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 ≤ z₁​) and P (Z ≤ z₂​)

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 (z₁​ < Z < z₂​) = P (Z ≤ z₂​) – P (Z ≤ z₁​)

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:

• Ensure that the Z scores are correctly identified and interpreted (negative or positive direction).
• The resulting probability between two Z scores provides insights into the likelihood of observations falling within that range in a standard normal distribution.