If both the mean and the standard deviation of 50 observations x _ 1 , x _ 2 , , x _ 50 are equal to 16 , then the mean of ( x _ 1 - 4) ^ 2 , ( x _ 2 - 4) ^ 2 , , ( x _ 50 - 4) ^ 2 is

Correct answer: C. 400 For observations x _ 1 , x _ 2 ,........... x _ 50 Mean, - x = ∑ x _i^ 50 = 16 ..... (i) Variance, σ ^ 2 = ∑ x _i^ 2 50 - ( - x ) ^ 2 = 16 ^ 2 ⇒ ∑ x _i^ 2 50 = 16 ^ 2 + ( - x ) ^ 2 = 16 ^ 2 + 16 ^ 2 = 512 ..... (ii) So, the mean value of ( x _ 1 - 4 ) ^ 2 , ( x _ 2 - 4 ) ^ 2 ,....... ( x _ 50 - 4 ) ^ 2 will be = ∑ ( x _ i - 4 ) ^ 2 50 = ∑ x _i^ 2 - 8 ∑ x _ i + 16 × 50 50 = ∑ x _i^ 2 50 - 8 ∑ x _ i 50 + 16 = 512 - 8 × 16 + 16 = 400 & (using (i) and (ii))

NTA Abhyas 2020

Exam: NTA Abhyas
Year: 2020
Subject: Mathematics
Chapter: Statistics

NTA Abhyas 2020 · Mathematics · Statistics

If both the mean and the standard deviation of 50 observations x _...

If both the mean and the standard deviation of

50

observations

{x}_{1}, {x}_{2}, \ldots , {x}_{50}

are equal to

16

, then the mean of ${\left({x}_{1} - 4\right)}^{2}, {\left({x}_{2} - 4\right)}^{2}, \ldots , {\left({x}_{50} - 4\right)}^{2} is

Options

A $525$
B $480$
C $400$
D $380$

Correct answer

C. $400$

Step-by-step solution

For observations

{x}_{1}, {x}_{2},........... {x}_{50}

Mean,

\overset{-}{x} = \dfrac{∑ {x}_i^{}}{50} = 16

..... (i)
Variance,

{\sigma}^{2} = \dfrac{∑ {x}_i^{2}}{50} - {( \overset{-}{x} )}^{2} = {16}^{2}

⇒ \dfrac{∑ {x}_i^{2}}{50} = {16}^{2} + {( \overset{-}{x} )}^{2} = {16}^{2} + {16}^{2} = 512

..... (ii)
So, the mean value of

{( {x}_{1} - 4 )}^{2}, {( {x}_{2} - 4 )}^{2},....... {( {x}_{50} - 4 )}^{2}

will be

= \dfrac{∑ {( {x}_{i} - 4 )}^{2}}{50} = \dfrac{∑ {x}_i^{2} - 8 ∑ {x}_{i} + 16 \times 50}{50}

= \dfrac{∑ {x}_i^{2}}{50} - 8\dfrac{∑ {x}_{i}}{50} + 16 = 512 - 8 \times 16 + 16 = 400

& (using (i) and (ii))

Practice Statistics on Quantrex Academy →

More from Statistics

Consider a data consisting of 10 observations x_1, x_2, …, x_ 10 , whose mean is 5 and… JEE Advanced 2026 A set of four observations has mean 1 and variance 13 . Another set of six observations… JEE Main 2026 Let the mean and the variance of seven observations 2, 4, α, 8, β, 12, 14 , α < β , be 8… JEE Main 2026 A data consists of 20 observations x_1, x_2, …, x_ 20 . If ∑_ i=1 ^ 20 (x_i + 5)^2 = 2500… JEE Main 2026 A variable X takes values 0, 0, 2, 6, 12, 20, …, n(n-1) with frequencies ^nC_0, ^nC_1… JEE Main 2026 The mean deviation about the mean for the data x_i 5 7 9 10 12 15 f_i 8 6 2 2 2 6 is… JEE Main 2026 For 10 observations x_1, x_2, …, x_ 10 , if ∑_ i=1 ^ 10 (x_i+2)^2=180 and ∑_ i=1 ^ 10… JEE Main 2026 Suppose that the mean and median of the non-negative numbers 21, 8, 17, a, 51, 103, b… JEE Main 2026 The mean and variance of n observations are 8 and 16 , respectively. If the sum of the… JEE Main 2026 If the mean of the data Class 5-10 10-15 15-20 20-25 25-30 30-35 Frequency 2 k 28 54 k+1… JEE Main 2026 The mean and variance of 10 observations are 9 and 34.2, respectively. If 8 of these… JEE Main 2026 Let X=\ x ∈ ~N: 1 ≤ x ≤ 19\ and for some a, b ∈ ℝ, Y=\ a x+b: x ∈ X\ . If the mean and… JEE Main 2026 The mean and variance of a data of 10 observations are 10 and 2, respectively. If an… JEE Main 2026 If the mean and the variance of the data Class 4-8 8-12 12-16 16-20 Frequency 3 λ 4 7 are… JEE Main 2026 Let the mean and variance of 8 numbers -10,-7,-1, x, y, 9,2,16 be (7)/(2) and (293)/(4)… JEE Main 2026 If the mean deviation about the median of the numbers k, 2 k, 3 k, …. ., 1000 k is 500… JEE Main 2026 Full Statistics list Statistics JEE Maths PYQs JEE Mathematics PYQs All NTA Abhyas JEE Main PYQs