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.
