One of the most basic problems that we need to solve in statistics is comparing the means from two (or more) groups. It's tempting to just take those means and compare them while ignoring all of the statistical theory. The central problem is that, if we did that, we would not have a reference level that we can compare that difference against (we wouldn't know whether that difference is large or small).

The statistical approach provides a foundation for this comparison, providing us with critical values that we should do this comparison against. In essence, this comparison depends on the variability in the data (the noisier the data is, the greater this difference needs to be to be deemed significative) and on how certain we want to be that a non-significative difference is considered significative (this is called the  value, which is also known as type 1 error). 

This test can be extended to more complex scenarios, such as comparisons between paired observations (that is, measurements before and after treatment is applied to certain subjects), multiple groups (this is called ANOVA), or multivariate data. In this chapter, we will review certain techniques that are relevant for these situations.