Let x 1 , x 2 , …… , x n be n observations, and let x be their arithematic mean and σ 2 be their variance. Statement 1 : Variance of 2 x 1 , 2 x 2 , …… , 2 x n is 4 σ 2 . Statement 2 : Ari

Question

Let x 1 ​ , x 2 ​ , …… , x n ​ be n observations, and let x be their arithematic mean and σ 2 be their variance. Statement 1 : Variance of 2 x 1 ​ , 2 x 2 ​ , …… , 2 x n ​ is 4 σ 2 . Statement 2 : Arithmetic mean of 2 x 1 ​ , 2 x 2 ​ , … .. , 2 x n ​ is 4 x .

TestHub · Accepted Answer

σ 2 = ∑ n x i 2 ​ ​ − ( ∑ n x i ​ ​ ) 2 Variance of 2 x 1 ​ , 2 x 2 ​ , … .. , 2 x n ​ = ∑ n ( 2 x i ​ ) 2 ​ − ( ∑ n 2 x i ​ ​ ) 2 = 4 [ ∑ n x i 2 ​ ​ − ( ∑ n x i ​ ​ ) 2 ] = 4 σ 2 Statement 1 is true. A.M. of 2 x 1 ​ , 2 x 2 ​ , …… , 2 x n ​ = n 2 x 1 ​ + 2 x 2 ​ + ⋯ + 2 x n ​ ​ = 2 ( n x 1 ​ + x 2 ​ + ⋯ + x n ​ ​ ) = 2 x ˉ Statement 2 is false.

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JEE Main 2028 (Class 11)