What is the standard deviation of Q, a set of consecutive

[Guest]

From MGMAT CAT #3

What is the standard deviation of Q, a set of consecutive integers?

(1) Q has 21 members.

(2) The median value of set Q is 20.

The answer is A and part of the explanation states: "The only reason we can find the difference between each term in the set and the mean is because we know that the set abides by a specified consecutive pattern and because we are told the number of terms in this set."

Would the statement in bold be true if statement 1 had told us that there were an even number of integers? OR do we only know the statement in bold is true b/c there are 21 (an odd number) of members?

Thanks!

[Guest]

It wouldn't matter how many consecutive as long as they are consecutive.

To see this, remember that standard deviation is the measure of how far the numbers are from 'mean' of the set. In this case, we know the mean (since we know they are consecutive and we know how many there are), we can calculate how far they each are. So (A) is sufficient.

If you remember the formula for calculating standard dev, it is std dev = sqrt[1/n * sum ((x - mean)^2) ]. All we need to know is how far each element is from the mean (to calculate x - mean) and how many numbers are there (n).

[JadranLee]

Just to clarify the comment below, you can figure out the standard deviation of a set if, as in this problem, you are told the mean and the fact that the set consists of n consecutive integers, where the number n is specified. It doesn't matter whether n is odd or even, as explained below.

It wouldn't matter how many consecutive as long as they are consecutive.

To see this, remember that standard deviation is the measure of how far the numbers are from 'mean' of the set. In this case, we know the mean (since we know they are consecutive and we know how many there are), we can calculate how far they each are. So (A) is sufficient.

If you remember the formula for calculating standard dev, it is std dev = sqrt[1/n * sum ((x - mean)^2) ]. All we need to know is how far each element is from the mean (to calculate x - mean) and how many numbers are there (n).

[rschunti]

Can you give exact example that help me understand the concept as to why option "A" will be true?

[RonPurewal]

It wouldn't matter how many consecutive as long as they are consecutive.

not true as written.

possibility (a) - literal interpretation of what you wrote:
IF you're trying to say that the standard deviation is the same number, regardless of the number of consecutive integers in the set, then that's wrong.
you could use the formula you wrote to prove this, but here's a more conceptual way to think about it:
say we have a set of 7 consecutive integers.
then these integers are 3 less than, 2 less than, 1 less than, equal to, 1 more than, 2 more than, and 3 more than the mean.
now let's say we have a set of 9 consecutive integers.
then that's the same as the set of 7 consecutive integers - except we've added numbers that are 4 less than and 4 more than the mean. since these new numbers are farther from the mean than any of the pre-existing numbers, it follows that the standard deviation must be a bigger number once we've added those numbers.

possibility (b)
IF you're trying to say that it's good enough to be given the EXACT NUMBER of consecutive integers in the set, REGARDLESS of what that number actually is, then you're right.
in other words:
any set of, say, 7 consecutive integers must have the same standard deviation as any other set of 7 consecutive integers.
(the reason is because, as mentioned above, any such set consists of numbers that are 3 less than, 2 less than, 1 less than, equal to, 1 more than, 2 more than, and 3 more than the mean, regardless of the actual value of the mean.)

[RonPurewal]

Can you give exact example that help me understand the concept as to why option "A" will be true?

take a look at the post directly preceding this one.

basically, rschunti, the idea is that you don't necessarily need the actual VALUES of the numbers in a set in order to calculate the set's standard deviation; rather, all you need is knowledge of those numbers' DISTANCES FROM THE MEAN.
if you have a set of a known number of consecutive integers, then you know all those distances.

specific example: same one as in the previous post - if you have a set of 7 consecutive integers, then they are 3 less than, 2 less than, 1 less than, equal to, 1 more than, 2 more than, and 3 more than the mean.
those distances will give a fixed standard deviation.

[Jazmet]

Some more help -

The procedure for finding the standard deviation for a set is as follows: 1) Find the difference between each term in the set and the mean of the set. 2) Average the squared "differences." 3) Take the square root of that average. Notice that the standard deviation
hinges on step 1: finding the difference between each term in the set and the mean of the set. Once this is done, the remaining steps are just calculations based on these "differences." Thus, we can rephrase the question as follows: "What is the difference between each term in the set and the mean of the set?"

[RonPurewal]

The procedure for finding the standard deviation for a set is as follows: 1) Find the difference between each term in the set and the mean of the set. 2) Average the squared "differences." 3) Take the square root of that average.

while this is of course true, this is exactly the kind of thing that you don't have to know at all on the gmat exam.

for the gmat exam, the only knowledge you need regarding the standard deviation is the vague / intuitive concept of what it measures. (by contrast, you need a quite exact idea of what, say, the mean and median are all about.)

in essence, you just need to know that ...
* the SD is a measure of how spread out the data are.
* if you include more data points at the mean (or at least closer to the mean than most of the existing values), the SD will decrease.
* if you include more data points far from the mean, the SD will increase.

... and that's it; that's the only stuff you have to know here. (i actually don't know from memory how to calculate standard deviations, so any question depending on the whole "add squares, square root" thing would stump me. but the good news is that they just won't give you that.)

in this problem, all you need is the first of these three statements (since you aren't changing the data set).
consider any set of 21 consecutive integers, and then consider the set of the integers 1 through 21.
no matter what, the spacing, or "spread", in those data sets will be exactly the same -- i.e., you can form any set of 21 consecutive integers by taking 1 through 21 and just "sliding" them up or down the number line by however many units.
since the spread of all the numbers is the same, the standard deviation of all such sets will always be the same. there's no need to calculate it (or even to know how!).

--

Thus, we can rephrase the question as follows: "What is the difference between each term in the set and the mean of the set?"

not exactly; you can have the same standard deviation in some cases where the answers to this question will differ. (conceptually, if you move some data points closer to the mean, but then move others farther away from it, then you can keep the same SD if you move them by exactly the right amount.)

[Jazmet]

Ron, Thank You for your inputs!

[RonPurewal]

you're welcome

[rfpchua]

Hi All,

Can you please tell me whether the GMAT considers the Stand deviation to have a denominator of n or n-1

The Quant Review on Page 27 says you divide by n, i.e.
Square root of [(Sum of squared differences]/n

I am certain that in proper statistics you need to divide by n-1

[RonPurewal]

Hi All,

Can you please tell me whether the GMAT considers the Stand deviation to have a denominator of n or n-1

this is not something you will ever have to know on the gmat exam, so it's not worth worrying about. (in statistics it can be either; the decision depends on how much is known, vs. how much is being inferred, about the overall situation/data set.)

as far as what you do need to know about the standard deviation for this exam -- scroll up 3 posts from your post, and check out what i wrote there.