Re: [Phys-L] determine k

[John Denker <jsd@av8n.com>]

On 02/08/2015 08:45 PM, Paul Lulai wrote:

> I plan to have students determine spring constants by both standard
> hooke's law relationships and the oscillation of a mass on the
> spring.

First the caveats, then some constructive suggestions:

The first thing to get across is that each raw data point
has *no* uncertainty and *no* error bars.  If you have
a large number of data points, there will be some spread
in the distribution, and you (sometimes) can describe the
/distribution/ in terms of its mean and standard deviation.
Still, though, the width is a property of the distribution,
not of any particular raw data point.

Let's suppose a given student has two observations of k.
It is possible to calculate the mean of those two numbers,
and also possible to calculate the standard deviation.
It is however hardly worth the trouble, because you are
using two numbers to describe two numbers.  You could
report the mean and standard deviation, but it would
be easier and in some ways better to just report k1
and k2.

-------

We can make good progress as follows:  Suppose there
are N students (or maybe N teams) each of which has a
reading for k1.  Collect all N of these readings and
and plot them.

  I recommend a diaspogram (tm).  It takes a bit of
  work to produce one, but the result is beautiful
  and easy to interpret.
    https://www.av8n.com/physics/probability-intro.htm#sec-diaspogram
  If you can find an easier and/or better way to 
  visualize the data, that's great.  The point is
  that it is important to find /some/ good way of
  visualizing the data.

In any case, there is a mean and a standard deviation
for the /distribution/ over k1.  Call this the "cooked"
k1 value.

Do the same for k2.

Now things get exciting.  You can take a /weighted/
average of the cooked values.  If one of the distributions
has less spread than the other, it gets more weight in
the weighted average.
  https://www.av8n.com/physics/uncertainty.htm#sec-weigh-evidence

=============

I think the process outlined above is pretty much
squeaky-clean in terms of physics and statistics.

Key points
 -- Visualize the data.
 -- The mean and standard deviation are properties
  of the distribution ... not of any particular 
  point that might have been drawn from the
  distribution.
 -- You can take a weighted average of distributions.