This paper presents theoretical studies of formation regularities for the statistical mean and variance of normally distributed random values with the unlimited argument values subjected to nonlinear transformations of functions $\sqrt{|X|}$ and $\sqrt{X}$. It is shown that for nonlinear square root transformation of a normally distributed random variable, the integrals of higher order mean $n>1$ satisfy the inequality $\overline{(y-\overline{Y})^n}\neq 0$. On the basis of the theoretical research, the correct boundaries $m,\sigma \to \infty$ of error transfer formulas are suggested.