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What is the standard deviation of a, b and c?

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
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What is the standard deviation of a, b and c?  [#permalink]

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26 Sep 2018, 23:55
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[Math Revolution GMAT math practice question]

What is the standard deviation of $$a, b$$ and $$c$$?

$$1) a^2+b^2+c^2 = 77$$
$$2) a+b+c =15$$

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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 $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" GMATH Teacher Status: GMATH founder Joined: 12 Oct 2010 Posts: 544 Re: What is the standard deviation of a, b and c? [#permalink] Show Tags 27 Sep 2018, 15:23 1 MathRevolution wrote: [Math Revolution GMAT math practice question] What is the standard deviation of $$a, b$$ and $$c$$? $$1) a^2+b^2+c^2 = 77$$ $$2) a+b+c =15$$ Very nice problem, congrats Max! $$? = \sigma \left( {a,b,c} \right)$$ $$\left( 1 \right)\,\,{a^2} + {b^2} + {c^2} = 77\,\,\,\,\,\left\{ \begin{gathered} \,{\text{Take}}\,\,\left( {a;b;c} \right) = \left( {\sqrt {\frac{{77}}{3}} \,;\sqrt {\frac{{77}}{3}} \,;\sqrt {\frac{{77}}{3}} \,} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,{\text{?}}\,\,{\text{ = }}\,\,{\text{0}}\,\, \hfill \\ \,\,{\text{Take}}\,\,\left( {a;b;c} \right) = \left( {\sqrt {77} \,;0\,;0\,} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,{\text{?}}\,\, \ne \,\,{\text{0}}\,\,\,\, \hfill \\ \end{gathered} \right.$$ $$\left( 2 \right)\,\,a + b + c = 15\,\,\,\,\,\left\{ \begin{gathered} \,{\text{Take}}\,\,\left( {a;b;c} \right) = \left( {5\,;5\,;5\,} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,{\text{?}}\,\,{\text{ = }}\,\,{\text{0}}\,\, \hfill \\ \,\,{\text{Take}}\,\,\left( {a;b;c} \right) = \left( {15\,;0\,;0\,} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,{\text{?}}\,\, \ne \,\,{\text{0}}\,\,\,\, \hfill \\ \end{gathered} \right.$$ $$\left( {1 + 2} \right)\,\,\,\,\mu = \frac{{a + b + c}}{3} = 5$$ $$? = \sqrt {\frac{{{{\left( {a - \mu } \right)}^2} + {{\left( {b - \mu } \right)}^2} + {{\left( {c - \mu } \right)}^2}}}{3}} \,\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\boxed{\,\,\,? = {{\left( {a - 5} \right)}^2} + {{\left( {b - 5} \right)}^2} + {{\left( {c - 5} \right)}^2}\,\,\,}$$ $$?\,\,\, = \,\,\,{\left( {a - 5} \right)^2} + {\left( {b - 5} \right)^2} + {\left( {c - 5} \right)^2}\,\,\, = \,\,\,\,\underbrace {{a^2} + {b^2} + {c^2}}_{77} - 10\underbrace {\left( {a + b + c} \right)}_{15} + 3 \cdot 25\,\,\,\,\,{\text{unique}}$$ The correct answer is (C), indeed. This solution follows the notations and rationale taught in the GMATH method. Regards, Fabio. _________________ Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version) Course release PROMO : finish our test drive till 30/Dec with (at least) 50 correct answers out of 92 (12-questions Mock included) to gain a 50% discount! Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6639 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: What is the standard deviation of a, b and c? [#permalink] Show Tags 30 Sep 2018, 21:51 => 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. The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. The mean of $$a, b$$ and $$c$$ is $$m = \frac{( a + b + c )}{3}$$. So, $$a + b + c = 3m.$$ The variance of a, b and c, that is, the square of the standard deviation of a, b and c, is given by $$VAR = SD^2 = \frac{{ ( a – m )^2 + ( b – m )^2 + ( c – m )^2 }}{3}$$ $$= \frac{{ a^2 - 2am + m^2 + b^2 - 2bm + b^2 + c^2 - 2cm + m^2 }}{3}$$ $$= \frac{{ a^2 + b^2 + c^2 – 2m(a+b+c) + 3m^2 }}{3}$$ $$= \frac{{ a^2 + b^2 + c^2 – 2m*3m + 3m^2 }}{3}$$ $$= \frac{{ a^2 + b^2 + c^2 – 3m^2 }}{3}$$ $$= \frac{( a^2 + b^2 + c^2 )}{3} – m^2$$ $$= \frac{( a^2 + b^2 + c^2 )}{3} – { \frac{( a + b + c )}{3} }^2$$ This value can be calculated using the information given in conditions 1) and 2). Thus, both conditions together are sufficient. Therefore, C is the answer. Answer: C When we have a sum of data values and a sum of the squares of data values, we can always calculate the standard deviation. _________________ 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$99 for 3 month Online Course"
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Re: What is the standard deviation of a, b and c? &nbs [#permalink] 30 Sep 2018, 21:51
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What is the standard deviation of a, b and c?

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