How can you tell if a z score is unusual?

Answer 1

z scores are assumed to have come from a normal population with mean zero and standard deviation one. So the farther a z score deviates from the mean of zero the more significant, or unusual, it must be taken to be.

The measure of 'how significant' is the probability of getting a bigger z score.

For example, the probability of getting a z score bigger than 1.96 is about 0.025, the probability of getting one bigger that 3.5 about 0.000233.

Now the obvious problem is, how small is small? For a lot of scientific work either 0.05 is considered 'small enough'. Sometimes 0.01 is demanded. I notice that for recent Higg's boson work scientists have only accepted results associated with even smaller probabilities for some reason.

When you need to calculate probabilities such as these you can use published tables, or there are various kinds of software for desktop use. I have just used wolframalpha.com on the web with the following kind of input:

P[X<3.5] for X~normal with mean 0 standard deviation 1

(You will notice the z score of 3.5 in this expression.)