frank1 wrote:If Q is a set of consecutive integers, what is the standard deviation of Q?
(1) Set Q contains 21 terms.
(2) The median of set Q is 20.
OA later
Standard deviation is a measure of how the set is
dispersed around its mean. thus, two sets with with the same dispersal patterns around their respective means will have the same standard deviation, even if the mean itself is different.
Take a simpler example of 3 consecutive integers: two sets, {1,2,3} and {101, 102, 103}. The mean for both sets is the median - the number in the middle, 2 and 102, but the standard deviation of both sets will be the same: For both sets, the three terms present the same "deviation" or dispersal from the respective mean
one term "1 below" the mean (1 and 101, respectively)
one term equal to the mean
one term "1 above" the mean (3 and 103, respectively).
and the SD is sqrt( ((-1)^2 + 0^2 + (1)^2)/3 ).
The bottom line is this: For a set of consecutive integers, The mean is always the median (the middle term for an odd number of terms). But the mean, in effect, is meaningless: to find the SD, all you need is the deviation of each member of the set from the mean.
Stat. (1) whatever these 21 terms are (1-21, 101-121, 165-181, whatever), the mean will be the 11th term, and the dispersal pattern around it is the same: 10 cons. terms above, 10 cons. terms below. The STD can be calculated (not that you'd be insane enough to do so)
sqrt( (-10)^2 + (-9)^2 + ..........9^2 + 10^2)/21 )
and it's the same regardless of whether the median is 20, 40 or 10,000. Sufficient.
Stat. (2) alone tells you nothing about the STD, because you don't even know how many terms are there in the set around the median of 20. We can build two sets with median 20:
{19,20,21}
{18,19,20,21,22}
and each of these will have a different STD. Insufficient.
Answer is
A.
