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