I like the idea of marking quartiles on a dotplot. I would suggest (a)
using "Q" rather than "Q1" or "Q3" as it is smaller and there 
is no chance of confusion; (b) putting these symbols on the axis, so 
instead of 

>                                           :
>                                           :                     .
>                       .         .         : .  : :    :         :
>                  . :  :    :    :    .    : :  : :    :  .    . :
>               :  : :  :    : :  : :  :    : :  : :  : :  :    : :
>               :  : :  : .  : :  : :  :    : :  : :  : :  : .  : :
>               :  : :  : :  : :  : :  : :  : :  : :  : :  : :  : :
>               :  : :  : :  : :  : :  : :  : :  : :  : :  : :  : :
>            ---+---------+---------+---------+---------+---------+---C1
>            20.0      24.0      28.0      32.0      36.0      40.0
>                       Q1           M         Q3

putting

                                           :
                                           :                     .
                       .         .         : .  : :    :         :
                  . :  :    :    :    .    : :  : :    :  .    . :
               :  : :  :    : :  : :  :    : :  : :  : :  :    : :
               :  : :  : .  : :  : :  :    : :  : :  : :  : .  : :
               :  : :  : :  : :  : :  : :  : :  : :  : :  : :  : :
               :  : :  : :  : :  : :  : :  : :  : :  : :  : :  : :
            ---+---------+--Q------+-------M-+-------Q+---------+---C1
            20.0      24.0      28.0      32.0      36.0      40.0

        (note corrected locations! This is *not* just a quibble - I was
able to read off the median and quartiles accurately enough from the
dotplot to notice -without any deliberate attempt to check - that the
values were incorrect [mailer tab interpretation?].

Thus, while your claim 

>BUT, they were not designed to report on the median (though sometimes you 
>can find it), nor the quartiles (though sometimes you can find them)...

        is no doubt technically correct (in terms of "design"), that's not
really the point.)

I must say I don't like the name "doxplot" very much, although I admit
that
Tukey, coiner (it seems) of most of the really flamboyant neologisms in
statistics today, might just have... this is partially because I am a
strong proponent of boxplots for outlier detection, so this device does
not have the "boxplot nature" to my mind.

        -Robert Dawson
.
.
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