One of the easiest statistical tests to carry out using pen and paper is the Paired (Dependent) t Test. This is a test to find a significant difference between two sets of sample data (normal distribution assumed) - the easiest way I know to remember what is needed for this type of test is that you have one group measured twice. Most commonly this would be a "before" and "after" situation. This is different from an unpaired (independent) t Test where you would have two groups measured once each.
You don't neet to calculate the mean, variance, or standard deviation, which can be time-consuming (especially in an exam situation). The solution is based on the differences between each pair - so if you can add, subtract, divide, and get the square root of a number, you can do a paired t Test. Here's my short video showing how to do this test with a small sample by hand: