Help using Normalized z-score Data for Statistical Significance

Hello! I am working with cognitive assessment data (Trail A, Rey Figure, Hopkins... etc) and am looking at a sample of individuals to assess their cognitive profile. I am hoping to develop a cognitive profile of the individuals at the end of my analysis. The data I am working with has all been converted to the respective z-scores for each individual (each individual has a z-score for each of the tests that they took rather than the raw score). There is no control in the data I am working with (since all of the data is in the z-score format it is inherently compared to a control). I am hoping that someone can help me with my statistical analysis. Without having a control sample set, is it possible to draw any conclusions about the significance of my population with their Z-scores?

It seems to me that, knowing only 6.68% should have a Z-score of >1.5, if 60% of my population has this Z >1.5 would be significant, but I cannot figure out what statistical analysis to use in order to make this claim (all tests I have used in the past all required the control size).

Thank you for any assistance you can offer; I greatly appreciate any suggestions you have!