There are others on this list who can say much more sensible things on
this type of thing than I, so I forwarded this to the list (this is in
general the preferred way of responding on the statalist, see point 5
in the statalist FAQ)
-- Maarten
--- Brooks Taggert J wrote to me privately:
> Thanks for the email. I realize part of the problem may in fact be
> the term fractional median. I looked at the link you sent, but I
> can't seem to find a matching statistic among the L-moments and
> Quantiles. I'll give another look because I could be missing it.
>
> Here is the explanation of fractional median my university uses.
>
> 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).
>
> -----Original Message-----
> From: [email protected]
> [mailto:[email protected]] On Behalf Of Maarten
> buis
> Sent: Saturday, January 17, 2009 9:18 AM
> To: [email protected]
> Subject: Re: st: Fractional Median
>
> --- 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?
>
> I am not quite sure what you mean with fractional median, but it
> sounds
> similar to what Nick Cox has done with -hdquantile-, see:
> -ssc describe hdquantile- and Nick's talk at the 2007 Nordic and
> Baltic
> Stata Users' Group Meetings:
> http://ideas.repec.org/p/boc/nsug07/1.html
>
> Hope this helps,
> Maarten
>
> -----------------------------------------
> Maarten L. Buis
> Department of Social Research Methodology
> Vrije Universiteit Amsterdam
> Boelelaan 1081
> 1081 HV Amsterdam
> The Netherlands
>
> visiting address:
> Buitenveldertselaan 3 (Metropolitan), room N515
>
> +31 20 5986715
>
> http://home.fsw.vu.nl/m.buis/
> -----------------------------------------
>
>
>
> *
> * For searches and help try:
> * http://www.stata.com/help.cgi?search
> * http://www.stata.com/support/statalist/faq
> * http://www.ats.ucla.edu/stat/stata/
>
>
>
-----------------------------------------
Maarten L. Buis
Department of Social Research Methodology
Vrije Universiteit Amsterdam
Boelelaan 1081
1081 HV Amsterdam
The Netherlands
visiting address:
Buitenveldertselaan 3 (Metropolitan), room N515
+31 20 5986715
http://home.fsw.vu.nl/m.buis/
-----------------------------------------
*
* For searches and help try:
* http://www.stata.com/help.cgi?search
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/
