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Manhattan GMAT - Standard Deviation

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krishna239455: Master | Next Rank: 500 Posts; Posts: 114; Joined: Mon Jun 13, 2011 9:10 am; Thanked: 1 times

by krishna239455 » Mon May 21, 2012 4:49 am

If Q is 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

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Source: Beat The GMAT — Data Sufficiency | Mon May 21, 2012 4:49 am

GmatKiss: Legendary Member; Posts: 2789; Joined: Tue Jul 26, 2011 12:19 amFollowed by: 43 members; Location: Chennai, India; Thanked: 206 times; GMAT Score:640

by GmatKiss » Mon May 21, 2012 5:24 am

Question frame:

Does consecutive integers mean the following,

1,2,3,4 ....
13,14,15,16,...

If so, answer should be D.

1) S.D is 1
2) S.D is 1

If consecutive integers mean,

1,2,3,4,5,.... and also
5,10,15,20,... and also
12,24,36,48,...

Then answer will be E

P:S. I have seen MGMAT Forum stating A as the answer.
Please help me if i am missing something basic here!

TIA,
GK

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Mon May 21, 2012 5:49 am

krishna239455 wrote:If Q is 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

SD describes how much a set of data DEVIATES from the mean.

Statement 1: Set Q contains 21 terms.
Any set of 21 consecutive integers will deviate from the mean exactly the same way.
Thus, the SD can be determined.
SUFFICIENT.

Statement 2: Median = 20.
With evenly spaced numbers, mean = median.
Thus, the mean of the set is 20.
But without knowing the number of terms, we can't determine how much the set DEVIATES from the mean.
If there is only one term -- if Q = {20} -- then there is NO deviation from the mean.
If there are 101 terms, then there is quite a bit of deviation from the mean.
Thus, the SD can be different values.
INSUFFICIENT.

The correct answer is A.

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ubhanja: Junior | Next Rank: 30 Posts; Posts: 19; Joined: Sun Oct 30, 2011 6:52 pm

by ubhanja » Tue May 22, 2012 10:19 am

how can this be A? This has to be C.

A - Tells that it can be determined. But we don't know the exact value till we combine both the statements. Please clarify.

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agarwalva: Master | Next Rank: 500 Posts; Posts: 152; Joined: Mon Feb 20, 2012 12:56 pm; Location: New york; Thanked: 9 times

by agarwalva » Tue May 22, 2012 2:26 pm

ubhanja wrote:how can this be A? This has to be C.

A - Tells that it can be determined. But we don't know the exact value till we combine both the statements. Please clarify.

example
consider list 1 , 2, 3, 4, 5 [5 consecutive integers]
mean is 3.
(1-3)^2 = 4 (5-3)^2 = 4
(2-3)^ = 1 (4-3)^2 = 1
(3-3)^2 = 0
SD = SqRoot (4+1+01+4)/5 = 2

Now another list 6,7,8,9,10
Mean is 8

(6-8)^2 = 4 (10-8)^2 = 4
(7-8)^2 = 1 (9-8)^2= 1
(8-8)^2 = 0

SD = SqRoot (4+1+01+4)/5 = 2

Standard deviation in case of any 5 consecutive integers is 2. Similarly SD in case of any 21 consecutive integers is same.
We just need the number of terms.

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heymayank08: Master | Next Rank: 500 Posts; Posts: 181; Joined: Fri Feb 17, 2012 9:36 amFollowed by: 1 member; Thanked: 1 times; GMAT Score:620

by heymayank08 » Tue May 22, 2012 5:53 pm

GMATGuruNY wrote:
krishna239455 wrote:
Statement 2: Median = 20.
With evenly spaced numbers, mean = median.
Thus, the mean of the set is 20.
But without knowing the number of terms, we can't determine how much the set DEVIATES from the mean.
If there is only one term -- if Q = {20} -- then there is NO deviation from the mean.
If there are 101 terms, then there is quite a bit of deviation from the mean.
Thus, the SD can be different values.
INSUFFICIENT.

The correct answer is A.
mitch but it is written that Q is a set of consecutive itegers so there ought to be more than one integer in the set..
isn't it???
pls correct me if i am wrong..

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vikram4689: Legendary Member; Posts: 1325; Joined: Sun Nov 01, 2009 6:24 amFollowed by: 14 members; Thanked: 105 times

by vikram4689 » Tue May 22, 2012 6:09 pm

Standard deviation means how much the values in a set differ from the mean value. To calculate we need range of integers and integers itself.

A gives us total numbers = 21 and consecutive integers means difference = 1 so we can form a sequence. Remembers S.D. does not vary by adding or subtracting a particular no. to all members of a set, so we need relative value as given in option A.

B says median is 20 but what about range 19,20,21 OR 18,19,20,21,22 ...both have different S.D. as they have different range.

Answer is A

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ronnie1985: Legendary Member; Posts: 626; Joined: Fri Dec 23, 2011 2:50 amFollowed by: 10 members; Location: Ahmedabad; Thanked: 31 times

by ronnie1985 » Wed May 23, 2012 7:02 am

Std dev = root mean square of deviations from mean

If the set starts from a and ends in a+n-1 containing n terms,

mean = a+(n-1)/2

dev1 = -(n-1)/2
dev2 = 1-(n-1)/2
dev3 = 2-(n-1)/2
.
.
.
devn = n-1-(n-1)/2

Hence the sum of squares of deviations will also be a function of n only. It is independent of a, the first term.

Hence number of terms is sufficient to find the std deviation of an AP.

S1 Gives no of terms hence sufficient

S2 gives median hence not sufficient

(A) is the answer

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