Calculating z-scores

 ZSCORE.83p ZSCORE.86p zscore.89p

Let Z be the standard normal distribution (having mean 0 and standard deviation 1). For a given probability p, we often need to know the value z such that P(Z <= z) = p, or P(z <= Z) = p, or P(-z <= Z <= z) = p. Such a value z is called a z-score, or critical value. The most commonly used z-scores are 1.645, 1.96. 2.326, and 2.576, where

 P(-1.645 <= Z <= 1.645) = 0.90 P(-1.96 <= Z <= 1.96) = 0.95 P(-2.326 <= Z <= 2.326) = 0.98 P(-2.576 <= X <= 2.576) = 0.99

Using the ZSCORE Program

We can calculate these or any other critical values using the ZSCORE program. To execute the program, first enter 1, 2, or 3 to specify that you want the critical value from a left probability, or the critical value from a right probability, or the critical values from a middle probability.

If you enter 1, then the program will find the value z such that P(Z <= z) = p. If you enter 2, then the program will find the value z such that P(z <= Z) = p. If you enter 3, then the program will give the values -z and z such that P(-z <= Z <= z) = p. In all cases, the z-scores are rounded to three decimal places.

After specifying either 1, 2, or 3, simply enter the desired probability p.

Example. Find the critical value z that satisfies:

(a) P(Z <= z) = 0.15
(b) P(z <= Z) = 0.035
(c) P(-z <= Z <= z) = 0.80

Solution. (a) After calling up the ZSCORE program, enter 1 to specify a left probability, then enter .15 for this left probability. We receive a critical value of -1.036. Thus, P(Z <= -1.036) = 0.15.

(b) First enter 2 to specify a right probability, then enter .035 for this right probability. We receive a critical value of 1.812. Hence, P(1.812 <= Z) = 0.035

(c) First enter 3 to specify a middle probability, then enter .8 for this middle probability. We receive the critical values -1.282 and 1.282. Hence, P(-1.282 <= Z <= 1.282) = 0.80.