Exp 1 - eqn on page 11 and sig. fig.

[Zellmer, Robert]

I'm getting questions about the eqn on page 11 for the actual density of water
and s.f. in the calculations, particularly for the error and % error and what
happens when you subtract two numbers and you get zero to the correct
number of s.f.  Also, setting s.f. in your data table and graph.

The eqn on page 11 uses two numbers, -0.00030 and 1.0042.  People
have asked if these are exact.  No they aren't.  Someone took temperature
and density data between 20 and 30 degC and fit the data to a linear eqn.
In doing so the sig. fig. for these numbers were determined from the s.f.
for the temp and density used.  You need to use the s.f. in these numbers
and the s.f. for your temperature to determine the proper s.f. for the
actual density.  This can be rather tricky since you need to use the rules
for mult/div and add/subtr. in the same calculation (mult. rule first followed
by add. rule).  I can pretty much make assurances you won't get 1 or 1.0.
Just so all of you know, water has it's greatest density of 1.00000 g/mL at
4.0 Celsius. It's density is 1.0 g/mL (two s.f.,1 decimal place) from 0 C to 100 C.
You actually encountered this in the pre-lab.

You need to report the proper number of s.f. in the table for all the numbers,
including the error and % error.  This applies even if you use Excel.  You will
have to set decimal places in order to get the correct sig. fig. in Excel since it
doesn't understand sig. fig.  I've explained how to do this on my web page.

While I'm at it, what happens when you subtract two numbers and you get
zero?  For example, lets say you subtract two numbers and at least one of
them is only to the 3rd decimal place and the result rounds to 0.000 in the
third decimal place.  This would be the proper way to report it.  This means
"one" s.f., in the last decimal place.  Carry any extra digits to the right of the
third decimal place to the next calculation, if there is one, remembering if
you are doing a multiplication or division in the next steps to report the
final result to 1 s.f.  This could very well occur for your error and % error
columns.

You also need to use correct s.f. on the axes for your graphs.  The s.f.
in your density and intercept are determined by the s.f. in your mass and
volume being plotted. Since doing a best-fit line averages out the error in
your actual data points, if you have enough data points you can usually
gain one s.f. from a graph.  For instance, with enough data points, if you
had 3 s.f. for the mass and 3 s.f. for the volume you could report 4 s.f. for
the slope and intercept.  Two or three data points is not enough to gain a
s.f.  when taking an average or from a graph. Why? Think about this from
the perspective of plotting a best-fit line. The purpose of a best-fit line is
to "average out" the error in the data points.  If you have only two data
points the best-fit line will go right through them and will not average out
the error in the data.  Adding one more (total of 3 data points) isn't much
better.  The more data points you have the safer it is to claim an extra s.f.
in the numbers from the best-fit line (slope, intercept).  How many should
you have before you can say it's safe to gain a s.f.   That's hard to say.  It
depends on the data you have.  For our purposes in lab we'll say you
need at least four.

This also applies when taking an average of data.  I explained this in lecture
and it's related to what I discussed above.  Technically, it's not safe to claim
an extra s.f. when averaging only 2 data points since it really doesn't effectively
average out the experimental error in the data (think about fitting a best-fit
line to only two data points as explained above).  Adding a third data point
doesn't help much.  Again, for our purposes in lab you can only gain a s.f. if
you have more than 3 data points., for either an average or numbers from a
graph.

In reality, this is simplifying error analysis.  Error analysis is much more
complicated than this but beyond the scope of this class.  Depending on
your data and the "noise" in your data it could take hundreds of data points
in order to safely gain a s.f.

Finally, if you set the s.f. properly in your Excel table before making a graph
they will carry over to the graph.  This is at least true if using the PC version
of Excel.  I'm not sure if this is true if using a version for another operating
system or a knockoff (Google Docs, OpenOffice, etc.).  You can set s.f.
on the axes by editing the axes.

You also need to use the proper number of s.f. in the data entry for exp 1 (SCM).
Do your Excel table first and then just copy those values over to the data
entry program.

Dr. Zellmer
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