EX 4.14 percentile of standard normal distribution

The 99th percentile of the standard normal distribution is that value on the horizontal axis such that the area under the $z$ curve to the left of the value is .9900. Appendix Table A. 3 gives for fixed $z$ the area under the standard normal curve to the left of $z$ , whereas here we have the area and want the value of $z$. This is the “inverse” problem to $P\left(Z\le z\right)=$ ? so the table is used in an inverse fashion: Find in the middle of the table .9900; the row and column in which it lies identify the 99th $z$ percentile. Here .9901 lies at the intersection of the row marked 2.3 and column marked .03, so the 99th percentile is (approximately) $z=2.33$ . (See Figure 4.17.)

Figure 4.17 Finding the 99th percentile

By symmetry, the first percentile is as far below 0 as the 99th is above 0, so equals $-2.33(1\%$ lies below the first and also above the 99th). (See Figure 4.18.)

Figure 4.18 The relationship between the 1st and 99th percentiles