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The standard deviation of some consecutive integers is found to be 2. Which of the following statements best describes the nature of the consecutive integers?
1. The integers are any set of eight consecutive integers
2. The integers are any set of eight consecutive positive integers
3. The integers are any set of seven consecutive integers
4. None of the above

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Correct Answer - Option 3 : The integers are any set of seven consecutive integers

Concept:

Mean (x̅) = Sum of elements/Total number of elements

Standard deviation\(\rm \sqrt{\frac{\bar x-x_i}{N}}\)

 

Calculation:

Let, seven consecutive integers be x, x+1, x+2, x+3, x+4, x+5, x+6

Now, mean = \(\rm \frac{x+x+1+x+2+x+3+x+4+x+5+x+6}{7}=\frac{7x+21}{7}\)

= x+3

Now, standard deviation = \(\rm \sqrt{\frac{x+3-x+x+3-x-1 +x+3-x-2 +x+3-x-3 +x+3-x-4 +x+3-x-5 +x+3-x-6}{7}}\)

\(=\rm \sqrt{\frac{28}{7}}\)

\(=\rm \sqrt{4}\)

= 2

Hence, option (3) is correct.

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