If an observation has a z-score of $0$, which of the following is true?

• A. The observation is $1$ standard deviation above the mean.
• B. The observation is at the maximum value of the distribution.
• C. The observation is equal to the mean of the distribution.
• D. The observation is $1$ standard deviation below the mean.

Answer: (C)

Verified step by step guidance

1. Recall that a z-score represents how many standard deviations an observation is from the mean of the distribution. It is calculated by the formula: \(\displaystyle z = \frac{X - \mu}{\sigma}\), where \(X\) is the observation, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.
2. Understand that if the z-score is exactly 0, it means the numerator in the formula is zero, i.e., \(X - \mu = 0\).
3. From the equation \(X - \mu = 0\), deduce that \(X = \mu\), meaning the observation is exactly equal to the mean of the distribution.
4. Recognize that a z-score of 0 does not indicate the observation is above or below the mean, nor does it indicate it is at the maximum value of the distribution.
5. Therefore, the correct interpretation of a z-score of 0 is that the observation is equal to the mean of the distribution.