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The graph below shows the cross-subjects correlation between scores on two different tests, taken by each subject. The scores are z-transformed at group level, separately for each test.

My question is: shouldn't the z-scores for each test by definition add up to zero for each of the two tests? In other words, shouldn't the scatterplot (cloud) be symmetrical around the zero point of both axes?

Could there be a mistake in how this graph was made, or simply in reporting how the z-scores have been computed?

enter image description here

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It's likely that this is a subset of the data that the z-scores were calculated on.

As an example, what if I wanted to correlate two sections on a college admissions test - but I only had (1) the raw scores for a particular testing center, (2) the national mean, and (3) the national std dev.

I would probably normalize (z-score) according to the national average and std dev. If my particular testing center had students who were especially well prepared for the test, then I may see a graph like what you've shown above.

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  • $\begingroup$ That sounds about right, thanks. One related followup question: these authors also claim that the plot demonstrates increased individual differences in these test scores. They presumably base this claim on the fact that the z-scores are quite spread out. But isn't that argument circular, given that z-scores are always going to be spread out, even if the stddev will be low, simply by how they are defined? The fact that they don't mention which subsets are represented on each axis (the dodgy bit about z-scores not adding up to zero) makes the individual differences claim even harder to verify. $\endgroup$
    – z8080
    Feb 22, 2016 at 11:22

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