posted by rachel .
suppose a data set has n elements with n greater than or equal to 6. as you know, if n is even, then the median of the set does not have to be an element of the set. for what values of n does the first quartile not have to be an element of the set?
The answer is any number that isn't one of the following: 7, 11, 15, 19 etc, that is, any integer that can't be expressed in the form (4k-1) for k greater than or equal to 2 (as you've said that n has to be greater than or equal to 6). Having said that, I'd have enormous difficulty proving it!