Re: another check on quality of Mark IV ensemble photometry

[Michael Richmond <richmond@stupendous.cis.rit.edu>]

  Tom asked questions about my table.  Sorry that I wasn't very clear.
Let's look at one small piece of the first table.


#  min   max  N        unclipped      clipped      median   interquartile
#                      mean  stdev    mean  stdev              mean  stdev
#----------------------------------------------------------------------------
       V-band

    7.0   8.0  1398   0.032 0.016    0.030 0.012    0.029     0.029 0.005
#----------------------------------------------------------------------------

  This single line means the following:

     - isolate all measurements of stars with mean ensemble magnitudes
               between V=7.0 and V=8.0. 
     - there are 1398 such stars in all the patches 
     - ignore the columns labelled "stdev" in the table, and
               pay attention only to the columns labelled
               "mean" or "median"
   
  Now, my primary goal in this work was to minimize the 
night-to-night scatter in measurements due to residual large-scale
"flatfield" issues.  I wanted to see how small the scatter
in a star's magnitude might be if one could consider it and just
the nearest neighbors, since some of the systematic errors will
affect all neighbors equally, and so cancel out.

  If you look at a plot of "scatter-vs-magnitude", you
typically see a curve which starts small, for bright stars,
and increases towards the faint end of the graph.
I wanted to know the typical value of the curve at each
bin in magnitudes -- that's what the "mean" or "median"
column in the table above indicates.  The _thickness_ of
the points around the curve at a given magnitude is 
what the columns above labelled "stdev" show.

  For most purposes, it's the value of the curve that's
important, not the thickness of the distribution around it.
So I'll ignore the "stdev" columns.

  The "unclipped" column shows that the standard deviation from
the mean magnitude of stars in this bin was 0.032 magnitudes.
That is, one particular star might have a mean value of V=7.32,
and then vary by about 0.032 mag from one night to the next
to the next...

  I worried that occasional image defects might cause a star
to appear much brighter or much fainter than usual -- and that
these outliers could cause ordinary statistics to appear
much worse than the usual case.  That is, if a star fell on
a bad pixel in one image, then it might be V=8.56 on that
one image.  If we include that one very diffferent
value together with the others, all close to V=7.32, and
then calculate the standard deviation from the mean, we might
find a very large value.  That would not be representative
of the usual small errors of size 0.032 mag.

  So, the "clipped" column shows the value of the scatter
after one round of clipping: I discarded any magnitude
values more than 3-sigma from the mean.  In the table, you
can see that this doesn't make much difference: the scatter
is now 0.030 mag instead of 0.032 mag, so outliers evidently
don't have a very large effect on stars in this bin.

  The "median" column is the median value of all differences
from the mean magnitude.  That is, I calculated the ensemble
mean magnitude, then for every measurement, found the absolute
value of the difference between it and the mean; I sorted all these 
values, and picked the middle one.  This was 0.029 mag -- again,
no big difference from the 0.032 mag unclipped scatter.

  The final columns show one more way of computing the
scatter around the mean: I again sorted the differences
between individual magnitudes and the mean value,
threw out the bottom 25% of these differences, threw
out the top 25% of these differences, and computed
the average of the remaining differences.  Once again,
it's just about the same as the unsophisticated value of 0.032.

  The bottom line is that, in most cases, outliers don't
contaminate the measurements very much.  One can use
rather simple statistical methods to extract accurate
measures of the Mark IV data quality.

                          Michael