The cumulative distribution function (CDF) of the standard normal distribution plays a central role in statistical hypothesis testing, meta-analysis, and many applications involving the conversion of \(Z\)-scores into \(p\)-values. In this work, we introduce the normal-tanh coordinate \[ L(x)=\operatorname{atanh}\!\left(2\Phi(x)-1\right), \tag{1} \] where \(\Phi(x)\) denotes the standard normal cumulative distribution function. The normal-tanh coordinate transforms the bounded normal CDF into an unbounded odd function and provides a natural target for polynomial approximation. We derive cubic and quintic approximations of \(L(x)\) using two different approaches: a Taylor expansion and a structure-preserving approximation based on the differential equation satisfied by the Mills ratio. A striking result is that both approaches yield exactly the same cubic coefficient, \[ a_3=-\frac{1}{6}+\frac{2}{3\pi}, \tag{2} \] whereas they diverge at quintic order. Numerical experiments show that the ODE-preserving quintic approximation provides a better approximation over a wider range of \(x\), while the Taylor approximation remains optimal in the immediate neighborhood of the origin. Motivated by this representation, we introduce a logistic-type family of probability distributions defined by \[ F(x)=\frac{1}{2}+\frac{1}{2}\tanh\!\left(f(x)\right). \tag{3} \] Within this framework, both the logistic and the standard normal distributions arise naturally as special cases. The cubic approximation defines a valid member of this family and possesses closed-form expressions for its cumulative distribution function, probability density function, and quantile function. By contrast, the ODE-preserving quintic approximation improves approximation accuracy but does not satisfy the global conditions required for membership in the logistic-type family. The coefficient \(-\frac{1}{6}+\frac{2}{3\pi}\) is also remarkably close to the empirical cubic coefficient used in the GELU approximation, suggesting a possible connection between the local structure of the normal distribution and the empirical success of GELU-type transformations.