Mark R Marino wrote:
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> Why do you think that the algorithms in these computer packages differ?
> Why aren't they universal?
> Any thoughts or comments?
The reason is that there is not and has never been (since five minutes
after the first person to think of using quartiles went down to the pub
and told his buddies about it) an agreement on the *definition* of
quartiles unless N = 4n+1.
The median is not well defined unless N is odd. Usually it is taken as
the mean of the two adjacent values, but this is artificial, goes
against the "non-arithmetic", transformation-invariant nature of the
median, and cannot be extended to ordinal data. It's realy an interval
for even N, but many people have a prejudice against returning an
interval instead of a number here ("dammit, I started with 11 numbers,
and you've given me an uncountable set, and you call that a
_summary_?!?!?!") so something arbitrary usually gets done.
Now, with quartiles, there are several approaches. If you think of Q1
and Q3 directly as quantiles, you may choose weighted means, based on
the idea that (N+1)/4 may be a quarter-integer so closer to one
neighbour than to the next; unweighted means, taking the "most generic"
member of the interval. If you think of "medians of lower/upper halves"
it is tempting to use the "fencepost" ambiguity in the middle to ensure
that this always exists.
I would like to make a small, possibly novel, proposal here; that for
EDA purposes this should be done, because the depiction (in a boxplot,
etc) of actual data is in the spirit of the exercise; and that when N is
even and the median not repeated, the box should be marked with both
data adjacent to the median quantile.
____ _ ____
Thus: -----------I____|_|____I---
-Robert Dawson
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