Questions
Let r be the range and be the SD of a set of observations , then
a
S≤rnn−1
b
S=rnn−1
c
S≥rnn−1
d
None of these
detailed solution
Correct option is A
We have r=maxxi−xjand S2=1n−1∑i=1n xi−x¯2 Now, xi−x¯2=xi−x1+x2+⋯+xnn2=1n2xi−x1+xi−x2+⋯+xi−xi−1+xi−x1+1+⋯+xi−xn≤1n2[(n−1)r]2 [∵]xi−xj∣≤r⇒ xi−x¯2≤r2⇒ ∑i=1n xi- x¯2≤nr2⇒ 1n−1∑i=1n xi−x¯2≤nr2(n−1)⇒ S2≤nr2(n−1)⇒ S≤rnn−1Talk to our academic expert!
Similar Questions
For (2n + 1) observations , and 0, where all x's are distinct, let SD and MD denote the standard deviation and median, respectively. Then which of the following is always true?
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