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Assume X~N(m ,s²) with m & s are unknown and that we draw a sample of size n from this population and denote the sample mean and sample S.D. by  and S respectively.  We define three useful sampling distributions:

FACT 1: The random variable T =  will follow a t distribution with (n-1) degrees of freedom.  NOTE: As n®¥,  the t distribution approaches the N(0,1) distribution.

FACT 2: The random variable  will follow a Chi-Square (C²)distribution with (n-1) degrees of freedom.

Assume X~N(m₁,s₁²) and Y~N(m₂,s₂²) with m₁,m₂ & s₁,s₂ unknown, and that we draw samples of size n₁ and n₂ respectively from these populations and denote the sample S.D. by S₁ and S₂ respectively.

FACT 3: The random variable (S₁²/s₁²)/(S₁²/s₂²) will follow an F distribution with n₁=(n₁-1) and n₂=(n₂-1) degrees of freedom.

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