Sam

• Jun 5th, 2007

 

Which of the following set of numbers has the highest Standard deviation?

a) 1,0,1,0,1,0

b)-1,-1,-1,-1,-1,-1

c) 1,1,1,1,1,1

d) 1,1,0,-1,0,-1

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Formula

Mean = (x1+x2+...+x6)/6
SD= SrRoot( 1/6 * [ x1- Mean]^2+[x2- Mean]^2+...[x6-Mean]^2 )

Note : The Number : 6 denotes no. of terms in a particular option and ^2 - denotes Square
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Take Option (a)

Mean(M) = (1+0+1+0+1+0)/6 = 3/6 = 1/2

SD = Sqroot( 1/6 * [ (1 - 1/2)^2 + (0-1/2)^2+ (1 -1/2)^2 + (0-1/2)^2 + (1 - 1/2)^2 + (0-1/2)^2 ] )
         = Sqroot( 1/6 *  [ 1/4 + 1/4 +1/4 + 1/4 + 1/4 +1/4] )
         = Sqroot( 1/6 *  6 * 1/4 )
         = Sqroot (1/4) = 1/2 = 0.5


For option  (b)

Mean = -6/6= -1

SD = Sqroot( 1/6 * [ (-1 - (-1))^2 + (-1 - (-1))^2 + (-1 - (-1))^2 + (-1 - (-1))^2 + (-1 - (-1))^2 + (-1 - (-1))^2 ] )
   = Sqroot ( 1/6 * [ 0 + 0 +0 + 0+ 0 + 0 ] )  
   = 0


For option  (c)

Mean = 6/6 = 1

SD = Sqroot (1/6 * [ (1-1)^2 +(1-1)^2 +(1-1)^2 + (1-1)^2 + (1-1)^2 + (1-1)^2 ] )
   = Sqroot (1/6 * 0 )
   = 0

For option  (d)

Mean = 0/6 = 0

SD = Sqroot ( 1/6 * ( [1-0]^2 + [1-0]^2 + [0-0]^2 + [-1-0]^2 + [0-0]^2 + [-1-0]^2 ) )
   = Sqroot ( 1/6 *  ( 1+1+0+1+0+1) )
   = Sqroot(2/3) =  0.816

    

Hence the Soln = Option (d)



For reference : en.wikipedia . org/wiki/Standard_deviation