<Dd> ∑ i = 1 n (X i − X _̄) 2 = ∥ X 1 − X _̄ ⋮ X n − X _̄ ∥ 2 . (\ displaystyle \ sum _ (i = 1) ^ (n) (X_ (i) - (\ bar (X))) ^ (2) = (\ begin (Vmatrix) X_ (1) - (\ bar (X)) \ \ \ vdots \ \ X_ (n) - (\ bar (X)) \ end (Vmatrix)) ^ (2).) </Dd> <P> If the data points X i (\ displaystyle X_ (i)) are normally distributed with mean 0 and variance σ 2 (\ displaystyle \ sigma ^ (2)), then the residual sum of squares has a scaled chi - squared distribution (scaled by the factor σ 2 (\ displaystyle \ sigma ^ (2))), with n − 1 degrees of freedom . The degrees - of - freedom, here a parameter of the distribution, can still be interpreted as the dimension of an underlying vector subspace . </P> <P> Likewise, the one - sample t - test statistic, </P> <Dl> <Dd> n (X _̄ − μ 0) ∑ i = 1 n (X i − X _̄) 2 / (n − 1) (\ displaystyle (\ frac ((\ sqrt (n)) ((\ bar (X)) - \ mu _ (0))) (\ sqrt (\ sum \ limits _ (i = 1) ^ (n) (X_ (i) - (\ bar (X))) ^ (2) / (n - 1))))) </Dd> </Dl>

What is degree of freedom in statistics pdf