Let x1,x2,…,x100 be 100 observations such that 100∑i=1xi=0,∑1≤i<j≤100|xixj|=80000 and mean deviation from their mean be 5. Then their standard deviation is
A
10
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B
30
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C
40
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D
50
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Solution
The correct option is B30 Given that ∑xi=0 ⇒¯¯¯x=∑xi100=0 Now, ∑|xi−¯¯¯x|100=5 ⇒∑|xi|=500 Squaring both the sides (∑|xi|)2=(500)2 ⇒∑x2i+2∑1≤i≤j≤100|xixj|=(500)2 [∵(∑|xi|)2=∑x2i+2∑|xixj|] ⇒∑x2i100=(500)2−2∑|xixj|100 ⇒∑x2i100=(500)2−2×80000100 ⇒∑x2i100=2500−1600=900 Standard deviation =√∑(xi−¯¯¯x)2100 =√900=30