# The standard deviation of the mean

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5 APPENDIX: Proof that E(Σ 2 ) = Startig with eq. (4), we shall compute E(Σ 2 ) = It is coveiet to rewrite the above equatio by otig that after usig E[(x i x) 2 ]. (8) Var(x i x) = E[(x i x) 2 ] [E(x i x)] 2 = E[(x i x) 2 ], Thus, eq. (8) ca be rewritte as E(x i x) = E(x i ) E(x) = µ µ = 0. E(Σ 2 ) = Var(x i x). To evaluate the above expressio, we shall use Var(cx) = c 2 Var(x) ad Var(x + y) = Var(x)+Var(y), where the latter holds uder the assumptio that x ad y are idepedet radom variables. Sice x i ad x are ot idepedet radom variables (sice x cotais x i i its defiitio), we must perform the followig maipulatio, x i x = x i ( ) x i = x i x j. Cosequetly, E(Σ 2 ) = = = = = = [ ( ) ] Var x i x j ( ) 2 Var(x i )+ Var(x 2 j ) ( ) ( ) 2 ( ) [+] = σ2 = σ2, which completes the proof. Note that this computatio justifies the presece of the deomiator factor rather tha i eq. (4). 5

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