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Question

If the mean and standard deviation of 75 observations is 40 and 8 respectively, find the new mean and standard deviation if

(i) Each observation is multiplied by 5.

(ii) 7 is added to each observation.


Solution

Given: ¯X=40, σ=8, N=75

(i) A "change of scale" affects the value of both ¯X and σ. A change of scale occurs when each observation is multiplied by a constant. As a result to that, ¯X and σ also get multiplied by that constant.

In our question, that constant is 5.

Hence,

New ¯X= Old ¯X×5

=40×5

=200

and,

New σ= Old σ×5

=8×5

=40

(ii) A "change of origin" affects the value of ¯X but not that of σ. A change of origin occurs when each observation is increased by a constant. As a result to that, ¯X also increases by the same constant, while σ remains unchanged.

In our question, that constant is 7.

Hence,

New ¯X= Old ¯X+7

=40+7

=47

while,

New σ= Old σ

=8


Economics

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