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# What is the standard deviation of the four numbers p, q, r, s?

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Retired Moderator
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What is the standard deviation of the four numbers p, q, r, s?  [#permalink]

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26 Feb 2011, 07:58
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Difficulty:

25% (medium)

Question Stats:

54% (01:01) correct 46% (01:20) wrong based on 132 sessions

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What is the standard deviation of the four numbers p, q, r, s?

(1) The sum of p, q, r and s is 24

(2) The sum of the squared differences of p, q, r and s from the mean is 224

The correct choice is (C).

My query however is regarding the formula used to derive the answer:

SD^2 = 1/N { SUM (Squares of numbers) - Avg(numbers)^2 }

I think I have not seen it before. Please share your opinions.

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Re: What is the standard deviation of the four numbers p, q, r, s?  [#permalink]

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26 Feb 2011, 09:14
S.D for 4 numbers p,q,r,s with mean m would be governed by the formula

S.D.^2 = 1/4*((p-m)^2+(q-m)^2+(r-m)^2+(s-m)^2)

Term on the right inside the bracket can be simplified as p^2+q^2+r^2+s^2-2m*(p+q+r+s)+4m^2

Now p+q+r+s = 4*m hence the term can simplify to p^2+q^2+r^2+s^2 - 8m^2+4m^2

or p^2+q^2+r^2+s^2 - 4m^2

Statement 1 tells us p+q+r+s = 24 and hence mean = 6, so we know that m = 6, and but this information would still not allow us to calculate SD as we do not know the value of p^2+q^2+r^2+s^2

Statement 2 tells us that p^2+q^2+r^2+s^2 = 224, but here we do not know the value of m, so insufficient.

Both statements combined give us m as well as p^2+q^2+r^2+s^2 and hence sufficient. Answer C
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Re: What is the standard deviation of the four numbers p, q, r, s?  [#permalink]

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26 Feb 2011, 17:46
subhashghosh wrote:
My query however is regarding the formula used to derive the answer:

SD^2 = 1/N { SUM (Squares of numbers) - Avg(numbers)^2 }

I think I have not seen it before. Please share your opinions.

You'll never need that formula on the GMAT. GMAT questions about standard deviation are conceptual, not computational - you'll want to know, in general terms, what standard deviation measures, and what phrases like 'one standard deviation above the mean' mean. You don't need to calculate standard deviations on the test, and you don't need to know obscure formulas like the one used in this question. Where is it from?
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Re: What is the standard deviation of the four numbers p, q, r, s?  [#permalink]

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26 Feb 2011, 19:34
Two things are necessary to find SD. mean and actual data i.e. numbers
1) gives mean
2) gives the numbers

1) + 2) Sufficient. Hence C.
If you see this question was written to take away your time in statement 2).
2) will give you probably two solutions since 2) is quadratic equation. But 1) will fix the issue since the sum should be 24. Thats my guess ! I will NOT solve this question - I will guess and move on !
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Re: What is the standard deviation of the four numbers p, q, r, s?  [#permalink]

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26 Feb 2011, 22:51
I saw it here:

http://www.questionbank.4gmat.com/mba_p ... 2805.shtml
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Re: What is the standard deviation of the four numbers p, q, r, s?  [#permalink]

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27 Feb 2011, 00:08
2
1
Good one ! Standard deviation = sqrt (Mean of squares - square of mean)

Just want to prove this hypothesis with two numbers. Lets say number are p,q

m = Mean = (p + q)/2 or 2m = p + q ---------------(1)
SD = sqrt(variance)

variance = [(p-m)^2 + (q-m)^2]/2 = [p^2 + m^2 - 2pm + q^2 + m^2 - 2qm]/2
lets say variance = numerator / 2
The numerator = (p^2 + m^2) - 2m(p+q) + 2 m^2 -----------(2)
From (1) and (2)
numerator = (p^2 + m^2) - 2m^2
variance = numerator/2 = (p^2 + m^2)/2 - m^2
Hence variance = Average of squares - Square of mean
SD = sqrt (Average of squares - Square of mean).Hence proved !
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Re: What is the standard deviation of the four numbers p, q, r, s?  [#permalink]

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21 Jul 2015, 18:13
beyondgmatscore,

Doesn't Statement 2 mean that the (p-m)^2 + (q-m)^2 + (r-m)^2 + (s-m)^2 is 224.
"The sum of the squared differences of p, q, r and s from the mean is 224"

beyondgmatscore wrote:
S.D for 4 numbers p,q,r,s with mean m would be governed by the formula

S.D.^2 = 1/4*((p-m)^2+(q-m)^2+(r-m)^2+(s-m)^2)

Term on the right inside the bracket can be simplified as p^2+q^2+r^2+s^2-2m*(p+q+r+s)+4m^2

Now p+q+r+s = 4*m hence the term can simplify to p^2+q^2+r^2+s^2 - 8m^2+4m^2

or p^2+q^2+r^2+s^2 - 4m^2

Statement 1 tells us p+q+r+s = 24 and hence mean = 6, so we know that m = 6, and but this information would still not allow us to calculate SD as we do not know the value of p^2+q^2+r^2+s^2

Statement 2 tells us that p^2+q^2+r^2+s^2 = 224, but here we do not know the value of m, so insufficient.

Both statements combined give us m as well as p^2+q^2+r^2+s^2 and hence sufficient. Answer C
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Re: What is the standard deviation of the four numbers p, q, r, s?  [#permalink]

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22 Jul 2015, 07:50

S.d is square root of Variance.

Variance is sum of sum of the squared differences of p, q, r and s from the mean, i.e 224. divided by total number , i.e 7
We have the necessary info here.
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What is the standard deviation of the four numbers p, q, r, s?  [#permalink]

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23 Jul 2015, 01:32
2
SD=sum of squared differences=224/4 (number of elements)=56 and sqrt=7.......

So B is SUFF
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Re: What is the standard deviation of the four numbers p, q, r, s?  [#permalink]

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04 Sep 2018, 19:42
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Re: What is the standard deviation of the four numbers p, q, r, s?   [#permalink] 04 Sep 2018, 19:42
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