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The mean square deviations of a set of observations `x_(1),x_(2), …, x_(n)` about a point c is defined to be `(1)/(n) sum_(i=1)^(n)(x_(i)-c)^(2)`. The mean square deviations about -1 and +1 of a set of observations are 7 and 3, repectively. Find the standard deviation of this set of observations. |
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Answer» Correct Answer - `sqrt(3)` Mean square deviations `=(1)/(n) sum_(i=1)^(n)(x_(i)-c)^(2)`, about c. Also, given that mean square deviation about -1 and +1 are 7 and 3, respectiveyl. `rArr (1)/(n)sum_(i=1)^(n)(x_(i)+1)^(2)=7 and (1)/(n)sum_(i=1)^(n)(x_(i)-1)^(2)=3` `rArr sum_(i=1)^(n)x_(i)^(2)+2sum_(i=1)^(n)x_(i)+n=7n and sum_(i=1)^(n)x_(i)^(2)-2sum_(i=1)^(n)x_(i) +n-3n` `rArr sum_(i=1)^(n)x_(i)^(2)+2sum_(i=1)^(n)x_(i) =6n and sum_(i=1)^(n)x_(i)^(2) -2sum_(i=1)^(n)x_(i)=2n` `rArr sum_(i=1)^(n)x_(i)-n rArr bar(x) =(sum_(i=1)^(n)x_(i))/(n)=1` `therefore` Standard deviation `=sqrt((1)/(n)sum_(i=1)^(n)(x_(i)-bar(x))^(2))=sqrt((1)/(n)sum_(i=1)^(n)(x_(i)-1)^(2))=sqrt(3)` |
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