Question 7

The mean is equal to zero with a standard deviation of 1.

A basic assumption of a z-score is that it is standardized in such a way that the mean of the distribution is zero and the standard deviation is one. This allows for comparisons across different datasets.

A) The mean is equal to zero with a standard deviation of 1.

This option correctly describes the fundamental property of z-scores. When calculating a z-score, the values are normalized so that the resulting distribution has a mean of zero and a standard deviation of one, facilitating easier comparison of data points across different scales.

B) Outlier data points are critical to a z-score calculation.

This option is incorrect because while outliers can affect the mean and standard deviation, they are not inherently critical to the calculation of a z-score itself. Z-scores can be calculated for any data point regardless of whether it is an outlier, but the presence of outliers may distort the interpretation of the z-scores.

C) Outlier data points must be eliminated from a z-score calculation.

This option is also incorrect. Outliers do not need to be eliminated when calculating z-scores; they can be included in the calculation. However, their presence may lead to skewed results in terms of the mean and standard deviation, which in turn affects the z-scores of all data points.

D) The mean is equal to zero with a standard deviation of 2.

This statement is incorrect as it misrepresents the fundamental characteristics of a z-score. The standard deviation of a z-score distribution is always 1, not 2, making this option fundamentally flawed.

Conclusion

The correct answer, that the mean is equal to zero with a standard deviation of 1, captures the essence of z-scores as a standardized measure. All other options fail to accurately describe the properties of z-scores, either by misrepresenting their calculation or by suggesting unnecessary conditions regarding outlier data points.