Stats question

[dhoomketu]

Q: Set A is composed of nine numbers, labeled A1 through A9. Set B is also composed of nine numbers, labeled B1 through B9. Set B is defined as follows: B1 = 1 + A1; B2 = 2 + A2; and so on, including B9 = 9 + A9. How much larger is the sum of set B's mean and range than the sum of set A's mean and range?


The choices were:
4
9
13
17
cannot be determined

IMHO the answer should be 13 even after reading the MGMAT explanation:

The question asks for [( B's Mean +B's Range) - (A's Mean +A's Range)]. The MGMAT explanation however only considers the range. I took a general case by without plugging any values for A. The general solution then gives 13.

I must be making some mistake but can someone try a general solution.

[rfernandez]

To get a value, it will be necessary to express the range and mean of set B in terms of the range and mean of set A. That will allow for quantities to cancel out and leave behind just constants.

Set B's mean can definitely be determined in terms of set A's mean:

A's mean = (A1 + A2 + A3 + A4 + A5 + A6 + A7 + A8 + A9) / 9

B's mean = [(A1 + 1) + (A2 + 2) + (A3 + 3) + (A4 + 4) + (A5 + 5) + (A6 + 6) + (A7 + 7) + (A8 + 8) + (A9 + 9)] / 9
= (A1 + A2 + A3 + A4 + A5 + A6 + A7 + A8 + A9 + 45) / 9
= (A1 + A2 + A3 + A4 + A5 + A6 + A7 + A8 + A9) / 9 + 5
= A's mean + 5

So we know that B's mean is 5 more than A's mean.

The range of B cannot be determined in terms of the range of A, however. To illustrate this, let's assign some values to the sets:

A = {1, 2, 3, 4, 5, 6, 7, 8, 9} (A1 is the first element, A2 is the second element, and so on)
mean = 5, range = 8
B = {2, 4, 6, 8, 10, 12, 14, 16, 18} (B1 is the first element, B2 is the second element, and so on)
mean = 10, range = 16

So, B's mean + B's range - (A's mean + A's range) = 10 + 16 - (5 + 8) = 13

Now, let's try another set of values:

A = {9, 8, 7, 6, 5, 4, 3, 2, 1} (A1 is the first element, A2 is the second element, and so on)
mean = 5, range = 8
B = {10, 10, 10, 10, 10, 10, 10, 10, 10} (B1 is the first element, B2 is the second element, and so on)
mean = 10, range = 0

So, B's mean + B's range - (A's mean + A's range) = 10 + 0 - (5 + 8) = -3


I think that in your solution, you assumed that the values A1, A2, ... , A9 were listed in increasing size. If that's true, then I agree that the answer would be 13. But that must be explicitly stated and cannot be assumed.

Rey

[dhoomketu]

Thanks I did understand my mistake.
I was trying to come up with a general solution - the reason for this was that I didn't assume that A1-A9 were consecutive.

Anyways if i assumed them to be consecutive I understand the mistake in my general solution. I was taking for range:
Range of A=absolute(A1-A9)
Range of B=absolute[(A1-A9) + 8] = absolute(A1-A9) +8; which is incorrect

Do you think the question should say consecutive numbers in the statement? otherwise it is impossible to compute range for A and problem is much simpler...

thanks

[UPA]

To get a value, it will be necessary to express the range and mean of set B in terms of the range and mean of set A. That will allow for quantities to cancel out and leave behind just constants.

Set B's mean can definitely be determined in terms of set A's mean:

A's mean = (A1 + A2 + A3 + A4 + A5 + A6 + A7 + A8 + A9) / 9

B's mean = [(A1 + 1) + (A2 + 2) + (A3 + 3) + (A4 + 4) + (A5 + 5) + (A6 + 6) + (A7 + 7) + (A8 + 8) + (A9 + 9)] / 9
= (A1 + A2 + A3 + A4 + A5 + A6 + A7 + A8 + A9 + 45) / 9
= (A1 + A2 + A3 + A4 + A5 + A6 + A7 + A8 + A9) / 9 + 5
= A's mean + 5

So we know that B's mean is 5 more than A's mean.

The range of B cannot be determined in terms of the range of A, however. To illustrate this, let's assign some values to the sets:

A = {1, 2, 3, 4, 5, 6, 7, 8, 9} (A1 is the first element, A2 is the second element, and so on)
mean = 5, range = 8
B = {2, 4, 6, 8, 10, 12, 14, 16, 18} (B1 is the first element, B2 is the second element, and so on)
mean = 10, range = 16

So, B's mean + B's range - (A's mean + A's range) = 10 + 16 - (5 + 8) = 13

Now, let's try another set of values:

A = {9, 8, 7, 6, 5, 4, 3, 2, 1} (A1 is the first element, A2 is the second element, and so on)
mean = 5, range = 8
B = {10, 10, 10, 10, 10, 10, 10, 10, 10} (B1 is the first element, B2 is the second element, and so on)
mean = 10, range = 0

So, B's mean + B's range - (A's mean + A's range) = 10 + 0 - (5 + 8) = -3


I think that in your solution, you assumed that the values A1, A2, ... , A9 were listed in increasing size. If that's true, then I agree that the answer would be 13. But that must be explicitly stated and cannot be assumed.

Rey

I got my calculation in my mind in less than 20 seconds as under:

increase in range is 9-1 = 8
increase in mean = (1+2+ ... +9)/9 = 5
so the difference = 8+5 = 13

[RonPurewal]

I got my calculation in my mind in less than 20 seconds as under:

increase in range is 9-1 = 8
increase in mean = (1+2+ ... +9)/9 = 5
so the difference = 8+5 = 13

check out what rey said above: you can't make that inference about the range unless you assume that a9 and b9 are the biggest numbers, and a1 and b1 the smallest numbers, in their respective sets.

here's a counterexample:
let a1 = 9, a2 = 8, a3 = 7, ..., a1 = 9.
then every single one of b1, b2, b3, ..., b9 is 10, so the new range of the b's is now zero. that ought to kill your range hypothesis in a hurry.

IF you make the aforementioned assumption about the min/max values of the sets, then your reasoning is impeccable.

--

also, note that the part of your reasoning that deals with the means of the sets is perfectly correct.