The variance of 20 observations is 5. If each observation is multiplied by 2, the new variance of the resulting observations is

Let the observations be x1,x2,…,x20 and x¯ be their mean. Given that, variance = 5 and n =20. We know that Variance σ2=1n∑i=120 xi−x¯2 ,  Or  5=120∑i=120 xi−x¯2 Or  ∑i=120 xi−x¯2=100-------(1) If each observation is multiplied by 2 , and the new resulting  observations are yi , then yi=2xi i.e., xi=12yi Therefore, y¯=1n∑i=120 yi=120∑i=120 2xi=2⋅120∑i=120 xi e.. y¯=2x¯ or x¯=12y¯ Substituting the values of xi and x in (1), we get ∑i=120 12yi−12y¯2=100 , i.e., ∑i=120 yi−y¯2=400 Thus, the variance of new observations =120×400=20  ALTIER  :     Variance=   V a X +b = a2 VX                    so  V2 X =22  VX =45=20