How To Convert Z-Scores To Raw Scores (Step-by-Step) - - Statology

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by Zach Bobbitt Posted on Last updated on August 30, 2021

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

Zach Bobbitt

Hey there. My name is Zach Bobbitt. I have a Masters of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike.  My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.

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