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  • Consider the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. If 1 is added to each number, the variance of the numbers so obtained is A. 6.5 B. 2.87 C. 3.87 D. 8.25

Consider the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. If 1 is added to each number, the variance of the numbers so obtained is
A. 6.5
B. 2.87
C. 3.87
D. 8.25

Answers (1)

We know that the standard deviation of the first n natural numbers is  \sqrt {\frac{n^{2}-1}{12}}~ \\

Now for first 10 natural numbers n=10 

substituting this in the equation of standard deviation we get  \sigma =\sqrt {\frac{ \left( 10 \right) ^{2}-1}{12}}=\sqrt {\frac{100-1}{12}}= \sqrt {\frac{99}{12}}= \sqrt {8.25} =2.87 \\

Now when 1 is added to each numbers of 1 , 2, 3, 4, 5, 6,7 ,8 ,9, 10 ; we get new series as 1+1, 2+1, 3+1, 4+1, 5+1, 6+1, 7+1, 8+1, 9+1, 10+1 

Now we know if standard deviation of x series is s, then standard deviation of k+x series is s,

So the S.D of 1+1, 2+1, 3+1, 4+1, 5+1, 6+1, 7+1, 8+1, 9+1, 10+1 series is also same as the S.D of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 series.

\\ \sigma =\sqrt {8.25}~ \\

Now for variance we will square on both the sides  \sigma ^{2}=8.25 \\

 

 

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