Standard normal with area property

Standard normal distribution: the normal distribution with mean 0 and variance 1. Z = (X − μ)/σ ~ N(0, 1), with pdf

f(z) = (1/√(2π)) e^(−z²/2), −∞ < z < ∞

Area property

The total area under the curve is 1 and the curve is symmetric about z = 0. The standard normal table gives the area between 0 and z, P(0 < Z < z).

              .-'''-.
           .-'   |   '-.
        .-'      |      '-.
  _.--'          |          '--._
 ------+----+----+----+----+------
      -2   -1    0    1    2   z
        |<-0.3413->|<-0.3413->|

Key properties:

1. P(Z < 0) = P(Z > 0) = 0.5.
2. Symmetry: P(0 < Z < a) = P(−a < Z < 0), and P(Z > a) = P(Z < −a).
3. P(Z < a) = 0.5 + P(0 < Z < a) for a > 0.
4. P(a < Z < b) = P(0<Z<b) − P(0<Z<a) when both are positive; for a < 0 < b it is the sum of the two table areas.
5. Empirical rule:

Example: P(−1.2 < Z < 0.8) = P(0 < Z < 1.2) + P(0 < Z < 0.8) = 0.3849 + 0.2881 = 0.6730.