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?



Not a robit
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.