If Q is a set of consecutive integers, what is the standard

Question

If Q is a set of consecutive integers, what is the standard deviation of Q?

(1) Set Q contains 21 terms.
(2) The median of set Q is 20.

OPEN DISCUSSION OF THIS QUESTION IS HERE: if-q-is-a-set-of-consecutive-integers-what-is-the-standard-151794.html

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Official Answer

A

Bunuel · Accepted Answer

Two very important properties of standard deviation: If we add or subtract a constant to each term in a set: Mean will increase or decrease by the same constant. SD will not change. If we increase or decrease each term in a set by the same percent (multiply all terms by the constant): Mean will increase or decrease by the same percent. SD will increase or decrease by the same percent. You can try it yourself: SD of a set: {1,1,4} will be the same as that of {5,5,8} as second set is obtained by adding 4 to each term of the first set. That's because Standard Deviation shows how much variation there is from the mean. And when adding or subtracting a constant to each term we are shifting the mean of the set by this constant (mean will increase or decrease by the same constant) but the variation from the mean remains the same as all terms are also shifted by the same constant. Back to the original question: If Q is a set of consecutive integers, what is the standard deviation of Q? (1) Set Q contains 21 terms --> SD of ALL sets with 21 consecutive integers will be the same, as any set of 21 consecutive integers can be obtained by adding constant to another set of 21 consecutive integers. For example: set of 21 consecutive integers {4, 5, 6, ..., 24} can be obtained by adding 4 to each term of another set of 21 consecutive integers: {0, 1, 2, ..., 20}. So we can calculate SD of {0, 1, 2, ..., 20} and we'll know that no matter what our set actually is, its SD will be the same. Sufficient. (2) The median of set Q is 20. Clearly insufficient. Answer: A. OPEN DISCUSSION OF THIS QUESTION IS HERE: if-q-is-a-set-of-consecutive-integers-what-is-the-standard-151794.html

cdowwe · Answer

(1) You can calculate SD from this. You know the difference between the mean and each number will be 1,2,3,4,5,6,7,8,9,10,0,10,9,8,7,6,5,4,3,2,1 no matter what the mean is, so you can calculate SD from this.

(2) You can not calculate SD from this. The median = mean when consecutive #s.

Since we do not know how many #s there are, we do not know SD.

For example, say set is 19,20,21. 
SD is square root of (1+0+1)/3 or .816

If set is 18,19,20,21,22
SD is square root of (4+1+0+1+4)/5 or 1.41

So answer is A?

badgerboy · Answer

A for me too. 
Since given the # of terms, and the fact that they're consecutive, it does not matter what the terms are. Std Dev will be the same.

From   median is 21, but if there are 5 terms each side of the median, then the std dev will be more than if there are 2 terms each side of the mean.

rao · Answer

If you know the basics then these questions become very easy.

Standard Deviation is (in generic terms) how far the term is from its median.

If you know the series is consecitvie even series then STD is same because the diff or any term from median is gonna be same no matter what two number you chose from series. ex. distance of 30 from 31 is same as distance form 4 to 5.

srini123 · Answer

So how we use this knowledge to attack this problem ? I knew that SD is how far the term is from mean (is it median, of course for consecutive numbers its the same), but I am not able to find an easy way to solve this Q, athough I could solve it by actually checking for #s for S2 19,20,21 or 18,19,20,21,22.

hogann · Answer

OA is A
Official explanation from the Manhattan GMAT

1) SUFFICIENT: From the question, we know that Q is a set of consecutive integers. Statement 1 tells us that there are 21 terms in the set. Since, in any consecutive set with an odd number of terms, the middle value is the mean of the set, we can represent the set as 10 terms on either side of the middle term x:

Notice that the difference between the mean (x) and the first term in the set (x – 10) is 10. The difference between the mean (x) and the second term in the set (x – 9) is 9. As you can see, we can actually find the difference between each term in the set and the mean of the set without knowing the specific value of each term in the set!
 
(The only reason we are able to do this is because we know that the set abides by a specified consecutive pattern and because we are told the number of terms in this set.) Since we are able to find the "differences," we can use these to calculate the standard deviation of the set. Although you do not need to do this, here is the actual calculation: 
 
Sum of the squared differences:
102 + 92 + 82 + 72 + 62 + 52 + 42 + 32 + 22 + 12 + 02 + (-1)2 + (-2)2(-3)2 + (-4)2 + (-5)2 + (-6)2(-7)2 + (-8)2 + (-9)2 + (-10)2 = 770
Average of the sum of the squared differences:

770/21 = 36 2/3
 
The square root of this average is the standard deviation: ≈ 6.06

rao · Answer

there are 21 consecutive terms i.e. (x-10)(x-9)...(x)....(x+9)(x+10)

No matter what the x is, you can still find the difference of each term from mean: 10,9..0..9,10. Hence you can calculate the SD without even knowing the terms just because you already know the difference from mean.

If Q is a set of consecutive integers, what is the standard

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If Q is a set of consecutive integers, what is the standard

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