Symmetry Properties of the Standard Normal Distribution

The probability density function of the standard normal distribution $Z \sim N(0,1)$ is symmetric about $0$. That is,

$$\phi(-z) = \phi(z)$$

for every real number $z.$

This symmetry means the area under the curve to the left of $-z$ equals the area to the right of $+z{:}$

$$P(Z < -z) \;=\; P(Z > z) \;=\; 1 - P(Z < z).$$

As a special case, since the total area is $1$ and the curve is symmetric about $0,$

$$P(Z < 0) \;=\; P(Z > 0) \;=\; 0.5.$$

For example, given that $P(Z < 1) = 0.8413,$ we can find $P(Z < -1)$ without consulting a new table entry:

$$P(Z < -1) \;=\; 1 - P(Z < 1) \;=\; 1 - 0.8413 \;=\; 0.1587.$$

This is why standard $z$-tables only list values for $z \geq 0$ — the negative side is recovered by symmetry.