Square Root Transformation Revisited

Properties of the transformation z = √(x + c) of a Poisson variable x are studied. Exact values of var (z) were determined by computer analysis, for various values of c and m, the Poisson parameter. It is shown that c = 0·386 is optimal in the sense that var (z) then converges fastest toward 14 without ever exceeding that value. In a mathematical analysis, the variance was approximated by using the relation var (z) = m + c ‐ {E(z)}2 and developing E(z) (but not E(z2)) into a Taylor series. The approximation obtained appears to be somewhat better than that published earlier in the literature. Tentatively, an analogy is drawn from the physical laws governing damped vibration, suggesting at least a possibility for a closed form for var (z).