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