Area under Normal Distribution Curve

Question

What is the formula that determines the Z-score table? More specifically, what formula can be used the equate the area underneath the normal distribution curve, without using the table?

Answer 1

See this section on Wikipedia

The values in the table come from evaluating the integral $$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^{-t^2/2}\mathop{dt}.$$

It is hard to evaluate this integral, which is why we rely on tables instead.

Answer 2

I have a (easy) formula, which approximate the standard normal distribution quiet good:

${\Phi(x) \approx 0.5 \cdot \Bigg( 1 + \sqrt{1-e^{- (\sqrt{\frac{\pi}{8}} \cdot x^2)}} \Bigg)}$

The diagram below shows the values of the Standard normal distribution ($\color{red}{ \textrm{red}}$) in comparision to the values of the approximation ($\color{blue}{ \textrm{blue}} $) .