Standard deviation (lost in the 2nd option)

El Cucu wrote:There is a set of numbers, the mean is m, the standard deviation is R, if add a number x to this set, is the new standard deviation bigger than R?

1) x=m
2) m<x<m+R

I'm not sure where the problem is from, but it's not worth spending any time on. The two statements are clearly contradictory for one thing (m can't both be equal to x and less than x), so it's impossible to consider them together - that never happens on a real GMAT question. It also tests properties of standard deviation that no test taker could possibly be expected to know.

Stuart Kovinsky wrote: In this question, the mean is m and the SD is R. Any number closer to the mean than a distance of R will reduce the SD; any number further from the mean than a distance of R will increase the SD.

That isn't true, unfortunately. If you add a single new number to a set which is different from the mean, all kinds of things happen: the mean changes, the number of elements changes, and all of the distances to the mean change. You can certainly add an element to a set which is more than one standard deviation from the mean, and have a lower standard deviation after doing so - try adding the element 21, for example, to the set {0,0,20,20}.

What *is* true, and what is sometimes tested in GMAT questions, is that if you add *two* elements to a set which have the same average as the set itself, and which are each exactly one standard deviation from the mean, the standard deviation will not change. That is, if you have a set with a mean of 0 and a standard deviation of 5, if you add the elements 5 and -5 to the set, the standard deviation will remain equal to 5. By extension, if you add pairs of elements which are less than one standard deviation from the mean, and have the same mean as the set itself, the standard deviation will go down (and analogously if the elements are more than one standard deviation from the mean, the standard deviation will go up). That's a much simpler situation, since the mean remains unchanged, and we only need to consider the distances to the mean of the newly added elements.

There's no easy way a GMAT test taker could evaluate Statement 2 in this question, given what the GMAT expects test takers to know about standard deviation; it's completely unrealistic. I'd be curious to know where it's from.

El Cucu wrote: 1 So the rule is that without knowing the number being added we can not know if the new SD goes up or down unless we know that adding from one side and substracting the same number from other side will obiously produce the same result.

2 Your example applies when there is 0 or 1 or also with other numbers? I mean what would happen if it says numbers are positive or negative but not cero or one or they are all different numbers? Set of numbers means that there is an order or not?

3. Regarding SD what would be the kind of concepts that can be tested in the exam? How do we know which concepts are out or wich are in? The OG mentions something about they?

Tksvm

Re 1, there are certainly many other rules about standard deviation in mathematics, but none that you would need to know for the test. You do need to know, conceptually, what standard deviation measures, and what would make the standard deviation of one set larger than the standard deviation of another. There is a real GMATPrep question which is related to the specific example I described:


www.beatthegmat.com/from-prep-standard- ... 32483.html

Re 2 - standard deviation is based only on the distances from each element in a set to the mean of the set. It doesn't matter whether the elements are positive or negative, large or small - these three sets have the same standard deviation, because the distances between elements are the same:

{1, 2, 3}
{-11, -10, -9}
{944.4, 945.4, 946.4}

I'm not sure I understand your question about 'order', but sets are not ordered; the set {1, 2, 3} is the same as the set {2, 3, 1}. We often choose to write our sets in a particular order (typically increasing order), because this makes it easier to identify the median. However, strictly speaking, sequences are ordered, and sets are not.

Re 3 - the best way to see what the GMAT might test is to see as many real, retired GMAT standard deviation questions as possible. The Official Guides and GMATPrep will give you a very good idea of what might be tested on the real GMAT. Standard deviation questions from other sources I have often found to be very poor - many test standard deviation in a way the GMAT never will, and are not worth studying.