Questions: In a distribution of Z scores, the mean is always

Solution Steps

A \( Z \)-score, also known as a standard score, measures how many standard deviations a data point is from the mean of the distribution. It is calculated using the formula: \[ Z = \frac{X - \mu}{\sigma} \] where \( X \) is the data point, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.

When calculating \( Z \)-scores for an entire dataset, the mean of the \( Z \)-scores is always 0. This is because the \( Z \)-score formula centers the data around the mean by subtracting \( \mu \) from each data point.

To confirm, consider the average of all \( Z \)-scores: \[ \text{Mean of } Z = \frac{1}{n} \sum_{i=1}^{n} Z_i = \frac{1}{n} \sum_{i=1}^{n} \left( \frac{X_i - \mu}{\sigma} \right) \] Since \( \mu \) is the mean of \( X \), the sum \( \sum_{i=1}^{n} (X_i - \mu) \) equals 0. Therefore: \[ \text{Mean of } Z = \frac{1}{n} \cdot 0 = 0 \]

Final Answer