The mean and standard deviation of six observations are 8 and 4, respectively. If each observation is multiplied by 3, find the new mean and new standard deviation of the resulting observations.

Explanation

Let observations are `x_(1),x_(2),x_(3),x_(4),x_(5),x_(6)`.
Now `n=6`
Mean `barx=8`
`implies (sumx_(i))/6=8impliessumx_(1)=48`…………….1
Standard deviation `sigma=4implies sigma^(2)=16`
`implies (sum(x_(i)-barx)^(2))/6=16`
`implies sum(x_(i)-barx)^(2)=96`
Now, new observations `=3x_(1),3x_(2),3x_(3),3x_(4),3x_(5),3x_(6)`
New mean `barx=(3x_(1)+3x_(2)+3x(3)+3x_(4)+3x_(5)+3x_(6))/6`
`=(3(x_(1)+x_(2)+x_(3)+x_(4)+x_(5)+x_(6)))/6`
`=(3.sumx_(i))/6=(3xx48)/6=24`
and standard deviation `=sqrt((sum(3x_(i)-3barx)^(2))/6)`
`=sqrt((sum3^(2)(x_(i)-barx)^(2))/6)`
`=sqrt((9sum(x_(i)-barx)^(2))/6)=sqrt(3/2xx96)`
`=sqrt(144)=12`