GMATPREP Statistics question pls explain how to answer

[rschunti]

Pls can you explain what is the best approach to solve this problem.What is the concept behind solving this particular problem.?
The average of 100 numbers is 6, and the standard deviation is D, where D is positive. When added which of the following numbers, the new deviation will be less than D?

A. -6 and 0

B. 0 and 0

C. 0 and 6

D. 0 and 12

E. 6 and 6

[StaceyKoprince]

Please make sure to transcribe problems exactly. Something's missing from the second sentence of the question. Am I adding the numbers in the answer choices to the 100 numbers? Or am I adding them to the standard deviation? (I assume the former, or the problem wouldn't make much sense, but please confirm.)

[rschunti]

Thats correct. You have to add the numbers in the answer choices to the 100 numbers.

[RonPurewal]

well, standard deviation is a measure of how spread out the data are. in other words, the standard deviation gives you a rough idea of how far away the numbers are from the average (mean).

hence the name: standard (= typical) deviation (= difference from the norm). notice that standard deviation is not simply the average of how far away the data points are from the average, but that notion is more than sufficient for conceptual understanding.

in any case, you have no idea how big the standard deviation is. it could be something like 0.0000001, meaning that every number in the set is extremely close to the average value of 6. therefore, the only way to be absolutely sure you're reducing the standard deviation is to insert two numbers each of which is exactly the average. since these points are both 0 units away from the average, they will lower the standard deviation.

[shaji]

Quite right!!!. An elegant qualitative approach.

Here's a Quantititve .

Stand Dev= Sqrt(Average of the squares of numbers - (mean of numbers)^2.

The correct answer is therefore E.

well, standard deviation is a measure of how spread out the data are. in other words, the standard deviation gives you a rough idea of how far away the numbers are from the average (mean).

hence the name: standard (= typical) deviation (= difference from the norm). notice that standard deviation is not simply the average of how far away the data points are from the average, but that notion is more than sufficient for conceptual understanding.

in any case, you have no idea how big the standard deviation is. it could be something like 0.0000001, meaning that every number in the set is extremely close to the average value of 6. therefore, the only way to be absolutely sure you're reducing the standard deviation is to insert two numbers each of which is exactly the average. since these points are both 0 units away from the average, they will lower the standard deviation.

[rschunti]

Thanks Ron for nice explanation. Is it possible for you to take any other numbers besides 6 and 6 and show why the Standard deviation will not decline? Also what is the best approach to answer these types of questions in GMAT as we are time constrained.? Do I need to remember these concepts or would have solved this question by taking quantitative approach(using formula).Pls advise?

[StaceyKoprince]

It's highly unlikely that you will have the time to use a quant approach to standard deviation on the test. Without a calculator, this just isn't something you can do in 2 min. For SD, then, it's important to make sure you understand the concepts so that you can use Ron's qualitative approach - that will be sufficient for SD questions b/c they know the formula is too complicated to be done on paper in 2 min.

And to answer your first question - we don't actually know for sure that it will not decline (or that it will!) unless we know something about the values in the set. (This is true for all the other options, not E.)

The problem tells us that D is positive - that is, there is a standard deviation (the other option is for it to be zero, in which case every number in the set is identical). If there is a standard deviation, that means that the numbers are not all identical - at least one is different.

So, let's say I have 0, 6, and 12. The average is 6, and there is some positive standard deviation, because the numbers are not all equal to the average. Note that only the 0 and the 12 contribute mathematically to the positive standard deviation. The 6 contributes an "SD value" of zero, because it is identical to the average, and the SD is a measure of the difference between any number in the set and the entire set's average.

Now, if I add two more 6's to the set, I'll have 0, 6, 6, 6, and 12. This will still have a positive standard deviation, because all of the numbers are not 6, but the standard deviation will be smaller than it was in my first group of three numbers because there are three 6's in this group - so three out of five numbers are identical to the average now, and all three are contributing "SD values" of zero, while only two numbers (0 and 12) are contributing to some positive SD. So the overall SD here will be smaller than it was in the example in the previous paragraph because the three 6's are "outweighing" the 0 and the 12 (more so, at least, than one 6 does).

[shaji]

In this particular problem, the quant method is much faster to arrive at the correct answer.

The new variance is (1/100-1/102) of D^2 and is evident within 15 secs in case of E.

It's highly unlikely that you will have the time to use a quant approach to standard deviation on the test. Without a calculator, this just isn't something you can do in 2 min. For SD, then, it's important to make sure you understand the concepts so that you can use Ron's qualitative approach - that will be sufficient for SD questions b/c they know the formula is too complicated to be done on paper in 2 min.

And to answer your first question - we don't actually know for sure that it will not decline (or that it will!) unless we know something about the values in the set. (This is true for all the other options, not E.)

The problem tells us that D is positive - that is, there is a standard deviation (the other option is for it to be zero, in which case every number in the set is identical). If there is a standard deviation, that means that the numbers are not all identical - at least one is different.

So, let's say I have 0, 6, and 12. The average is 6, and there is some positive standard deviation, because the numbers are not all equal to the average. Note that only the 0 and the 12 contribute mathematically to the positive standard deviation. The 6 contributes an "SD value" of zero, because it is identical to the average, and the SD is a measure of the difference between any number in the set and the entire set's average.

Now, if I add two more 6's to the set, I'll have 0, 6, 6, 6, and 12. This will still have a positive standard deviation, because all of the numbers are not 6, but the standard deviation will be smaller than it was in my first group of three numbers because there are three 6's in this group - so three out of five numbers are identical to the average now, and all three are contributing "SD values" of zero, while only two numbers (0 and 12) are contributing to some positive SD. So the overall SD here will be smaller than it was in the example in the previous paragraph because the three 6's are "outweighing" the 0 and the 12 (more so, at least, than one 6 does).

[StaceyKoprince]

If you completely know how to calculate SD, then yes, you can do the calculations more quickly. But most people won't be able to do this approach efficiently enough - and it isn't necessary to learn how to do this for the test. Given everything else people have to learn, the quant method for SD should be low on the priority list, given that SD problems can be done just from understanding the concepts logically.