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J is a collection of four odd integers and the greatest [#permalink]
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30 Aug 2006, 03:28
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J is a collection of four odd integers and the greatest difference between any two integers in J is 4. The standard deviation of J must be one of how many numbers?
(A) 3 (B) 4 (C) 5 (D) 6 (E) 7
Last edited by kevincan on 30 Aug 2006, 05:13, edited 1 time in total.



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Interesting.
Will give Ex. cause it will be easier to explain.
Diff0 3,3,3,3
Diff2 and 0 3,3,3,5
Diff4 and 0 3,3,3,7
Diff2 and 4and 0 3,3,5,7
Diff 2 and 0 3,3,5,5
diff 4 and 0 3,3,7,7
Difference can not be 1 and 3 cause it will give even integer
So should be D)



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D  6
Let four numbers are a,b,c,d and are in increasing order. da = 4
So a and d are fixed. We can change b and c but there is only one number between a and d. Let that number be z
There are six possibilities.
a,a,a,d
a,d,d,d
a,z,z,d
a,a,z,d
a,z,d,d
a,a,d,d
Hence answer is 6.
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kevincan wrote: ps_dahiya wrote: D  6
Let four numbers are a,b,c,d and are in increasing order. da = 4 So a and d are fixed. We can change b and c but there is only one number between a and d. Let that number be z
There are six possibilities. a,a,a,d a,d,d,d a,z,z,d a,a,z,d a,z,d,d a,a,d,d
Hence answer is 6. Won't some of these have the same s.d.?
I know that two of these will have same mean but I thought the dispersion will be different.
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Do {1,1,1,5} and {1,5,5,5} have different standard deviations?



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ps_dahiya wrote: D  6
Let four numbers are a,b,c,d and are in increasing order. da = 4 So a and d are fixed. We can change b and c but there is only one number between a and d. Let that number be z
There are six possibilities. a,a,a,d a,d,d,d a,z,z,d a,a,z,d a,z,d,d a,a,d,d
Hence answer is 6.
I think 7. The other possibility is where the standard deviation is 0
aaaa or dddd or zzzz



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ps_dahiya wrote: D  6
Let four numbers are a,b,c,d and are in increasing order. da = 4 So a and d are fixed. We can change b and c but there is only one number between a and d. Let that number be z
There are six possibilities. a,a,a,d a,d,d,d a,z,z,d a,a,z,d a,z,d,d a,a,d,d
Hence answer is 6.
Agree with this explanation... D should be the answer.
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Hmmm  does that mean that the max difference can be 0, 2, or 4?
Because then we get 
a, a, z, d
a, z, z, d
a, z, d, d
a, a, d, d
a, d, d, d
a, a, a, d
a, z, z, z
a, a, a, z
a, a, z, z
z, z, z, d
z, d, d, d
z, z, d, d
SD = 0 for
a, a, a, a
z, z, z, z
d, d, d, d
Too many possible SDs (for the choices given)?
I think we need to assume that there always exists a pair in the set, for which difference is 4. The remaining choices 
a, a, z, d
a, z, z, d
a, z, d, d
a, a, d, d
a, d, d, d
a, a, a, d
6 SDs...
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Uh uh. I know what you're thinking. "Is the answer A, B, C, D or E?" Well to tell you the truth in all this excitement I kinda lost track myself. But you've gotta ask yourself one question: "Do I feel lucky?" Well, do ya, punk?



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The greatest difference between any two elements in a set is equal to the range of the set.



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kevincan wrote: The greatest difference between any two elements in a set is equal to the range of the set.
Is it A  3 then?
1. when all the elements are the same
2. when any three elements are the same
3. when two elements are the same



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I started with the premise that 4 IS the greatest difference, but there are two elements in the set with a difference of 4.
(so no aaaa, dddd etc sets)
Now there are only three elements in this set, or two, because of the constraints, e.g. if the two numbers are 5 and 9, then at most the other member of hte set can be 7.
Thus the members can be:
5559
5999
(same std dev, diff mean)
5599
5779
(same std dev, same mean)
5579
5799
(same std dev, diff means)
Thus, provided I havent missed anything, the max std dev should be 3. Interestingly, the max means can be only 5. Not 6.
MG



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I think there are 7 ways.
Basically we need to find out the different SDs we can get.
If x = n.SD^2 is unique, then SD must be unique
(1111): M = 1, X = 0
(1113): M = 1.5, X = 0.25+0.25+0.25+2.25 = 3
(1115): M = 2, X = 1+1+1+9 =12
(1135): M = 2.5, X = 2.25+2.25+0.25+6.25 = 11
(1133): M = 2, X = 1+1+1+1 = 4
(1155): M = 4, X = 4+4+4+4 = 16
(1355): M = 3.5, X = 6.25+0.25+2.25+2.25 = 11
(1335): M = 3, X = 4+4 = 8
From the above, we have X = {0,3,4,8,11,11,12,16} which gives us 8 values. Of these, 7 are distinct. Hence I choose E.



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Re: PS: Standard Deviation of J [#permalink]
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06 Sep 2006, 18:48
kevincan wrote: J is a collection of four odd integers and the greatest difference between any two integers in J is 4. The standard deviation of J must be one of how many numbers?
(A) 3 (B) 4 (C) 5 (D) 6 (E) 7
i found 4.
lets say the integers are 1, 3 and 5. the four integers can be:
1 1 3 5
1 1 1 5
1 1 5 5
1 3 3 5
1 3 5 5
1 5 5 5
but two pairs have the same SD. therefore finally there are 4 different SDs.



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is it true that we must have 1 and 5 in J
Because the quest just say the greatest diff between any two number = 4. It doesn't mean that there must be 2 num in J which their diff =4
I think this is a too tough quest and it wastes too much time.










