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# If m, s are the average and standard deviation of integers a, b, c

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Manager
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If m, s are the average and standard deviation of integers a, b, c  [#permalink]

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12 Mar 2017, 20:49
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55% (hard)

Question Stats:

42% (01:00) correct 58% (01:09) wrong based on 84 sessions

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If m, s are the average and standard deviation of integers a, b, c, and d, is s > 0?

1. m > a
2. a + b + c + d = 0

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Re: If m, s are the average and standard deviation of integers a, b, c  [#permalink]

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12 Mar 2017, 21:11
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STE 1: SD can +ve or 0. if all the elements in the set are 0 then SD will be zero otherwise a +ve value. given that m>a ie avg is greater than one element. all elements are not equal .Sufficient.

STE 2 : all the elements can be 0 or a negative value + a positive value. Not sufficient

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Re: If m, s are the average and standard deviation of integers a, b, c  [#permalink]

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29 Aug 2018, 06:55
Top Contributor
ashwink wrote:
If m, s are the average and standard deviation of integers a, b, c, and d, is s > 0?

1. m > a
2. a + b + c + d = 0

Target question: Is the standard deviation (s) greater than 0?

The test-makers LOVE testing students on the following facts about standard deviation:
If all of the values in a set are EQUAL, then the standard deviation is ZERO.
If all of the values in a set are NOT EQUAL, then the standard deviation is GREATER THAN ZERO.

Statement 1: m > a
In other words, the average (mean) of the set is greater than one of the values in the set (a)
If the mean is greater than one of the values in the set, then we can be certain that the values in the set are all NOT EQUAL (otherwise, the mean would equal all of the values in the set)
If all of the values in a set are NOT EQUAL, then the standard deviation is GREATER THAN ZERO
So, the answer to the target question is YES, the standard deviation (s) IS greater than 0
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: a + b + c + d = 0
There are several sets that satisfy statement 2. Here are two:
Case a: a = 0, b = 0, c = 0 and d = 0. In this case, the standard deviation is 0. So, the answer to the target question is NO, the standard deviation (s) is NOT greater than 0
Case b: a = 1, b = 1, c = -1 and d = -1. In this case, the standard deviation is positive. So, the answer to the target question is YES, the standard deviation (s) IS greater than 0]
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

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Re: If m, s are the average and standard deviation of integers a, b, c  [#permalink]

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30 Aug 2018, 06:17
ashwink wrote:
If m, s are the average and standard deviation of integers a, b, c, and d, is s > 0?

1. m > a
2. a + b + c + d = 0

$$a,b,c,d\,\,\,\,{\text{ints}}$$

$$m = \frac{{a + b + c + d}}{4}$$

$$s = \sigma \left( {a,b,c,d} \right)$$

$$s\,\,\mathop > \limits^? \,\,0\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\boxed{?\,\,\,:\,\,\,\,a = b = c = d\,\,\,\,{\text{false}}}\,$$

$$\left( 1 \right)\,\,\,m > a\,\,\,\, \Rightarrow \,\,\,\,a = b = c = d\,\,\,\underline {{\text{false}}} \,\,\,({\text{otherwise}}\,\,m = a)\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{YES}}} \right\rangle$$

$$\left( 2 \right)\,\,\,\left\{ \begin{gathered} \,{\text{Take}}\,\,\left( {a,b,c,d} \right) = \left( {0,0,0,0} \right)\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{NO}}} \right\rangle \,\,\, \hfill \\ \,{\text{Take}}\,\,\left( {a,b,c,d} \right) = \left( {1, - 1,0,0} \right)\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,\,\, \hfill \\ \end{gathered} \right.$$

The above follows the notations and rationale taught in the GMATH method.
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Re: If m, s are the average and standard deviation of integers a, b, c &nbs [#permalink] 30 Aug 2018, 06:17
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