Multiple choice

The mean square deviation of set of $n$ observations ${ x }{ 1 },{ x }{ 2 },.....{ x }{ n }$ about a point $c$ is defined as $\displaystyle \frac { 1 }{ n } \sum _{ i=1 }^{ n }{ { \left( { x }{ i }-c \right) }^{ 2 } } $ The mean square deviation about $-2$ and $2$ are $18$ and $10$ respectively, then standard deviation of this set of observations is

  1. $3$
  2. $2$
  3. $1$

  4. none of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Mean square deviation about c is E[(X-c)^2] = E[X^2] - 2cE[X] + c^2. Given E[(X+2)^2] = 18 and E[(X-2)^2] = 10. Expanding: E[X^2] + 4E[X] + 4 = 18 and E[X^2] - 4E[X] + 4 = 10. Subtracting: 8E[X] = 8, so E[X] = 1. Then E[X^2] + 4 + 4 = 18, so E[X^2] = 10. Variance = E[X^2] - (E[X])^2 = 10 - 1 = 9. Standard deviation = sqrt(9) = 3.

AI explanation

Use the mean square deviation identity, which is (1/n) times the sum of (x_i - c)^2, equal to variance plus the square of the difference between the mean and c. Setting up the equations for the given values yields 18 equals variance plus the square of the mean minus two, and 10 equals variance plus the square of the mean minus four. Subtracting the second equation from the first gives 8 equals four times the mean, so the mean is 2. Substituting this into the second equation gives 10 equals variance plus four, making the variance 6; since standard deviation is the square root of variance, it is the square root of 9, which is 3.

Keep practicing — related questions