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# If the average (arithmetic mean) of a, b and c is m, is their standard

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If the average (arithmetic mean) of a, b and c is m, is their standard  [#permalink]

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08 Aug 2018, 02:10
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[Math Revolution GMAT math practice question]

If the average (arithmetic mean) of a, b and c is m, is their standard deviation less than 1?

1) a, b and c are consecutive integers with a < b < c.
2) m = 2

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"Only $79 for 1 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" ##### Most Helpful Community Reply Senior Manager Joined: 18 Jun 2018 Posts: 267 Re: If the average (arithmetic mean) of a, b and c is m, is their standard [#permalink] ### Show Tags 08 Aug 2018, 03:02 2 3 MathRevolution wrote: [Math Revolution GMAT math practice question] If the average (arithmetic mean) of a, b and c is m, is their standard deviation less than 1? 1) a, b and c are consecutive integers with a < b < c. 2) m = 2 OA:A Given : $$\frac{a+b+c}{3}= m$$ is standard deviation of a,b and c less than 1? Statement 1 : $$a, b$$ and $$c$$ are consecutive integers with $$a < b < c$$. It means $$b$$ is mean, Standard Deviation, $$S.D =\sqrt{\frac{(a-b)^2+(b-b)^2+(c-b)^2}{3}}$$ $$a-b=-1,c-b=1$$ as $$a,b,c$$ are consecutive number. Standard Deviation, $$S.D =\sqrt{\frac{(-1)^2+(0)^2+(1)^2}{3}} =\sqrt{\frac{2}{3}}<1$$ Statement 1 alone is sufficient. 2) $$m = 2$$ Now Plugging in number,taking $$a=1,b=2$$ and $$c=3$$ $$S.D<1$$ as already seen in statement 1 that 3 consecutive number have S.D less than 1. or We could have taken $$a=2,b=2,c=2$$, then also m would have been $$2$$. In that case $$S.D=0$$ Is S.D<1 for a,b,c : Yes Now taking $$a=-4, b =2, c = 8$$, mean would be 2 as required by Statement 2. Standard Deviation, $$S.D =\sqrt{\frac{(-4-2)^2+(2-2)^2+(8-2)^2}{3}}= \sqrt{\frac{72}{3}}=2\sqrt{6}>1$$ Is S.D<1 for a,b,c : No Statement 2 alone is not sufficient. ##### General Discussion VP Joined: 09 Mar 2016 Posts: 1255 If the average (arithmetic mean) of a, b and c is m, is their standard [#permalink] ### Show Tags 08 Aug 2018, 06:06 Bismarck wrote: MathRevolution wrote: [Math Revolution GMAT math practice question] If the average (arithmetic mean) of a, b and c is m, is their standard deviation less than 1? 1) a, b and c are consecutive integers with a < b < c. 2) m = 2 OA:A Given : $$\frac{a+b+c}{3}= m$$ is standard deviation of a,b and c less than 1? Statement 1 : $$a, b$$ and $$c$$ are consecutive integers with $$a < b < c$$. It means $$b$$ is mean, Standard Deviation, $$S.D =\sqrt{\frac{(a-b)^2+(b-b)^2+(c-b)^2}{3}}$$ $$a-b=-1,c-b=1$$ as $$a,b,c$$ are consecutive number. Standard Deviation, $$S.D =\sqrt{\frac{(-1)^2+(0)^2+(1)^2}{3}} =\sqrt{\frac{2}{3}}<1$$ Statement 1 alone is sufficient. 2) $$m = 2$$ Now Plugging in number,taking $$a=1,b=2$$ and $$c=3$$ $$S.D<1$$ as already seen in statement 1 that 3 consecutive number have S.D less than 1. or We could have taken $$a=2,b=2,c=2$$, then also m would have been $$2$$. In that case $$S.D=0$$ Is S.D<1 for a,b,c : Yes Now taking $$a=-4, b =2, c = 8$$, mean would be 2 as required by Statement 2. Standard Deviation, $$S.D =\sqrt{\frac{(-4-2)^2+(2-2)^2+(8-2)^2}{3}}= \sqrt{\frac{72}{3}}=2\sqrt{6}>1$$ Is S.D<1 for a,b,c : No Statement 2 alone is not sufficient. ' Bismarck what if you took 20, 21, 22 for statement one Senior Manager Joined: 18 Jun 2018 Posts: 267 Re: If the average (arithmetic mean) of a, b and c is m, is their standard [#permalink] ### Show Tags 08 Aug 2018, 06:18 1 dave13 wrote: Bismarck wrote: MathRevolution wrote: [Math Revolution GMAT math practice question] If the average (arithmetic mean) of a, b and c is m, is their standard deviation less than 1? 1) a, b and c are consecutive integers with a < b < c. 2) m = 2 OA:A Given : $$\frac{a+b+c}{3}= m$$ is standard deviation of a,b and c less than 1? Statement 1 : $$a, b$$ and $$c$$ are consecutive integers with $$a < b < c$$. It means $$b$$ is mean, Standard Deviation, $$S.D =\sqrt{\frac{(a-b)^2+(b-b)^2+(c-b)^2}{3}}$$ $$a-b=-1,c-b=1$$ as $$a,b,c$$ are consecutive number. Standard Deviation, $$S.D =\sqrt{\frac{(-1)^2+(0)^2+(1)^2}{3}} =\sqrt{\frac{2}{3}}<1$$ Statement 1 alone is sufficient. 2) $$m = 2$$ Now Plugging in number,taking $$a=1,b=2$$ and $$c=3$$ $$S.D<1$$ as already seen in statement 1 that 3 consecutive number have S.D less than 1. or We could have taken $$a=2,b=2,c=2$$, then also m would have been $$2$$. In that case $$S.D=0$$ Is S.D<1 for a,b,c : Yes Now taking $$a=-4, b =2, c = 8$$, mean would be 2 as required by Statement 2. Standard Deviation, $$S.D =\sqrt{\frac{(-4-2)^2+(2-2)^2+(8-2)^2}{3}}= \sqrt{\frac{72}{3}}=2\sqrt{6}>1$$ Is S.D<1 for a,b,c : No Statement 2 alone is not sufficient. ' Bismarck what if you took 20, 21, 22 for statement one dave13 taking $$a=20, b= 21$$ and $$c =22$$ Mean,$$m$$ $$= \frac{20+21+22}{3} =\frac{63}{3}=21$$, which is equal to $$b$$ $$S.D =\sqrt{\frac{(a-m)^2+(b-m)^2+(c-m)^2}{3}}$$ $$S.D =\sqrt{\frac{(20-21)^2+(21-21)^2+(22-21)^2}{3}}=\sqrt{\frac{(-1)^2+(0)^2+(1)^2}{3}}=\sqrt{\frac{2}{3}}<1$$ $$S.D<1$$ CEO Joined: 12 Sep 2015 Posts: 3922 Location: Canada Re: If the average (arithmetic mean) of a, b and c is m, is their standard [#permalink] ### Show Tags 08 Aug 2018, 15:03 Top Contributor MathRevolution wrote: [Math Revolution GMAT math practice question] If the average (arithmetic mean) of a, b and c is m, is their standard deviation less than 1? 1) a, b and c are consecutive integers with a < b < c. 2) m = 2 Target question: Is the standard deviation of a, b and c less than 1? Statement 1: a, b and c are consecutive integers with a < b < c. It's important to know that standard deviation is a measure of dispersion (how spread apart the values are). So, ANY 3 consecutive integers will have the same standard deviation. For example, the standard deviation of {1,2,3} = the standard deviation of {6,7,8} = the standard deviation of {23,24,25} etc So, IF we calculate the standard deviation of {1,2,3} then THAT value will provide sufficient info to the answer to the target question Since we COULD answer the target question with certainty, statement 1 is SUFFICIENT Statement 2: m = 2 There are several values a, b and c that satisfy statement 2. Here are two: Case a: a = 2, b = 2 and c = 2 (mean = 2 and standard deviation = 0). In this case, the answer to the target question is YES, the standard deviation IS less than 1 Case b: a = -100, b = 0 and c = 106 (mean = 2 and standard deviation = some number much greater than 1. In this case, the answer to the target question is NO, the standard deviation is NOT less than 1 Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT Answer: A RELATED VIDEO FROM OUR COURSE _________________ Test confidently with gmatprepnow.com Director Joined: 06 Jan 2015 Posts: 695 Location: India Concentration: Operations, Finance GPA: 3.35 WE: Information Technology (Computer Software) Re: If the average (arithmetic mean) of a, b and c is m, is their standard [#permalink] ### Show Tags 08 Aug 2018, 19:48 MathRevolution wrote: [Math Revolution GMAT math practice question] If the average (arithmetic mean) of a, b and c is m, is their standard deviation less than 1? 1) a, b and c are consecutive integers with a < b < c. 2) m = 2 Is their standard deviation less than 1? Here standard deviation is a measure of how spread out numbers are. or distance or spread from mean we need not to calculate the actual SD 1) a, b and c are consecutive integers with a < b < c. For Consecutive integers b=mean So we can find out the answer Hence A is Suff 2) m = 2 a,b,c are not given For ex: a=2 b=2 c=2 SD = 0 If a=1, b=2, c=3 SD varies Hence A _________________ आत्मनॊ मोक्षार्थम् जगद्धिताय च Resource: GMATPrep RCs With Solution Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 7766 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: If the average (arithmetic mean) of a, b and c is m, is their standard [#permalink] ### Show Tags 10 Aug 2018, 02:31 => Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution. We have $$m = \frac{( a + b + c )}{3}$$ from the original condition. Since we have 4 variables and 1 equation, E is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first. Conditions 1) & 2) Since $$a, b$$ and $$c$$ are consecutive integers, we can assume $$a = b – 1$$ and $$c = b + 1$$ and $$b = m$$. Since $$m = 2$$ by condition 2, we have $$a = 1, b = 2, c = 3.$$ The standard deviation is $$\sqrt{\frac{a-b^2+(b-b)^2+(c-b)^2}{3}}$$ =$$\sqrt{\frac{(-1)^2+0^2+1^2}{3}}$$ = $$\sqrt{\frac{2}{3}}$$ which is less than 1. Thus, both conditions are sufficient, when considered together. Since this question is a statistics question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B. Condition 1) Since $$a, b$$ and $$c$$ are consecutive integers, we can assume that $$a = b – 1$$ and $$c = b + 1$$ and $$b = m$$. Then $$a – b = -1$$ and $$c – b = 1$$, and the standard deviation is $$\sqrt{\frac{a-b^2+(b-b)^2+(c-b)^2}{3}}$$ =$$\sqrt{\frac{(-1)^2+0^2+1^2}{3}}$$ = $$\sqrt{\frac{2}{3}}$$ which is less than 1. Thus, condition 1) is sufficient on its own. Condition 2) Condition 2) is not sufficient since it gives us no information about $$a, b$$ and $$c$$. Therefore, A is the answer. Answer: A In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions on which the answer is A, B, C or D. _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$79 for 1 month Online Course"
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Re: If the average (arithmetic mean) of a, b and c is m, is their standard   [#permalink] 10 Aug 2018, 02:31
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