Area Under the Normal Curve — Using the Standard Normal Table

Once a raw value $x$ has been standardized to $z = \dfrac{x-\mu}{\sigma}$, the Standard Normal Table gives $\Phi(z)$ — the cumulative area to the LEFT of $z$ under the standard normal curve, which is exactly $P(Z \le z)$. This is the single tool used to answer every "less than / more than / between" probability or percentage question in this chapter.

A compact reference of standard normal areas (the values needed for this chapter's worked examples and exercises; a full table runs from $z=0.00$ to $z=3.99$ in steps of 0.01, and is available in the syllabus's prescribed statistical tables booklet):

Three rules that turn a table of "area to the left" into an answer for any type of question — this is the working method used throughout this chapter, and the basis of the independent "second method" check in every worked numerical:

1. "Less than" area: $P(X < x) = P(Z < z) = \Phi(z)$ — read directly from the table.
2. "More than" area (complement rule): $P(X > x) = 1 - \Phi(z)$, because the total area under the curve is 1.
3. "Between two values" area: $P(x_1 < X < x_2) = \Phi(z_2) - \Phi(z_1)$ — the larger cumulative area minus the smaller one.
4. Symmetry rule for a negative $z$: the table above is stated only for $z \ge 0$; for a negative $z$, use $\Phi(-z) = 1 - \Phi(z)$, a direct consequence of the curve's symmetry about zero (Section 2, property 1). For example, $\Phi(-1.00) = 1 - \Phi(1.00) = 1 - 0.8413 = 0.1587$. …