David Gale writes: << Mike Greenwald asks, how a median can be 3.8. Good question! . . . Anyhow, someone please help me out on those fractional medians
In my fairly extensive interaction with statisticians + stat textbooks, all definitions I've seen of a median of a finite sample of N real-valued data are identical: If N odd, median = middle value; if N even, median = average of the 2 middle values. BUT -- there is a more sophisticated definition that may exist: If N is even, and the two middle values are distinct but at least one of them occurs with multiplicity > 1, one might prefer to use the "asimov median", defined as follows: Assume the data sorted by size: x(1) <= x(2) <= . . . <= x(N). Let J be the smallest interval of indices that is symmetrical about the real number N/2, such that J contains a) all repetitions of x((N+1)/2) (if N is odd), or b) all repetitions of both data x(N/2) and x(N/2 + 1) (if N is even). Now define the "weighted median" as the average of all values whose indices lie in J. Thus if N = 200 and the sorted data looks like this: sort #: . . . 94 95 96 97 98 99 100 101 102 103 104 105 106 . . . datum: . . . 2 3 3 3 4 4 4 4 4 4 4 4 5 . . . then one may wish to say the asimov median is the average of data whose indices k satisfy 100.5 - 5 <= k <= 100.5 + 5, i.e., 96 <= k <= 105, so we get a.m. = (3+3+4+4+4+4+4+4+4+4)/5 = 3.8 . --Dan