This old message claims that <I>/<s> = <I/s> "weighted" by s.
If counting error is significant, s will be larger for stronger reflections, which are likely to
have small I/s, so in general <I>/<s> > unweighted <I/s>. As Werten points
out.
However this seems to be the opposite of the OP's situation, if both measures
refer to the same outer shell.
EAB
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b...@freesurf.fr wrote:
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From these we calculate a weighted mean <Ih> and an error estimate of
the mean sd(<Ih>), and a ratio for that unique reflection
<Ih>/sd(<Ih>)
What is printed as Mn(I)/sd is the mean value of that ratio for all
reflections (possibly in a resolution bin) ie
<<Ih>/sd(<Ih>)>
Which is very different (!) from anything one could extract from a
Scalepack/HKL2000 output file, which only lists average intensity and
average sigma (in the final table at the end). The ratio of those two
numbers, i.e. <I>/<sigma>, tends to be considerably more "optimistic" than
a proper <I/sigma>.
Werten leaves it as an exercise for the reader
to verify this. Here is my submission:
The (ratio of means) is equivalent to a weighted
(mean of ratios):
<I>/<s> = sum(I)/n / sum(s)/n
= sum (I) / sum(s)
= sum(s*(I/s)) / sum(s)
= <I/s> "weighted" by s
This means <I>/<s> looks like <I/s> but with more
weight given to those measurements with large s.
If the standard deviation is related to counting
error, then s is larger for strong reflections
(yes of course as a percentage it is smaller
for strong reflections, but in absolute value
it is larger).
Thus the statistic <I>/<s> is weighted toward
strong reflections, and will be more optimistic
than unweighted <I/s>.
The next-to-last table of truncate output lists
<I/s>, <I>/<s>, <F/s>, <F>/<s> vs resolution.
In the critical last shells <I/s> and <I>/<s>
tend to be the same, perhaps because all
reflections are around the noise level and s
is dominated by background counts so that all
reflections are weighted equally.
for R-type factors (error/magnitude) the ratio of
the means (as in R-merge) is more optimistic
than the mean of the ratios (as in std deviation),
and by a much larger extent: weighted by I,
whereas "s" varies at most as the sqrt(I).
If we used the mean ratio the result would be
to horrible to contemplate, with all those weak
reflections at the noise level weighted equally
with the strong! But seriously, it makes some
sense for a statistic to be weighted toward the
strong reflections, since they contribute more
to the map. Who cares if a reflection measurement
has 70% error if it is so weak that it could be
omitted completely with no visible effect on
the map?
Ed
Best regards,
S. Werten
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