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Peter,

> > > (Bill: this is a  good example of how averaging works:
> > > the individual results are so-so (as shown by the sigma)
> > > but the average is excellent).
>
> Taking a series of observations with random error,
> isn't the size of
> the average related to the volume?

I agree that this "good example"
is somewhat extreme. I mean it is an accident that so poor
individual observations gave such a nice average in
the end.

> Given enough data the average error
> error, if random, should trend towards zero?

There is a mathematical theorem which
(under certain precise conditions) says that this is
so. But of course this is not so in practice:-)
Because all mathematical models are only approximations
to reality. In practice, you cannot decrease the
error of your average indefinitely.

I mean if you average 1,000,000 observations with
an ordinary sextant, you are not going to obtain
the accuracy of 1/1000 of a minute.
Application of mathematics
to the real world always has limits.

> The average seems to tell us little
> except that a great number of sights have been considered.

It depends on what you want to know.
If your goal is to evaluate the sextant performance
or the observer's performance, then the average is useless.
You better use standard deviation for this purpose.

But if your goal is to know your position in the sea,
the average is not useless: it tells you your position.
The average of 9 observations tells you your position
about 3 times more accurately than one observation.

> And that there does not seem to be
> significant systematic error.

This is another important reason why the average
is useful, and why I post the average error,
not only the standard deviation.
The average error tells us something about systematic
error, while the standard deviation tells us about
random error.

Both types of error are of interest.

Alex.


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