*56*

A **z-score** tells us how many standard deviations away a value is from the mean. We use the following formula to calculate a z-score:

Z-Score =(x – μ) / σ

where:

**x:**A raw data value**μ:**The mean of the dataset**σ:**The standard deviation of the dataset

To convert a z-score into a raw score (or “raw data value”), we can use the following formula:

Raw Score =μ + zσ

The following examples show how to convert z-scores to raw scores in practice.

**Example 1: Annual Incomes**

In a certain city, the mean household annual income is $45,000 with a standard deviation of $6,000.

Suppose a certain household has an annual income with a z-score of 1.5. What is their annual income?

To solve this, we can use the raw score formula:

- Raw score = μ + zσ
- Raw score = $45,000 + 1.5*$6,000
- Raw score = $54,000

A household with a z-score of 1.5 has an annual income of **$54,000**.

**Example 2: Exam Scores**

For a certain math exam, the mean score is 81 with a standard deviation of 5.

Suppose a certain student has an exam score with a z-score of -2. What is their exam score?

To solve this, we can use the raw score formula:

- Raw score = μ + zσ
- Raw score = 81+ (-2)*5
- Raw score = 71

A student with a z-score of -2 received an exam score of **71**.

**Example 3: Plant Heights**

The mean height of a certain species of plant is 8 inches with a standard deviation of 1.2 inches.

Suppose a certain plant has a height with a z-score of 0. What is the height of this plant?

To solve this, we can use the raw score formula:

- Raw score = μ + zσ
- Raw score = 8+ 0*5
- Raw score = 8

A plant with a z-score of 0 is **8** inches tall.

**Additional Resources**

How to Interpret Z-Scores (With Examples)

5 Examples of Using Z-Scores in Real Life