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Question

If ¯¯¯x1 and ¯¯¯x2 are the means of two distributions such that ¯¯¯x1<¯¯¯x2 and ¯¯¯x is the mean of the combined
distribution, then

A
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B
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C
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D
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Solution

The correct option is B
Let n1 and n2 be the number of observations in two groups having means ¯¯¯¯¯x1 and ¯¯¯¯¯x2
Respectively. Then,¯¯¯x=n1¯¯¯x1+n2¯¯¯x2n1+n2
Now, ¯¯¯x¯¯¯x1=n1¯¯¯x1+n2¯¯¯x2n1+n2¯¯¯x1
=n2(¯¯¯x2¯¯¯x1)n1+n2=0,[¯¯¯x2>¯¯¯x1]
¯¯¯x>¯¯¯x1.....(i)
And¯¯¯x>¯¯¯x2=n(¯¯¯x1¯¯¯x2)n1+n2<0,[¯¯¯x2>¯¯¯x1]
¯¯¯x>¯¯¯x2....(ii)
From (i) and (ii) ¯¯¯x1<¯¯¯x<¯¯¯x2.

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