When you just have a few data points but have reason to believe they come
from a Gaussian distribution, I know you really should use the Student t
distribution to estimate your uncertainty. But is that ever done in
practice in the physics world? And how would it be done? Find some
equivalent sigma_t with the same 68% meaning as the sigmas that are
usually reported? Just use a standard deviation and let the reader make
of it what he will?


--
"The average person, during a single day, deposits in his or her underwear
an amount of fecal bacteria equal to the weight of a quarter of a peanut."
-- Dr. Robert Buckman, Human Wildlife, p119.

"Gregory L. Hansen" wrote in message
...

When you just have a few data points but have reason to believe they
come
from a Gaussian distribution, I know you really should use the Student
t
distribution to estimate your uncertainty. But is that ever done in
practice in the physics world? And how would it be done? Find some
equivalent sigma_t with the same 68% meaning as the sigmas that are
usually reported? Just use a standard deviation and let the reader
make
of it what he will?


--
"The average person, during a single day, deposits in his or her
underwear
an amount of fecal bacteria equal to the weight of a quarter of a
peanut."
-- Dr. Robert Buckman, Human Wildlife, p119.

You pick your desired assurance level. I have a completely worked out
example (re repeated rate-of-climb tests in a stochastic atmosphere) in
Appendix A of Performance of Light Aircraft, pp 425-427.

On Dr. Buckman's point: good reason not to wear underwear. Or anything
else, for that (non-fecal) matter.

In article .net,
John T Lowry wrote:

"Gregory L. Hansen" wrote in message
...

When you just have a few data points but have reason to believe they
come
from a Gaussian distribution, I know you really should use the Student
t
distribution to estimate your uncertainty. But is that ever done in
practice in the physics world? And how would it be done? Find some
equivalent sigma_t with the same 68% meaning as the sigmas that are
usually reported? Just use a standard deviation and let the reader
make
of it what he will?


--
"The average person, during a single day, deposits in his or her
underwear
an amount of fecal bacteria equal to the weight of a quarter of a
peanut."
-- Dr. Robert Buckman, Human Wildlife, p119.

You pick your desired assurance level. I have a completely worked out
example (re repeated rate-of-climb tests in a stochastic atmosphere) in
Appendix A of Performance of Light Aircraft, pp 425-427.

In physics, the desired assurance level is typically one standard
deviation. That is, about 68%. It's a convention that makes it
straightforward to combine data in weighted means with weighted standard
deviations, and the rest of the statistical stuff. Ideally I wouldn't
just pick out some random figure, like a neutron wavelength with 95%
confidence, but that uncertainty doesn't propagate with the standard
deviations without some further processing.

My thinking is to either go with the standard deviation and let the reader
decide what to do with that, or report a 68.xx confidence interval based
on the Student t distribution. I think that should propagate as usual,
but none of my data reduction books seems to really discuss it.


On Dr. Buckman's point: good reason not to wear underwear. Or anything
else, for that (non-fecal) matter.

The story I've heard is that underwear was invented so that sailors
wouldn't have to wash their pants as often.

--
"Let us learn to dream, gentlemen, then perhaps we shall find the
truth... But let us beware of publishing our dreams before they have been
put to the proof by the waking understanding." -- Friedrich August Kekulé

In article ,
(Gregory L. Hansen) wrote:
In article .net,
John T Lowry wrote:
snip

On Dr. Buckman's point: good reason not to wear underwear. Or anything
else, for that (non-fecal) matter.

The story I've heard is that underwear was invented so that sailors
wouldn't have to wash their pants as often.

Exactly. It's also cheaper.

/BAH

Subtract a hundred and four for e-mail.

(Gregory L. Hansen) wrote in message ...
When you just have a few data points but have reason to believe they come
from a Gaussian distribution, I know you really should use the Student t
distribution to estimate your uncertainty. But is that ever done in
practice in the physics world? And how would it be done? Find some
equivalent sigma_t with the same 68% meaning as the sigmas that are
usually reported? Just use a standard deviation and let the reader make
of it what he will?

In engineering applications, typically what is done is there is an
estimate of the error from other means.

For example: if you were measuring the density of some object, and
it was very expensive (or time consuming or whatever) to produce
many objects (or measure them or whatever) then you measure a few,
and use information about the manufacturing process to estimate the
size of the error bars.

Or, if you were measuring a radioactive decay, and could not for
some reason, hang around measuring long enough to get a good sample
of the actual distribution. You would use estimates based on the
amount of the sample, and the existing knowledge about the decay,
to get you a handle on the expected decay.

Typically, the result is you quote a main measured value, an error
bar in the value, and sometimes an accuracy estimate in the error
bars. Or, it may be the case that the error bars are widened a bit
to account for the uncertainty in the error bars themselves.

There is quite a bit of literature on this. When it is difficult
(or expensive, or dangerous, etc.) to take individual measurements,
there is quite a lot of prior art on how to make the measurements you
can do count for as much as possible. The general topic to look under
is "experiment design."

I was involved in this for reactor safety. One of the test labs
I've worked at involves deliberately removing the coolant from a
fuel channel in a nuclear reactor. Such experiments are very hard
to do, very expensive, and can potentially produce drastic consequences
in terms of exposure of personnel, damage to the reactor, release
of isotopes, etc. So they want to get the most possible data, and
the most possible bennefit, out of each one.
Socks

In article ,
wrote:
(Gregory L. Hansen) wrote in message
...
When you just have a few data points but have reason to believe they come
from a Gaussian distribution, I know you really should use the Student t
distribution to estimate your uncertainty. But is that ever done in
practice in the physics world? And how would it be done? Find some


There is quite a bit of literature on this. When it is difficult
(or expensive, or dangerous, etc.) to take individual measurements,
there is quite a lot of prior art on how to make the measurements you
can do count for as much as possible. The general topic to look under
is "experiment design."

I have a few books on data reduction, like Bevington and Lyons. But they
haven't really gone into this. I'd like to find a book that's a step
beyond Bevington, but maybe not two steps beyond. Statisticians tend to
talk just like mathematicians when they think they're writing for their
peers.
--
"Beer is living proof that God loves us and wants us to be happy."
-- Benjamin Franklin

(Gregory L. Hansen) wrote in message ...
When you just have a few data points but have reason to believe they come
from a Gaussian distribution, I know you really should use the Student t
distribution to estimate your uncertainty. But is that ever done in
practice in the physics world? And how would it be done? Find some
equivalent sigma_t with the same 68% meaning as the sigmas that are
usually reported?
No.

Just use a standard deviation and let the reader make
of it what he will?

You could do that, with a reasonably prominent note of the small
sample size.

Or you could use Student's t distribution to make a confidence
interval. That's what I'd do if I were plotting error bars.

In article ,
Patrick Powers wrote:
(Gregory L. Hansen) wrote in message
...
When you just have a few data points but have reason to believe they come
from a Gaussian distribution, I know you really should use the Student t
distribution to estimate your uncertainty. But is that ever done in
practice in the physics world? And how would it be done? Find some
equivalent sigma_t with the same 68% meaning as the sigmas that are
usually reported?
No.

No, but...


Just use a standard deviation and let the reader make
of it what he will?

You could do that, with a reasonably prominent note of the small
sample size.

Or you could use Student's t distribution to make a confidence
interval. That's what I'd do if I were plotting error bars.

Sociologists and quality controllers want a 95% or 99% or something
confidence interval. I'm interested in a physical measurement that will
be an intermediate result of other measurements, with an uncertainty that
must propagate in a meaningful and preferably conventional way. As, in a
simple case, two measurements (lengths or masses or other physical
quantities), x and y, assumed coming from a Gaussian distribution, with
uncertainties conventionally defined as the standard deviations of the
means, sigma_x and sigma_y, with the sample size large enough that we
can assume the sample standard deviation is a good approximation of the
true standard deviation. In that case, the synthesized result

z = x + y

has uncertainty

sigma_z = sqrt(sigma_x^2 + sigma_y^2)

But suppose x comes from many measurements, while y comes from, say,
five. The uncertainty in the uncertainty in y will be pretty big, so
an actual uncertainty in y will be larger than the usual recipie
predicts. So my question is how to combine that with x. I don't know
what to do with a 95% confidence interval from a t distribution in that
case. Small data sets don't really seem to be addressed very thoroughly
by the likes of Bevington and Lyons, which are the books I've been
working with. And what I've seen that deals with Student's t distribution
doesn't seem to deal with propagation of uncertainties.

My feeling is that if we assume the underlying distribution is Gaussian,
then the uncertainties will propagate in the same way. So find a figure
that corresponds to the same confidence interval as one sigma from a
large data set. But I'm not really sure that's right, and I don't recall
ever seeing that done by others.

--
"Beer is living proof that God loves us and wants us to be happy."
-- Benjamin Franklin