The Standard Normal Variate and Reading Areas from the Normal Table

Every normal distribution — whatever its mean $\mu$ and standard deviation $\sigma$ — can be converted into the same reference distribution, called the standard normal distribution, which has mean $0$ and standard deviation $1$. The conversion is done using the standard normal variate:

$Z$ measures how many standard deviations a value $X$ lies above (positive $Z$) or below (negative $Z$) the mean. Once a business variable is converted to $Z$, its probability can be read off from a single, universal standard normal table, which lists the area under the standard normal curve between $Z = 0$ and any given positive value of $Z$. Because the curve is symmetric, the same table serves negative $Z$ values too (the area between $0$ and $-z$ equals the area between $0$ and $+z$), and the total area on either side of the mean is exactly $0.5$.

A few standard, widely-published table values used repeatedly in this chapter (area between $Z = 0$ and the stated $Z$):

To find a probability such as $P(X > a)$, $P(X < a)$, or $P(a < X < b)$ for a normally distributed variable $X$:

1. Convert every raw value to its $Z$-score using $Z = \dfrac{X-\mu}{\sigma}$.
2. Identify which region of the curve the question is asking for — a single tail, a range spanning the mean, or a range on one side of the mean. …