### Critical Values of z

 “Critical" values of z are associated with interesting central areas under the standard normal curve. For instance, as seen in the picture to the right, there is an 80% probability that any normal variable will have a z-score between –1.28 and 1.28. In other words, there is an 80% probability that any normal variable will fall within 1.28 standard deviations of its mean. So we say that 1.28 is the critical value of z that corresponds to a central area of 0.80. To any central area there corresponds a “tail area." (This is yellow in the picture.) Since there are two “tails", the central area is always 1 - 2(tail area), and the tail area is always 0.5(1 – central area). Critical z values are often denoted by zα, where the subscript α (alpha) is the tail area. For instance, the picture on the right indicates that z.10 = 1.28. The four pictures below illustrate other important critical values of z.     These five critical values of z are summarized in the following table. α = tail area central area = 1 – 2α zα 0.10 0.80 z.10 = 1.28 0.05 0.90 z.05 = 1.645 0.025 0.95 z.025 = 1.96 0.01 0.98 z.01 = 2.33 0.005 0.99 z.005 = 2.58