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**.

**Additional Resources**

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