Question

A data consists of n observations:
x₁, x₂,......,xₙ . If ⁿ∑ᵢ₌₁ (xᵢ + 1)² = 9n and ⁿ∑ᵢ₌₁ (xᵢ - 1)² = 5n, then the standard deviation of this data is:
(A) 5 (B) √5
(C) √7 (D) 2

Answer 1

Option B: The standard deviation is $\sqrt{5}$

Explanation:

Given that the data consists of n observations.

Also, given that $^n_{i=1}\sum (x_i+1)^2=9n$ and $^n_{i=1}\sum (x_i-1)^2=5n$

We need to determine the value of the standard deviation of the data.

The formula to determine the standard deviation is given by

$SD=\sqrt{Variance}$

Thus, we have,

$^n_{i=1}\sum (x_i+1)^2=9n$ --------(1)

$^n_{i=1}\sum (x_i-1)^2=5n$ --------(2)

Adding the equations (1) and (2), we have,

$^n_{i=1}\sum (x_i+1)^2+^n_{i=1}\sum (x_i-1)^2=9n+5n$

Expanding the terms, we get,

$^n_{i=1}\sum (x_i^2+2x_i+1)+^n_{i=1}\sum (x_i^2-2x_i+1)=14n$

Simplifying the terms, we get,

$^n_{i=1}\sum x_i^2+2^n_{i=1}\sum x_i+n+^n_{i=1}\sum x_i^2-2^n_{i=1}\sum x_i+n=14 n$

Adding, we get,

$2(^n_{i=1}\sum x_i^2+n)=14 n$

    $^n_{i=1}\sum x_i^2+n=7 n$

           $^n_{i=1}\sum x_i^2=6 n$

             $\frac{^n_{i=1}\sum x_i^2}{n} =6$  ----------(3)

Similarly, subtracting the equations (1) and (2), we have,

               $^n_{i=1}\sum (x_i+1)^2-^n_{i=1}\sum (x_i-1)^2=9n-5n$

$^n_{i=1}\sum (x_i^2+2x_i+1)-^n_{i=1}\sum (x_i^2-2x_i+1)=4n$

Simplifying the terms, we get,

$^n_{i=1}\sum x_i^2+2^n_{i=1}\sum x_i+n-^n_{i=1}\sum x_i^2+2^n_{i=1}\sum x_i-n=4 n$

Adding, we get,

$4^n_{i=1}\sum x_i=4 n$

  $\frac{^n_{i=1}\sum x_i}{n} =1$  -----------(4)

The variance can be determined by subtracting (3) and (4)

Thus, we have,

Variance = 6 - 1 =5

Thus, the standard deviation is given by

$SD=\sqrt{5}$

Thus, the standard deviation is $\sqrt{5}$

Therefore, Option B is the correct answer.

Learn more:

(1) Compute variance and standard deviation of the following data observations. 9,12,15,18,21,24,27

brainly.in/question/14753803

(2) The standard deviation of 20 observations is √6. If each observation is multiplied by 3, find

the standard deviation and variance of the resulting observations.

brainly.in/question/15168963