1966 Volume 32 Issue 4 Pages 107-114
The angular transformationθ=Arcsin√P transforms a probability p varying from 0 to 1 into an angle varying from 0 to π/2 radians and is remarkable in that the amount of information concerning θ is constant. Arcsin√Pmay be written Arcsin√P=√P+1/2 p/3 √P+1·3/2·4 p2/5 √P+……+1·3……(2n-1)/2·4…………2n pn/2n+1 √P+… [Taylor expantion of f (x) about x=p] If a probability p were as small as a death rate, the second term p2/3/2·3 and the following terms 3p2/5/2·4·5+…… would be negligible altogether. Thus, normal deviate for the mortality of any sample community and its standard error Et are calculated respectively as follows: t=2Σ[NA(√PA)/ΣNA], and Et=1/√ΣNA P : age specific death rate for age A among sample community. Pa : age specific death rate for the same age among the standard population. N : the number of persons A years old among sample community. Normal deviate for the mortality of 46 prefectures has been investigated using vital statistics of Japan of 1960: these values were further comparatively studied with the crude or corrected death rates respectively. Naturally, correlation between normal deviate for the mortality and corrected death rate was far higher than that between normal deviate and crude death rate both in male and female populations. It has been deducted by the author that normal deviate for mortality is a useful indicator for evaluating levels of health of communities.