why is sample mean an unbiased estimator of population mean
Bias of an estimator
X  ̄ = 1 n ∑ i = 1 n X i S 2 = 1 n ∑ i = 1 n ( X i − X  ̄ ) 2 ( \ displaystyle ( \ overline ( X ) ) \ , = ( \ frac ( 1 ) ( n ) ) \ sum _ ( i = 1 ) ^ ( n ) X_ ( i ) \ qquad S ^ ( 2 ) = ( \ frac ( 1 ) ( n ) ) \ sum _ ( i = 1 ) ^ ( n ) ( \ big ( ) X_ ( i ) - ( \ overline ( X ) ) \ , ( \ big ) ) ^ ( 2 ) \ qquad )
X  ̄ = 1 n ∑ i = 1 n X i S 2 = 1 n ∑ i = 1 n ( X i − X  ̄ ) 2 ( \ displaystyle ( \ overline ( X ) ) \ , = ( \ frac ( 1 ) ( n ) ) \ sum _ ( i = 1 ) ^ ( n ) X_ ( i ) \ qquad S ^ ( 2 ) = ( \ frac ( 1 ) ( n ) ) \ sum _ ( i = 1 ) ^ ( n ) ( \ big ( ) X_ ( i ) - ( \ overline ( X ) ) \ , ( \ big ) ) ^ ( 2 ) \ qquad )