The next figure gives an example of how to find inverse standard normal distribution. This means how to find interval limits given the total percentage P(a ≤ z ≤ b) of the population that should be covered by it.

Examples of finding the following quantities:

• Find interval limits [a, b] with that holds 90% of the values that follow a normal distribution with parameters N(0, 1).

Another way to see this question is that the b upper limit of the interval [a, b] is chosen in a manner that P(z ≤ b) = 1 − α/2 = 0.95 holds as shown in the next figure.

According to previous deductions for a standard normal distribution, the following holds: P(−1.6449 ≤ z ≤ 1.6449) = 0.90. 

Example of application on finding inverse standard normal distribution

Find the following quantities:

• Find interval limits [a, b] with that holds 90% of the values that follow a normal distribution with parameters N(40, 8).

• According to previous deductions for a standard normal distribution the following holds: P(−1.6449 ≤ z ≤ 1.6449) = 0.90. To find the interval that encompasses 90% of the values for the normal distribution with parameters N(40, 8):

– For z = −1.6449: x = μ + zσ = 40 − 1.6449·8 = 26.84.

– For z = 1.6449: x = μ + zσ = 40 + 1.6449·8 = 53.16.

The interval must be [26.84, 53.16].