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# J is a collection of four odd integers and the greatest

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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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30 Aug 2006, 04:21
Interesting.
Will give Ex. cause it will be easier to explain.
Diff-0 3,3,3,3
Diff-2 and 0 3,3,3,5
Diff-4 and 0 3,3,3,7
Diff-2 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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30 Aug 2006, 08:25
D - 6

Let four numbers are a,b,c,d and are in increasing order. d-a = 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

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30 Aug 2006, 10:13
kevincan wrote:
ps_dahiya wrote:
D - 6

Let four numbers are a,b,c,d and are in increasing order. d-a = 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

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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30 Aug 2006, 13:44
Do {1,1,1,5} and {1,5,5,5} have different standard deviations?
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30 Aug 2006, 14:38
ps_dahiya wrote:
D - 6

Let four numbers are a,b,c,d and are in increasing order. d-a = 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

I think 7. The other possibility is where the standard deviation is 0
aaaa or dddd or zzzz
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30 Aug 2006, 14:49
ps_dahiya wrote:
D - 6

Let four numbers are a,b,c,d and are in increasing order. d-a = 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

Agree with this explanation... D should be the answer.
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30 Aug 2006, 15:00
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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30 Aug 2006, 16:07
The greatest difference between any two elements in a set is equal to the range of the set.
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30 Aug 2006, 18:39
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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31 Aug 2006, 06:57
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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06 Sep 2006, 17:57
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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06 Sep 2006, 19:12
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.
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