Percentage Points of the Standard Normal Distribution

Use percentage point (critical value) notation $z_\alpha$ for the standard normal distribution to look up the cutoff value that produces a specified upper-tail probability, and combine this with symmetry to handle lower-tail, complement, and two-tailed (central) probability statements.

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Tutorial

Upper Percentage Points

In hypothesis testing and confidence intervals, we frequently need the $z$ -value that cuts off a specified probability in the right tail under the standard normal curve. Such a $z$ -value is called a percentage point (or critical value) of the standard normal distribution.

The upper $\alpha$ percentage point, written $z_\alpha$ , is defined by

P(Z > z_\alpha) = \alpha.

In words, $z_\alpha$ is the value with area $\alpha$ to its right under the standard normal density. The commonly tabulated values are:

\begin{array}{|c|c|}\hline \alpha & z_\alpha \\ \hline 0.10 & 1.2816 \\ 0.05 & 1.6449 \\ 0.025 & 1.9600 \\ 0.01 & 2.3263 \\ 0.005 & 2.5758 \\ 0.001 & 3.0902 \\ \hline \end{array}

For instance, $z_{0.05} = 1.6449$ tells us that

P(Z > 1.6449) = 0.05.

Notice that smaller tail probabilities $\alpha$ correspond to larger percentage points $z_\alpha$ , since pushing further out into the tail leaves less area to the right.