Standard devaiation

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treker: Senior | Next Rank: 100 Posts; Posts: 33; Joined: Fri Aug 22, 2008 9:32 pm; Thanked: 1 times

Standard devaiation

by treker » Fri Jan 15, 2010 10:49 pm

Set B = {x, x, y, 9, 10, 16, 16}

Set B has 7 members and x and y are distinct positive integers. If x is the mode of Set B and the mean of Set B is 12, which of the following is a valid value of x that would cause the standard deviation of the set to be greatest?
(A) 9
(B) 10
(C) 12
(D) 16
(E) 18

How do we solve such questions without complex calculations?

Thanks.

Quote

Source: Beat The GMAT — Problem Solving | Fri Jan 15, 2010 10:49 pm

sanju09: GMAT Instructor; Posts: 3650; Joined: Wed Jan 21, 2009 4:27 am; Location: India; Thanked: 267 times; Followed by:80 members; GMAT Score:760

by sanju09 » Sat Jan 16, 2010 12:59 am

For the standard deviation of the set to be greatest, there must be a considerably big difference between one or few data in the set and the arithmetic mean of the set, that is one has to look for the maximum possible absolute value of (x - A), where x is a random data from the set and A is mean of the set; which is 12 in our case. In this case, for the mode to be x, there must be at least 3 appearances of x as there are already two appearances of 16 there, which is not openly told the mode, which might well have been the mode if x were 16, in fact x can take any value given in set EXCEPT y, to be the mode of the set, but! We will take only that as x which maximizes the absolute value of (x - 12), of the given values, only 16 works, CANNOT take 18, as this won't make x the mode ahead of 16. Take D

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