Under which of the following conditions will a data value have a negative z-score when using the formula $z=\frac{x-\mu }{\sigma }$?

• A. When the standard deviation $\sigma$ is negative
• B. When the data value $x$ is equal to the mean $\mu$
• C. When the data value $x$ is less than the mean $\mu$
• D. When the data value $x$ is greater than the mean $\mu$

Answer: (C)

Verified step by step guidance

1. Recall the formula for the z-score: \(z = \frac{x - \mu}{\sigma}\), where \(x\) is the data value, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
2. Understand that the standard deviation \(\sigma\) is always a positive number because it measures the spread of data and cannot be negative.
3. Focus on the numerator \(x - \mu\): if \(x\) is less than \(\mu\), then \(x - \mu\) will be negative.
4. Since \(\sigma\) is positive, dividing a negative numerator by a positive denominator results in a negative z-score.
5. Therefore, a data value will have a negative z-score when it is less than the mean \(\mu\).