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Question

5 students of a class have an average height 150 cm and variance 18 cm2. A new student, whose height is 156 cm joined them. The variance (in cm2) of the height of these six students is:

A
20
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B
16
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C
18
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D
22
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Solution

The correct option is A 20
Let x1,x2,x3,x4,x5 be the height of the five students, so
5i=1xi=5×150=750
and 5i=1x2i51502=18
5i=1x2i=5×(18+1502)

Let x6 be the height of the new student.
So x6=156 cm
Now
6i=1xi=750+156=906 and 6i=1x2i=5×(18+1502)+1562

Then, new variance=6i=1x2i6⎜ ⎜ ⎜ ⎜6i=1xi6⎟ ⎟ ⎟ ⎟2=5×(18+1502)+15626(9066)2=5×(18+1502)+15626(151)2=20

Alternate solution :
From given data ,
Let ¯¯¯¯¯x1=150, ¯¯¯¯¯x2=156
σ21=18, σ22=0 (considering only one observation
n1=5,n2=1

¯¯¯x=5(150)+1566=151
Now using combined variance formulae,
σ2=1n1+n2[n1(σ21+d21)+n2(σ22+d22)]
where d1=¯¯¯x¯¯¯¯¯x1=151150=1
d2=¯¯¯x¯¯¯¯¯x2=151156=5σ2=16[5(18+1)+1(02+25)]σ2=1206=20

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