The mean and standard deviation of 10 observations x _ 1 , x _ 2 , x _ 3 ..... x _ 10 are ¯ x and σ respectively. Let 10 is added to x _ 1 , x _ 2 ..... x _ 9 and 90 is subtracted from x _ 10 . If still, the standard

Correct answer: B. 45 ∑ _ i = 1 ^ 10 ( x _ i - ¯ x )2 10 = σ ^ 2 … (1) ∑ _ i = 1 ^ 9 ( x _ i + 10 - ¯ x )2 + ( x _ 10 -90 - ¯ x )2 10 = σ ^ 2 … (2) From (1) and ( 2 ) ∑ _ i = 1 ^ 10 ( x _ i - ¯ x )2 10 = ∑ _ i = 1 ^ 10 ( x _ i - ¯ x )2 + 20 ∑ _ i = 1 ^ 9 ( x _ i - ¯ x ) - 180( x _ 10 - ¯ x ) + 100 × 9 + 8100 10 2 × ∑ _ i = 1 ^ 9 ( x _ i - ¯ x ) +2( x _ 10 - ¯ x ) -20( x _ 10 - ¯ x ) +90 + 810 = 0 ⇒ 900 = 20( x _ 10 - ¯ x ) ⇒ x _ 10 - ¯ x = 45

NTA Abhyas 2020

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

NTA Abhyas 2020 · Mathematics · Statistics

The mean and standard deviation of 10 observations x _ 1 , x _ 2 ,...

The mean and standard deviation of

10

observations

{x}_{1}, {x}_{2}, {x}_{3}..... {x}_{10}

are

\overset{¯}{x}

and

\sigma

respectively. Let

10

is added to

{x}_{1}, {x}_{2}..... {x}_{9}

and

90

is subtracted from

{x}_{10}.

If still, the standard deviation is the same, then

{x}_{10} - \overset{¯}{x}

is equal to

Options

A $35$
B $45$
C $55$
D $50$

Correct answer

B. $45$

Step-by-step solution

\mathop{∑}_{i = 1}^{10}\dfrac{ \left({x}_{i} - \overset{¯}{x}\right)2}{10} = {σ}^{2}

…

\left(1\right)

\dfrac{\mathop{∑}_{i = 1}^{9}\left({x}_{i} + 10 - \overset{¯}{x}\right)2 + \left({x}_{10} -90 - \overset{¯}{x}\right)2}

{10} = {σ}^{2}

… \left(2\right)

<br />From \left(1\right)

and ( 2 )

\dfrac{\mathop{∑}_{i = 1}^{10}\left({x}_{i} - \overset{¯}{x}\right)2}{10} = \dfrac{\mathop{∑}_{i = 1}^{10}\left({x}_{i} - \overset{¯}{x}\right)2 + 20\mathop{∑}_{i = 1}^{9}\left({x}_{i} - \overset{¯}{x}\right) - 180\left({x}_{10} - \overset{¯}{x}\right) + 100 \times 9 + 8100}{10}

2 \times \mathop{∑}_{i = 1}^{9}\left({x}_{i} - \overset{¯}{x}\right) +2\left({x}_{10} - \overset{¯}{x}\right) -20\left({x}_{10} - \overset{¯}{x}\right) +90 + 810 = 0

⇒ 900 = 20\left({x}_{10} - \overset{¯}{x}\right)

⇒ {x}_{10} - \overset{¯}{x} = 45

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