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Question

If the standard deviation of a variable X is σ, then the standard deviation of variable aX+bc, is

A
aσ
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B
acσ
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C
acσ
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D
aσ+bc
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Solution

The correct option is C acσ
If the S.D. of variate X is .
σ(X)=Ni=1(Xi¯¯¯¯¯X)2N
For (aX+b) ,
Mean, (aX1+b)+(aX2+b)+...+(aXN+b)N
=a(X1+X2+...+XN)N+b
=a¯¯¯¯¯X+b
σ(aX+b)=Ni=1(xi¯¯¯x)2N
=[(aX1+b)(a¯¯¯¯¯X+b)]2+[(aX2+b)(a¯¯¯¯¯X+b)]2+....+[(aXN+b)(a¯¯¯¯¯X+b)]2N
= a2[(X1¯¯¯¯¯X)2+(X2¯¯¯¯¯X)2+...+(XN¯¯¯¯¯X)2]N
=|a| Ni=1(Xi¯¯¯¯¯X)2N
=|a|σ(X)
For (aX+b)c

σ(aX+b)c=acσ(X)

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