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The conversion of z-scores to raw scores is done by using the formula z= (x-µ)/σ, where x is the raw score, µ is the mean, and σ is the standard deviation. The z-score is then solved for x, which is the raw score. This conversion is important in order to compare the raw score to the mean and standard deviation of the data set.
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.