z score of normal distribution

Question

Good day,

I want to ask about standard normal distribution. What is the highest and lowest value of $z$ score can be?

From the table of standard normal, the value $z$ score is only for -3.99 $\leq$ $z$ $\leq$ 3.99.

Can it be higher than the range? Is there any programme or formula to compute the probability of standard normal distribution for the range outside -3.99 $\leq$ $z$ $\leq$ 3.99?

Thank you for your help.

Answer 1

The table below shows the values given by R for $\Phi(x)=\Pr(X \le x)$ for some $x$. In addition $\Phi(-x)=-\Phi(x)$.

For $x$ large and negative, a reasonable approximation is $\dfrac{-x}{x^2+1}\dfrac{\exp\left(-x^2/2\right)}{\sqrt{2\pi}}$. For example, with $x=-15$, this gives 3.670825e-51

So for $x$ large and positive, a reasonable approximation is $1-\dfrac{x}{x^2+1}\dfrac{\exp\left(-x^2/2\right)}{\sqrt{2\pi}}$.

   x  Phi(x) =  Pr(X<=x)
  -15    3.670966e-51
  -14    7.793537e-45
  -13    6.117164e-39
  -12    1.776482e-33
  -11    1.910660e-28
  -10    7.619853e-24
   -9    1.128588e-19
   -8    6.220961e-16
   -7    1.279813e-12
   -6    9.865876e-10
   -5    2.866516e-07
   -4    3.167124e-05
   -3    1.349898e-03
   -2    2.275013e-02
   -1    1.586553e-01
    0    5.000000e-01
    1    8.413447e-01
    2    9.772499e-01
    3    9.986501e-01
    4    9.999683e-01
    5    9.999997e-01

Answer 2

$z$ score that is above $3.999$ or below $-3.999$ is considered highly unusual in Triola book which means that it very rarely occurs. Any other value of $z$ that is greater than $3.999$ is treated the same as $3.999$ or $-3.999$ in the case of negative $z$ because $P(|z| > 3.999) \approx 0$