For 10 observations x₁, x₂, , x₁₀ , if _ i=1 ¹⁰(x_i+2)^2=180 and _ i=1 ¹⁰(x_i-1)^2=90 , then their standard deviation is:

Step-by-step solution

Given _ i=1 ¹⁰(x_i+2)^2=180 _ i=1 ¹⁰ x_i^2 + 4 _ i=1 ¹⁰ x_i + 40 = 180 _ i=1 ¹⁰ x_i^2 + 4 _ i=1 ¹⁰ x_i = 140 Also given _ i=1 ¹⁰(x_i-1)^2=90 _ i=1 ¹⁰ x_i^2 - 2 _ i=1 ¹⁰ x_i + 10 = 90 _ i=1 ¹⁰ x_i^2 - 2 _ i=1 ¹⁰ x_i = 80 Subtracting the second equation from the first: 6 _ i=1 ¹⁰ x_i = 60 _ i=1 ¹⁰ x_i = 10 Substituting this value in the second equation: _ i=1 ¹⁰ x_i^2 - 2(10) = 80 _ i=1 ¹⁰ x_i^2 = 100 The standard deviation is given by: = x_i^2 n - ( x_i n )^2 = 100 10 - ( 10 10 )^2 = 10 - 1 = 9 = 3 Answer: 3