The mean of a z-score is always zero. Does this mean that half of a set of z scores will be negative and half positive? Explain.

No, the mean of a z-score being always zero does not necessarily mean that half of a set of z scores will be negative and half positive.

Z-scores are a measure of how many standard deviations an element is from the mean. A positive z-score indicates the data point is above the mean, while a negative z-score indicates it is below the mean.

However, the distribution of these scores above and below the mean is not necessarily even. It depends on the distribution of the original data. If the original data is symmetrically distributed around the mean, then yes, half of the z-scores would be positive and half would be negative. But if the original data is skewed, then the z-scores would reflect that skewness.

For example, if the original data is positively skewed (i.e., more data points are below the mean), then more z-scores would be negative. Conversely, if the original data is negatively skewed (i.e., more data points are above the mean), then more z-scores would be positive.

So, while the mean of the z-scores is always zero, the distribution of z-scores above and below the mean depends on the distribution of the original data.