12 Days of Christmas GMAT Competition - Day 8: Set X = {a, b, c, d, 9}

Question

12 Days of Christmas GMAT Competition with Lots of Fun

Set X = {a, b, c, d, 9}, where a, b, c, d and 9 are five distinct positive integers. If 9 is the only odd integer in the set, what is the average (arithmetic mean) of set X?

(1) The standard deviation of set X is 2.
(2) The range of {a, b, c} is 4 and the average (arithmetic mean) of {a, b, c} is 10.

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zhanbo · Accepted Answer

My answer is  (A) . It took me three minutes.

In order to calculate the average of set X = {a, b, c, d, 9}, we either need to EITHER figure out individual values for a, b, c, and d OR the sum of the 4 numbers.

(1) The standard deviation of set X is 2.
Because of the square root calculation, SD is typically an irrational number. Here, wow, it is actually an integer!
I am a bit surprised that GMAT expects us to know how to calculate SD (as opposed to some understanding to SD: SD is used to measure the degree of the dispersion of data distribution, for example.)
So, it is purely through intuition that I tried the following set:
6, 8, 9, 10, 12
(The intuition is inspired by the realization that the only odd number 9 must be the medium. )
I am astonished to find its SD is 2!
Other sets will have different SD (and an irrational number). 
SUFFICIENT

(2) The range of {a, b, c} is 4 and the average (arithmetic mean) of {a, b, c} is 10.
Here, we can determine that {a, b, c} are {8, 10, 12}
But we have no info about the value of d.
INSUFFICIENT

Therefore, answer is (A).

Biddu86 · Answer

X = {a,b,c,d,9}
Avg = (a+b+c+d+9)/5

1)SD=2
9-(a+b+c+d+9)/5=2 =>a+b+c+d=26
Avg=(26+9)/5=7  Sufficient
 
2)Range of {a,b,c} is 4 and avg. is 10
We don't have any info on value of d. Hence  Insufficient

Ans is A.

BIDJI23 · Answer

Nice question.

If we combine both propositions, we know that with B that a,b,c are 8,10,12. And because of the standard deviation is 2, d has to be 6. And then we can conclude and find the answer

But as  Math Revolution  says: check the single answer if C is easy to find!!!!! We can eliminate B because d can have many values.

But Answer A alone is interesting. 
If the SD is 2. It means that the sum of the square deviation from the mean is (2^2)*5 because there are 5 elements in the set. So it's 20.
So the sum of the difference is even and all the numbers in the set are even except 9, 9 is the mean. If 9 is not the mean then we can not have an even SD. If 9 is the mean and the rest of the numbers are integers, even and distinct, the only possibility is 6,8,9,10,12. Then...

PUSH A

azizgmat · Answer

The answer is A.

If standart deviation will be 2 and a, b, c, and d are all even and distinct there are not many options, clearly they all have to be close to 9 since SD is 2 and there is only one condition 6, 8, 10, and 12. I found the numbers using statement 2 but actually it is not needed.

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Deepakjhamb · Answer

Set X = {a, b, c, d, 9}, where a, b, c, d and 9 are five distinct positive integers. If 9 is the only odd integer in the set, what is the average (arithmetic mean) of set X?

(1) The standard deviation of set X is 2.
(2) The range of {a, b, c} is 4 and the average (arithmetic mean) of {a, b, c} is 10.

Solution :

Lets analyze contstraints in set first :

1) one nos =9 , total 5 distinct integers positive integers , also 9 is only odd number in set , therefore we infer we have other 4 postive even numbers which are different from one another
we need to find mean avg of set

Statement 1) Standard deviation of set A = 2

we have to understand the in case we have 4 consecutive even integers the standard deviation for these 4 number is minimum, ie = slightly above 2
 
only if we have 4 consecutive even integers and if the mean of even integers coincide with 9 , only then we can have standard deviation of set as 2

lets take example 6,8,10 ,12 and 9 we have mean =9 and SD =2

If we take any other 4 consecutive even integers the mean will shift and not coincide with 9 , in case we try to match not consecutive even numbers to fit in to match mean of 9 we will increase the standard deviation of set greater than 2 
and cannot match the standard deviation

so only possible even numbers that fits in standard deviation of 2 considering all distinct even numbers need to be there and one number is 9 is 6,8,10,12 and so mean of this set is 9 only

so statement 1 is sufficient 
statement 2 )The range of {a, b, c} is 4 and the average (arithmetic mean) of {a, b, c} is 10.

lets consider a<b< c as all 3 are distinct even numbers

we have c-a= 4 and a+b+c =30

so we have 2a+b = 26 , only possible set of even numbers that satisfy this is a=8 , b= 10 , c=12 but we know nothing about d - it could be 4 , 6 , 14 , 18 , any even positive number other than 8,10 or 12 , so the setA can have different means possible

so statement 2 is insufficient

so answer is a )Statement 1 alone is sufficient

rocky620 · Answer

Statement-1 
The SD of set X is 2

For SD to be 2, the Variance should be 4

Variance = Sigma (X - X')^2/n
X = set element
X' = Mean of the set
n = Number of terms (which we know is 5)

From this we get Sigma (X - X')^2 = 20, since (X - X')^2 is a perfect square we can only have 3 cases, which will have sigma (X-X')^2 as 20

Case-1:  4, 4, 4, 4, 4.....we can eliminate it since all terms are distinct and only 2 numbers have a square 4
 Case-2:  1, 9, 0, 9, 1.....we will keep it, as there are 5 numbers which have these as squares (-1, -3, 0, 3, 1)
 Case-3:  0,0,0,16,4.......we can eliminate it as well, same reason as Case-1

no other cases are possible.

For Case 2, the remaining 4 numbers should be equidistant from 9, which will be the middle term and hence the mean of the set X.
 SO set X = {6, 8, 9, 10, 12}
 Mean = 9

Hence sufficient

Statement-2 
We are given that range of a, b, c is 4 and mean is 10, so a, b, c will be 8, 10, 12.
But the mean of the set will change depending on the value of d.
  Hence Insufficient

IMO option A

DJ2911 · Answer

Correct Answer: A

Statement1: Std(X) = 2 
thus (sum(x - mu)^2)/5)^0.5 = 2
sum(x - mu)^2 = 20........(i)
Considering 9 is the only odd number, and the others are positive even numbers, we need to realize if there is a specific combination that will lead to the above equation (i) or whether it can be achieved by any other combination. 
By hit and trial we can see that one such combination is:

X= {6, 8, 9, 10, 12}
In this case, mean(mu) is 9 and the sum(x-mu)^2 = 20

We can see that there can't be any combination that will have a value of equation (i) less than 20. And any other combination of data such as (8,9,10,12,14}...etc. etc..will have the value of equation (i) more than 20. Thus we know the values of a,b,c,d will be one of {6,8,10,12}
Thus statement 1 alone is sufficient for us to find all the values of the set and hence the average of the set.

Statement 2: :The range of {a, b, c} is 4 and the average (arithmetic mean) of {a, b, c} is 10 
This tells us that a,b,c are one of {8,10,12} but doesn't give us any info about d. Thus this statement alone is not sufficient to find the average of the set X.

vv65 · Answer

My answer is A: Statement (1) is sufficient

Evaluating statement (2):  The range of {a, b, c} is 4 and the average (arithmetic mean) of {a, b, c} is 10

Therefore, {a, b, c} = {8, 10, 12}
d can be any even integer other than 8, 10, or 12
The mean will have different values for different values of d
Statement (2) is insufficient

Evaluating statement (1):  The standard deviation of set X is 2

The sum of the squares of differences from the mean will be 20

Set X has to be { 6, 8, 9, 10, 12}
This is the only set that satisfies the conditions and has a standard deviation of 2

Therefore the mean will always be 9
Statement (1) is sufficient

To get to this answer, I have used the numbers from Statement (2)

KarishmaB · Answer

The question is not GMAT-like because you need to calculate the SD of 5 numbers. GMAT expects you to understand the SD concept and use it effectively but not use the formula of calculating SD.

The answer of course is (A) and this is how I got it.

Set X = {a, b, c, d, 9}

a, b, c and d are distinct even integers so something like 6, 8, 10, 12 or 2, 6, 10, 12 etc.

To get the mean, we need to know all the 4 unknown numbers or at least mean of some of them and the other numbers.

(1) The standard deviation of set X is 2.

Normally, I would ignore and move on. There are many ways in which one can get a particular SD and I don't expect GMAT to make me calculate the SD. But the odd thing is that the SD is very small. Considering that all numbers must be distinct and not consecutive, they must be tightly packed so they would be able to take very few values.

The closest packed set can be 6, 8, 9, 10, 12 which gives an SD on 2 upon calculation. This means the moment I change any one number, the SD will increase. So the numbers must be these and their avg would be 9.

Sufficient.

(2) The range of {a, b, c} is 4 and the average (arithmetic mean) of {a, b, c} is 10.

This tells us that a, b, c are 8, 10, 12 (range 4 and avg 10 with 3 distinct even integers) but tells us nothing about d. 
Not sufficient

Answer (A)

Thekingmaker · Answer

This takes it to beat the system or the gmat system i should be assolutely honest this has to do something largely with intution
A gives us the SD which helps us to making the assumption that it could be 10 or 8 as it cannot be odd number hence i treaded casually between 6 and 12 respectively and whoosh and with 2 SD we are able to figure out the arithmatic mean too which's 9
B gives us a lot of information howecer we get different values when we try to pin down the value for d hence .....

IMO A hope it helps satisfaction overflowing

bumpbot · Answer

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12 Days of Christmas GMAT Competition - Day 8: Set X = {a, b, c, d, 9}

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12 Days of Christmas GMAT Competition - Day 8: Set X = {a, b, c, d, 9}

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