How to compare two sample populations?

I'm doing some education research and would love some help! I'm guessing I need to learn about Z-statistics, but I'm just a beginner at stats....

We're looking at the scores of students across a country. The average score is 0. It's pretty much a normal distribution, will students doing less well having a negative score. Most scores range from -1 to +1, the standard deviation is -0.36.

I have a group of a particular kind of students, let's call them SS, who have an average mean nationally of -0.47 (and a similar standard deviation to the main population). They aren't doing so well. In comparison, the non-SS students have a mean score of 0.19.

What is the probability that if I have a school with a total of say 100 students, and say X of them are SS, that the SS students do better (have a higher mean) than the non-SS students? Even though on average across the country, they do 0.47 worse?



Less is more. Stay pure. Stay poor.
Is your dependent variable standard normal? Meaning that the values represent how far their scores are away from the overall mean.
I fear that to do this exactly would require t distributions (not just cut points) and possibly non-central t, which is not something I work with.

To do it roughly, calculate the standard error of the difference of means from your population SD (which must be 0.36, not -0.36, since an SD cannot be negative). The SEDoM is the square root of (0.36^2/n1 + 0.36^2/n2) To be clear, ^2 = squared, of course.

Take as the population true mean difference 0.19 - -0.47 = 0.66.

Calculate a z score as (0.66 - 0)/SEDoM and figure out how likely you are to get a z score that large or larger. That is your probability. I think it will turn out to be very unlikely.

A more clearly valid but a bit less informative way to do it would be to put the data from your school into a t-test program and let it calculate a 95% confidence interval for the population difference of means. Then you can see how close it comes to what you know as the population difference for the whole country.