# How to Convert Z-Scores to Raw Scores (Step-by-Step)

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.

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