In statistics, a tells us how many standard deviations away a given value lies from a population mean.

We use the following formula to calculate a z-score for a given value:

z = (x – μ) / σ

where:

• x: Individual data value
• μ: Mean of population
• σ: Standard deviation of population

To find the area under a normal distribution that lies to the right of a given z-score, we can use one of two methods:

1. Use the .

2. Use the .

The following examples show how to use each of these methods in practice.

Example 1: Area to the Right of Negative Z-Score

The weight of a certain species of dolphins is normally distributed with mean μ = 300 pounds and standard deviation σ = 15 pounds. Approximately what percentage of dolphins weigh more than 284 pounds?

The z-score for a weight of 284 pounds would be calculated as z = (284 – 300)  / 15 = -1.07

We can use one of two methods to find the area to the right of this z-score:

Method 1: Use z table.

To find the area to the right of the z-score, we can simply look up the value -1.07 in the :

This represents the area to the left of z = -1.07. Thus, the area to the right is calculated as 1 – 0.1423 =  is 0.8577.

Method 2: Use Area to the Right of Z-Score Calculator

We can also use the to find that the area to the right of z = -1.07 is 0.8577.

Example 2: Area to the Right of Positive Z-Score

The scores on a certain exam are normally distributed with mean μ = 85 and standard deviation σ = 8. Approximately what percentage of students score greater than 87 on the exam?

The z-score for an exam score of 87 would be calculated as z = (87 – 85)  / 8 = 0.25

We can use one of two methods to find the area to the right of this z-score:

Method 1: Use z table.

To find the area to the right of the z-score, we can simply look up the value 0.25 in the :

The represents the area to the left of z = 0.25. Thus, the area to the right is calculated as 1 – 0.5987 = 0.4013. Applied to our scenario, this means approximately 40.13% of students score greater than 87 on this exam.

Method 2: Use Area to the Right of Z-Score Calculator

We can also use the to find that the area to the right of z = 0.25 is 0.4013.

The following tutorials provide additional information on how to work with z-scores: