You'd be interested in learning about the origin and application of Chi-Squared tests (or Fisher's exact test). Detecting changes in population means, and making sure those changes aren't just random noise, is quite straightforward and can be done by hand.
It won't be possible for me to explain the answer to your satisfaction, if you lack at least a basic education in probability & statistics. Maybe Khan Academy has an intro level course, or a "Stats for Humanities" variant that can help you get started.
