We can use the following process to find the probability that a normally distributed random variable *X* takes on a certain value, given a mean and standard deviation:

**Step 1: Find the z-score.**

A z-score tells you how many standard deviations away an individual data value falls from the mean. It is calculated as:

**z-score = (x – μ) / σ**

where:

**x:**individual data value**μ:**population mean**σ:**population standard deviation

**Step 2: Find the probability that corresponds to the z-score.**

Once we’ve calculated the z-score, we can look up the probability that corresponds to it in the z table.

The following examples show how to use this process in different scenarios.

**Example 1: Probability Less Than a Certain Value**

The scores on a certain test are normally distributed with mean μ = 82 and standard deviation σ = 8. What is the probability that a given student scores less than 84 on the test?

**Step 1: Find the z-score.**

First, we will find the z-score associated with a score of 84:

z-score = (x – μ) / σ = (84 – 82) / 8 = 2 / 8 = **0.25**

**Step 2: Use the z-table to find the corresponding probability.**

Next, we will look up the value **0.25 **in the z-table:

The probability that a given student scores less than 84 is approximately **59.87%**.

**Example 2: Probability Greater Than a Certain Value**

The height of a certain species of penguin is normally distributed with a mean of μ = 30 inches and a standard deviation of σ = 4 inches. If we randomly select a penguin, what is the probability that it is greater than 28 inches tall?

**Step 1: Find the z-score.**

First, we will find the z-score associated with a height of 28 inches.

z-score = (x – μ) / σ = (28 – 30) / 4 = -2 / 4 = **-.5**

**Step 2: Use the z-table to find the corresponding probability.**

Next, we will look up the value **-0.5**** **in the z-table:

The value that corresponds to a z-score of -0.5 is .3085. This represents the probability that a penguin is less than 28 inches tall.

However, since we want to know the probability that a penguin will have a height greater than 28 inches, we need to subtract this probability from 1.

Thus, the probability that a penguin will have a height greater than 28 inches is: 1 – .3085 = **0.6915**.

**Example 3: Probability Between Two Values**

The weight of a certain species of turtle is normally distributed with a mean of μ = 400 pounds and a standard deviation of σ = 25 pounds. If we randomly select a turtle, what is the probability that it weighs between 410 and 425 pounds?

**Step 1: Find the z-scores.**

First, we will find the z-scores associated with 410 pounds and 425 pounds

z-score of 410 = (x – μ) / σ = (410 – 400) / 25 = 10 / 25 = **0.4**

z-score of 425 = (x – μ) / σ = (425 – 400) / 25 = 25 / 25 = **1**

**Step 2: Use the z-table to find the corresponding probability.**

First, we will look up the value **0.4**** **in the z-table:

Then, we will look up the value **1**** **in the z-table:

Then we will subtract the smaller value from the larger value: **0.8413 – 0.6554 = 0.1859**.

Thus, the probability that a randomly selected turtle weighs between 410 pounds and 425 pounds is **18.59%**.

**Additional Resources**

How to Calculate a P-Value from a Z-Score by Hand

How to Convert Z-Scores to Raw Scores

How to Find Z-Scores Given Area