standard Deviations.

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Scale of Standard Deviation of the Population

The left middle finger samples show much the same features as those of the height, but as the grouping is not so large compared to the variability the curves fit the observations more closely. Diagrams III.^[1] and IV. give the standard deviations and the $z$’s for this set of samples. The results are as follows:—

1. There are three small mistakes in plotting the observed values in Diagram III., which make the fit appear worse than it really is.