How to Find Probability Between Two Z Scores? (original) (raw)

Last Updated : 23 Jul, 2025

Answer: Calculate the difference in the cumulative probabilities associated with each Z score to find the probability between two Z scores in a standard normal distribution.

Finding the probability between two Z scores in a standard normal distribution involves using the Z table to determine the cumulative probabilities associated with each Z score and then calculating the difference between them. Here's a detailed explanation:

Standard Normal Distribution

In a standard normal distribution (Z distribution), Z scores represent the number of standard deviations a data point is from the mean. The mean is 0, and the standard deviation is 1.

Z Table

A Z table provides the cumulative probabilities for various Z scores. Alternatively, statistical software can be used to directly calculate cumulative probabilities.

Steps to Find Probability Between Two Z Scores:

**Step 1: Identify Z Scores:
Determine the Z scores for the two values between which you want to find the probability.

**Step 2: Consult the Z Table:
Look up or calculate the cumulative probabilities associated with each Z score.

**P (Z ≤ z 1 ​) and **P (Z ≤ z 2 ​)

These represent the probabilities of obtaining a Z score less than or equal to z1​ and z2, respectively.

**Step 3: Calculate the Probability Between:
Find the probability between the two Z scores by taking the difference of the cumulative probabilities.

**P (z 1 ​ < Z < z 2 ​) = P (Z ≤ z 2 ​) – P (Z ≤ z 1 ​)

**Example: Suppose you want to find the probability between Z scores -1.5 and 1.8. Using a Z table or software, find:

**P( Z ≤ −1.5) and **P**( Z **≤ 1.8)

Then, calculate: **P(−1.5 < Z < 1.8) = P(Z ≤ 1.8)−P(Z ≤ −1.5)

**Note: