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Converting Z-scores to raw scores involves using a mathematical formula to transform standardized scores (Z-scores) into their original, unstandardized form (raw scores). This process is useful in statistics and data analysis, as it allows for comparison and interpretation of scores on different scales. To convert Z-scores to raw scores, one must first calculate the mean (μ) and standard deviation (σ) of the original dataset. Then, using a simple formula (X = Zσ + μ), the Z-score can be multiplied by the standard deviation and added to the mean to obtain the raw score. This process can be repeated for each Z-score in the dataset, resulting in a set of corresponding raw scores. By following these steps, one can accurately convert Z-scores to raw scores and use them for further analysis and interpretation.
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.