An Approximation to the Probability Integral
Joy Williams · The Annals of Mathematical Statistics · 1946
It is shown that $\frac{1}{\sqrt{2\pi}} \int_{-x}^x e^{-\frac{1}{2}t^2} dt \geq \lbrack 1 - e^{-(2/\pi)x^2}\rbrack^\frac{1}{2}$ and that the equality is never in error by as much as three-fourths of one percent. Other approximations are discussed.