[Phys-L] Re: Average earlier or average later?

[John Denker <jsd@AV8N.COM>]

Michael Edmiston wrote:
> General question... You make several measurements of a number that will
> be used in a calculation to obtain a final result.  You can average the
> measurements into one average measurement then use that average in the
> calculation, or you can do the calculation with each measurement then
> average the results of the calculation.  Depending on the type of
> calculation performed, and depending on the standard deviation of the
> measurements, the two methods may produce almost identical results, or
> may produce noticeably different results.  If the results are different,
> which method would you say is the correct or preferred method... average
> first, or average last?

That's an excellent question.

The answer is:  You should average earlier.  Or later.  Or both.  Or neither.

There are other ways of thinking about it.  In order of increasing
sophistication, we have
  -- averaging
  -- curve fitting
  -- data reduction in general.

Averaging is a particular type of curve fitting.  Curve fitting is a
particular type of data reduction.  You get the idea.

Averaging is appropriate if you think the quantity of interest is
the mean of the raw observations.  This is commonly the case, if
you have noise that is additive and independently distributed.

Meanwhile, in contrast, there are plenty of cases where averaging
is not appropriate, for instance if the noise is multiplicative
instead of additive.  For instance, once upon a time I was helping
a friend make a frequency-lock for her laser.  For a frequency
standard we used the doppler-free absorpion of the sodium D line.
The light coming out of the absorption cell was darker or
brighter depending on whether the light was on resonance or not,
but it was also darker or brighter because there were tremendous
fluctuations in the intensity of the light source.  The only way
to make the frequency-lock work was to measure the brightness
upstream of the cell, measure the brightness downstream of the
cell, and divide.  Actually what I did was take the logarithms
(using an analog circuit) and subtract them (using an op-amp).
It would have been madness to do any averaging before taking
the logarithms.

Whenever possible, I like to approach the problem by making
a mental model of what is the physics that I care about, and
what are the sources of noise.  Typically there are a few
parameters that describe the physics, so that task is to
figure out what values of the parameters are consistent with
the observed data.  This is in general an inverse problem,
because given the parameters and the noise model, I can
easily generate simulated data ... but data reduction involves
going the other way, i.e. going from data to model-parameters.

The more complex the model, the more unlikely it is that
averaging -- especially early averaging -- is the right
answer.  Averaging is a reasonable way of estimating the mean.
Sometimes that's exactly what you want, and sometimes not.
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