Thanks to Steve and Maarten for the suggestions. I might be missing
something but it doesn't seem as though there isn't an easy
implementation.
For clarification here is what is meant by fractional median.
Explanation of Fractional Median
Using the median becomes problematic when the data set contains large
numbers of repeated values. In the case of SEI scores, the student can
only choose 1 of 5 values, and so, by design, the set of SEI scores of
most classes contains large groups of the same value. And so, if the
regular median were calculated, there would only be a few possible
results for all classes/instructors. The fractional median is used to
provide a wider range of possible results, while still maintaining some
of the desirable properties of the regular median. In terms of the
mathematics, the fractional median provides a more continuous range of
outcomes instead of the discrete set possible with the regular median.
Here is the basic idea (and an example): To arrive at a continuous set
of outcomes, one assumes that each data value is the center of the true
set of values that could have been measured. For example, when a student
selects a 3 instead of a 2 or a 4, one can assume that if the student
were allowed to choose any numerical value from the real line, they
would have selected something between 2.5 and 3.5, and since they were
not allowed to list their exact observation, they selected the nearest
choice, in this case a 3.
We call this range of values associated with each observable measurement
(choice) a bin or a cell. The cell for the choice 1 is .5 to 1.5, for a
2 it is 1.5 to 2.5, etc. We now want to calculate the fractional median,
which estimates what the median would have been if the student could
have selected any real value (not just the 5 choices given). First we
determine what cell the standard median lives in. The fractional median
will be a value from the cell that contains the standard median. We then
determine how far into the cell the median actually is (again assuming
they could have selected any value in the cell). This gives the
fractional median.
Example Data Set: two 2's, nine 3's, eight 4's, and eight 5's. (I picked
an odd number of values because it is a little tricky, the even number
case is a bit easier.)
This is a total of 27 measurements (student scores). Half of 27 is 13.5,
and so if we look for the thirteen and a half value, we end up looking
between two 4's. So the median is a 4, which comes from the cell ranging
from 3.5 to 4.5, and so the fractional median will be between 3.5 and
4.5.
The fractional median in this case will be 3.5 plus the percentage of
the distance into the cell the middle value represents. So if it were
the case that the true median was the middle 4, then the percentage of
the distance into the cell would be 50%= .5, thus the fractional median
would be a 3.5+.5=4 (so the median equals the fractional median in this
situation). In our example, the 4 that represents the true median would
be between the 2nd and 3rd four (the 2.5th four, let's say) of the eight
4's in the cell, which is 2.5/8 ths of the way into the cell. Now, since
2.5/8=.3125, the fractional median for this example would be
3.5+.3125=3.8125. Note: if more of the 3's were 4's, then the "middle 4"
would be a greater distance into the cell, resulting in a higher
fractional median (but the regular median would still be 4).
TJ
-----Original Message-----
From: [email protected]
[mailto:[email protected]] On Behalf Of Steven
Samuels
Sent: Saturday, January 17, 2009 2:58 PM
To: [email protected]
Subject: Re: st: Fractional Median
-pctile- with the -altdef- option gives an interpolated percentile
different from that in -sum-. See if that's what you need.
-Steve
--- Brooks Taggert J <[email protected]> wrote:
> My university uses the fractional median when calculating scores from
> student evaluation of instructors. I'm helping us move electronically
> and in the process was using Stata to do some of the statistics.
> Strangely I can't find an easy method (ie a pre-built ado) to
> calculate the fractional median, nor can I find a reference anywhere.
> Am I missing something?
>
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