Theorem 1: Let x̄ and ȳ be the sample means of two sets of data of size nx and ny respectively. If x and y are normal, or nx and ny are sufficiently large for the Central Limit Theorem to hold, and x and y have the same variance, then the random variable
has distribution T(nx + ny – 2) where
Proof: Let σ be the common standard deviation of x and y. Then x̄ – ȳ has a normal distribution with mean µx – µy and standard deviation
Defining z as follows, we know that z has distribution N(0, 1).
We also know that has distribution χ2(nx – 1) and has distribution χ2(ny – 1), and so
has distribution χ2(nx + ny – 2).
Defining t = z/u, where m = nx + ny – 2, it follows by Property A of Basic Concepts of t Distribution that t has distribution T(m).
where s is defined as in the statement of the theorem.