Statistics: Comparing z scores across distributions

[Quantum1990]

Homework Statement

This problem is taken from the 2011 AP Statistics Exam, which we reviewed in class. The exact problem can be found here, and solution after:

http://apcentral.collegeboard.com/apc/public/repository/ap11_frq_statistics.pdf
http://apcentral.collegeboard.com/apc/public/repository/ap11_statistics_scoring_guidelines.pdf
Its Question 1, part c.

However, my problem is more general.

Suppose I have two players, each with measured running times and weight lifting amounts. It has already been concluded that at least one of these distributions is not normal. The z score table is reproduced(approximately) below:
A B
Amount Held 2.4 2.6

Running Time -1.2 -.2

According to the solution, A is slightly worse at weight lifting, but much greater at running, thus A is the overall better player. However, in concluding that, we saw that the B led A in amount held by .2, and A led B in running time by 1.0. We compared the two z scores, and determined A's ability in running overshadows B's ability in amount held. But by doing this, aren't we comparing z scores(or differences in them) across distributions? Specifically, how do we know that a 1.0 lead in running is more impressive than a .2 lead in Amount held, if we don't know the skewness of the distributions?

This was a long question, and I wasn't quite sure how to phrase it. Any help in understanding this would be great.

Homework Equations

The Attempt at a Solution

 

[nate808]

it is important to always consider the limitations and assumptions of any statistical analysis. In this case, the conclusion that player A is overall better is based on the assumption that the distributions of weight lifting and running times are both approximately normal. However, if this assumption is not met, then the conclusion may not be valid.

Additionally, the z-score table only allows for comparisons within a single distribution, not between different distributions. Therefore, comparing the z-scores of weight lifting and running times may not accurately reflect the true differences between the two players' abilities.

Furthermore, as you mentioned, the skewness of the distributions is also an important factor to consider. If the distributions are highly skewed, then the z-scores may not accurately represent the players' abilities. In this case, it may be more appropriate to use non-parametric methods of analysis that do not rely on the assumption of normality.

In conclusion, while the analysis presented in the solution may provide some insights, it is important to consider the limitations and assumptions of the statistical methods used and to be cautious in drawing conclusions based on them. It may be beneficial to explore alternative methods of analysis and to further investigate the distributions of the players' abilities.