Question

If the scores on a test have a mean of 26 and a standard deviation of 4, what is the z-score for a score of 18?

• A. 2
• B. 11
• C. -2
• D. -41

Hint:

A positive z-score indicates a value above the mean, while a negative z-score indicates a value below the mean.
A z-score of 0 is exactly equal to the mean.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
A z-score (or standard score) measures how many standard deviations an individual raw score (\(X\)) lies above or below the population mean (\(\mu\)).
It is a dimensionless value used to compare scores from different normal distributions.
Key Formula or Approach:
The formula used to calculate a z-score is:
\[ z = \frac{X - \mu}{\sigma} \]
Where:
- \(X\) is the individual raw score.
- \(\mu\) is the population mean.
- \(\sigma\) is the population standard deviation.

Step 2: Detailed Explanation:
We are given the following values from the problem:
- Mean (\(\mu\)) = 26
- Standard deviation (\(\sigma\)) = 4
- Raw score (\(X\)) = 18
Now, we substitute these values into the z-score formula:
\[ z = \frac{18 - 26}{4} \]
Calculate the difference in the numerator:
\[ 18 - 26 = -8 \]
The negative sign indicates that the raw score of 18 lies below the population mean of 2
Now, divide by the standard deviation in the denominator:
\[ z = \frac{-8}{4} = -2 \]
A z-score of -2 means that the score of 18 is exactly two standard deviations below the population mean.

Step 3: Final Answer:
Therefore, the z-score for a score of 18 is -