A set S contains exactly four distinct positive integers: A, B, C, D

Question

A set S contains exactly four distinct positive integers: A, B, C, D. What is the standard deviation of set S?

(1) A, B, C, D are four consecutive odd integers.

(2) If each element of set S is increased by 3, the standard deviation of the new set so formed will be √5.

u1983 · Answer

(SD) =  \sigma = \sqrt{variance} = \sqrt{\frac{\sum(x_i - x_{av})^2}{N}}

(1) A, B, C, D are four consecutive odd integers.

The mean is mid point of B & C(say x), & A-x=D-x=|3| & B-x=C-x=|1| ( as A,B ,C D are consecutive integers)
Hence,  \sigma =\sqrt{\frac{\sum(x_i - x_{av})^2}{N}}= \sqrt{\frac{3^2+1^2+1^2+3^2}{4}} = \sqrt{\frac{20}{4}} = \sqrt{5} ...... . Thus sufficient.

(2) If each element of set S is increased by 3, the standard deviation of the new set so formed will be √5.

As (SD) =  \sigma = \sqrt{variance} = \sqrt{\frac{\sum(x_i - x_{av})^2}{N}}

The SD changes when the expression  (x_i - x_{av})^2  changes.
But  as all observations are increased by 3 the mean will also increase by 3 . Thus The new expression   ^2 = (x_i - x_{av})  
Thus new SD= old SD............................................................................................................................................................ .Thus sufficient

Hence I would go for  option D .

alldaytestprep · Answer

A set S contains exactly four distinct positive integers: A, B, C, D. What is the standard deviation of set S?

(1) A, B, C, D are four consecutive odd integers.

(2) If each element of set S is increased by 3, the standard deviation of the new set so formed will be √5.

You NEVER should have to do any calculations regarding standard deviation on the GMAT. You just need to understand it conceptually in order to answer pretty much any question they give you that has to do with SD.

Standard deviation is simply a measure of how far on average each term in a set is from the mean. That is all you need to know to answer this.

1) Consecutive set means that mean = median. Four consecutive odd integers can be represented as: (x), (x + 2), (x + 4), (x + 6)
- mean = median = (x + 3) 
- the two outer terms are 3 away from the mean. the two inner terms are 1 away from the mean. 
- since we know how far each term is from the mean we can definitely get the average how far they are ---> SUFFICIENT

2) If all the terms are increased by the same amount their relative distance to the mean would not change at all and therefore the SD of the new set would be equal to the SD of the old set ----> SUFFICIENT

Answer: D

CareerGeek · Answer

(1) A, B, C, D are four consecutive odd integers.

Odd numbers are represented by "2n + 1"

Let A = 2n + 1. So, B = 2n + 3, C = 2n + 5, D = 2n + 7

Mean =  \frac{(2n + 3 + 2n + 5)}{2}  = 2n + 4

Deviation from Mean = -3, -1, 1, 3

So, Standard deviation can be found out -->  Sufficient

(2) If each element of set S is increased by 3, the standard deviation of the new set so formed will be √5.

Property: Standard Deviation doesnt not change with addition or subtraction of a constant to all the terms

Given SD of A+3, B+3, C+3, D+3 is  \sqrt{5}

--> SD of A, B, C, D is also  \sqrt{5}

So,  Sufficient

Option D

Pls Hit Kudos if you like the solution

A set S contains exactly four distinct positive integers: A, B, C, D

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GMAT Data Sufficiency (DS) Questions

A set S contains exactly four distinct positive integers: A, B, C, D

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