cubicle_bound_misfit wrote:Hi Ian,
This may be really stupid but if we don't know about anything about set of 21 integers how come their standard deviation could be measured?
suppose, it is a set of 21 negative integers or a set of 21 positive integers
Remember what we use to calculate the standard deviation: the distance from each number to the average. I'll illustrate with an example - rather than use 21 consecutive integers, I'll just use seven for simplicity. Say you have this set:
0, 1, 2, 3, 4, 5, 6
The average of this set is 3. Now, to find the standard deviation, we first find how far each element is from the average, by subtracting (and making the result positive). Starting from 0, the distances from each element to the average (3) are:
3, 2, 1, 0, 1, 2, 3
These distances then go into the standard deviation formula, which you don't need to know in detail for the GMAT - and to save time, you'd definitely not want to go any further with the calculation. (Though if you want to know the calculation, you first square each distance, to get: 9, 4, 1, 0, 1, 4, 9; you then average this set to get (9+4+1+0+1+4+9)/7 = 28/7 = 4; and finally you take the square root to get 2).
Notice that it's the distances between elements in the set that matters here, and not the elements themselves. If we take a different set of seven consecutive integers, say:
990, 991, 992, 993, 994, 995, 996
then the average is 993, and the distances are again:
3, 2, 1, 0, 1, 2, 3
So plugging these distances into the standard deviation formula, we'll get the same answer as we got with the set {0,1,2,3,4,5,6}, since the distances are identical. The same would happen with a set of seven consecutive negative integers, or any other set of seven consecutive integers.
The upshot is this: whenever you know all of the distances between elements in a set, you can find the standard deviation. You don't actually need to know how large or small the elements are; you only need to know how they are separated. Every set of seven consecutive integers will have the same standard deviation, for example.
