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

 

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