# How to aggregate multiple Z-scores

Is there a way to aggregate multiple Z-scores to get a single Z-score that corresponds to the probability that no null hypotheses are rejected?

Backstory: We have a tool that has some error E that it adds to each measurement. Assume E is normally distributed with mean 0 and standard deviation 1. Given multiple readings from the tool, I want to find the Z-score that corresponds to the probability that none of the true values exceed a certain threshold T.

Let's assume that the error in each measurement is independent.

EDIT: Here is an example. Suppose I have a slightly imperfect scale where the difference (in lbs) between the reported and true weights follows a standard normal distribution.

I weigh 3 objects and get readings of 1,2, and 3 lbs. What is the likelihood that at least one of the objects weighs 4 or more lbs?

Is there a way to solve this question with only a single Z-score lookup at the end?

• There is such a mixture of different concepts and procedures here that it is difficult to tell what is being asked. Could you perhaps articulate one clear hypothesis you would like to test or describe very clearly one event whose probability you would like to compute? – whuber Nov 27 '18 at 0:39
• @whuber Added example. Does that help? – Jack Nov 27 '18 at 17:35
• Yes, it does. It looks like your question is equivalent to several others that have been asked: namely, to find the distribution of the maximum of independent but non-identically distributed Normal variables. Would that be a correct interpretation? – whuber Nov 27 '18 at 17:55
• @whuber Kind of. Can you explain a little bit more? Also, I'm trying to limit the number of times I have to lookup a Z-score. – Jack Nov 27 '18 at 18:25