Conditions initially designed to yield successful transformation in time series modeling are considered. For these cases, this article proposes an approach aimed at exploring the range at which the expectation of the untransformed attains equilibrium with the transformed and maintains variance as a proportional product of the untransformed. Methodologically, new results are obtained, and relations in comparison of normality are compared. The square transformation has a moderate effect on distribution shape. The square   transformed  could be used to reduce left skewness as squaring makes sense even if the variable concerned is zero or positive, given that   and   are identical. It is the motive of this paper to primarily investigate the limits of the square transformation with a view to eliminating distributional assumptions that may impede expectations for the confidence intervals. We compared means and variances of both the transformed and untransformed to establish the bounding curves.

Keywords: Transformation, Statistical properties, distributional assumptions, confidence intervals.