Difficulty: Easy

Average Score: 100%

What is a basic assumption of a z-score?

Rationale

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

A basic assumption of a z-score is that it standardizes the distribution of data such that the mean becomes zero and the standard deviation becomes one. This allows for the comparison of individual scores to the overall distribution, facilitating the understanding of how far a score deviates from the mean in terms of standard deviations.

A (The mean is equal to zero with a standard deviation of 2.) YOUR ANSWER

This statement is incorrect because the standardization process of a z-score specifically involves adjusting the data so that the standard deviation is 1, not 2. A standard deviation of 2 would indicate a broader spread of data, which is not consistent with the properties of z-scores.

B (The mean is equal to zero with a standard deviation of 1.) CORRECT

This statement correctly reflects the fundamental principle of z-scores in statistics. When data is transformed into z-scores, the resulting distribution will always have a mean of 0 and a standard deviation of 1, allowing for easier interpretation and comparison of different data sets.

C (Outlier data points are critical to a z-score calculation.) YOUR ANSWER

While outliers can affect the calculation of z-scores, they are not critical. In fact, outliers can skew the mean and standard deviation, potentially leading to misleading z-scores. This choice misrepresents the role of outliers in the context of z-score calculations.

D (Outlier data points must be eliminated from a z-score calculation.) YOUR ANSWER

This statement is misleading because while removing outliers may sometimes improve the accuracy of statistical analyses, it is not a requirement for calculating z-scores. Z-scores can be computed with outliers present, but their influence on the mean and standard deviation should be acknowledged.

Conclusion

A z-score transforms a data set such that its mean is 0 and its standard deviation is 1, enabling clear comparisons across different distributions. The only correct assumption among the options provided is that the mean is zero and the standard deviation is one. The treatment of outliers can vary depending on the analysis context, but it does not alter the fundamental definition of a z-score.

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