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<div><big><font size="2" face="Times New Roman"><big>I'm getting
questions about the eqn on page 9 for the actual density of
water<br>
and s.f. in the calculations.<br>
<br>
The eqn on page 9 uses two numbers, -0.00030 and 1.0042.
People<br>
have asked if these are exact. No they aren't. Someone
took temperature<br>
and density data between 20 and 30 degC and fit the data to
a linear eqn.<br>
In doing so the sig. fig. for these numbers were determined
from the s.f.<br>
for the temp and density used. You need to use the s.f. in
these numbers<br>
and the s.f. for your temperature to determine the proper
s.f. for the<br>
actual density. This can be rather tricky since you need to
use the rules<br>
for mult/div and add/subtr. in the same calculation (mult.
rule first followed<br>
by add. rule). I can pretty much make assurances you won't
get 1 or 1.0.<br>
<font size="3">Just so all of you know, water has it's
greatest density of </font> </big></font></big><big><big><font
size="3" face="Times New Roman">1.00000 g/mL at<br>
4.0 Celsius. It's density is 1.0 g/mL (two s.f., 1 decimal
place) from 0 C to 100 C.<br>
<br>
</font></big></big><big><big><big><font size="2" face="Times
New Roman"><big>You need to report the proper number of
s.f. in the table for all the numbers,<br>
including the error and % error. This applies even if
you use Excel. You will</big></font></big></big></big></div>
<div><big><big><big><font size="2" face="Times New Roman"><big>have
to set decimal places in order to get the correct sig.
fig. in Excel since it</big></font></big></big></big></div>
<div><big><big><big><font size="2" face="Times New Roman"><big>doesn't
understand sig. fig. I've explained how to do this on
my web page.</big></font></big></big></big></div>
<div><big><big><big><font size="2" face="Times New Roman"><big><br>
</big></font></big></big></big></div>
<div><big><big><big><font size="2" face="Times New Roman"><big>While
I'm at it, what happens when you subtract two numbers
and you get</big></font></big></big></big></div>
<div><big><big><big><font size="2" face="Times New Roman"><big>zero?
For example, lets say you subtract two numbers and at
least one of</big></font></big></big></big></div>
<div><big><big><big><font size="2" face="Times New Roman"><big>them
is only to the 3rd decimal place and the result rounds
to 0.000 in the</big></font></big></big></big></div>
<div><big><big><big><font size="2" face="Times New Roman"><big>third
decimal place. This would be the proper way to report
it. This means</big></font></big></big></big></div>
<div><big><big><big><font size="2" face="Times New Roman"><big>"one"
s.f., in the last decimal place. Carry any extra digits
to the right of the</big></font></big></big></big></div>
<div><big><big><big><font size="2" face="Times New Roman"><big>third
decimal place to the next calculation, if there is
one, remembering if</big></font></big></big></big></div>
<div><big><big><big><font size="2" face="Times New Roman"><big>you
are doing a multiplication or division in the next steps
to report the</big></font></big></big></big></div>
<div><big><big><big><font size="2" face="Times New Roman"><big>final
result to 1 s.f. This could very well occur for your
error and % error</big></font></big></big></big></div>
<div><big><big><big><font size="2" face="Times New Roman"><big>columns.<br>
</big></font></big></big></big></div>
<div><big><big><big><font size="2" face="Times New Roman"><big><br>
</big></font></big></big></big></div>
<div><big><big><font size="3" face="Times New Roman">You also need
to use correct s.f. on the axes for your graphs. The s.f.</font></big></big></div>
<div><big><big><font size="3" face="Times New Roman">in your density
and intercept are determined by the s.f. in your mass and</font></big></big></div>
<div><big><big><font size="3" face="Times New Roman">volume being
plotted. Since doing a best-fit line averages out the error
in</font></big></big></div>
<div><big><big><font size="3" face="Times New Roman">your actual
data points, if you have enough data points you can usually</font></big></big></div>
<div><big><big><font size="3" face="Times New Roman">gain one s.f.
from a graph. For instance, with enough data points, if you</font></big></big></div>
<div><big><big><font size="3" face="Times New Roman">had 3 s.f. for
the mass and 3 s.f. for the volume you could report 4 s.f.
for</font></big></big></div>
<div><big><big><font size="3" face="Times New Roman">the slope and
intercept. Two or three data points is not enough to gain a
</font></big></big></div>
<div><big><big><font size="3" face="Times New Roman">s.f. when
taking an average or from a graph. Why? Think about this
from</font></big></big></div>
<div><big><big><font size="3" face="Times New Roman">the perspective
of plotting a best-fit line. </font></big></big><big><big><font
size="3" face="Times New Roman">The purpose of a best-fit
line is</font></big></big></div>
<div><big><big><font size="3" face="Times New Roman">to "average
out" the error in the data points. If you have only two
data</font></big></big></div>
<div><big><big><font size="3" face="Times New Roman">points the
best-fit line will go right through them and will not
average out</font></big></big></div>
<div><big><big><font size="3" face="Times New Roman">the error in
the data. Adding one more (total of 3 data points) isn't
much</font></big></big></div>
<div><big><big><font size="3" face="Times New Roman">better. The
more data points you have the safer it is to claim an extra
s.f.</font></big></big></div>
<div><big><big><font size="3" face="Times New Roman">in the numbers
from the best-fit line (slope, intercept). How many should</font></big></big></div>
<div><big><big><font size="3" face="Times New Roman">you have before
you can say it's safe to gain a s.f. That's hard to say.
It</font></big></big></div>
<div><big><big><font size="3" face="Times New Roman">depends on the
data you have. For our purposes in lab we'll say you<br>
need at least four.<br>
<br>
This also applies when taking an average of data. I
explained this in lecture<br>
and it's related to what I discussed above. Technically,
it's not safe to claim<br>
an extra s.f. when averaging only 2 data points since it
really doesn't effectively<br>
average out the experimental error in the data (think about
fitting a best-fit<br>
line to only two data points as explained above). Adding a
third data point<br>
doesn't help much. Again, for our purposes in lab you can
only gain a s.f. if<br>
you have more than 3 data points., for either an average or
numbers from a</font></big></big></div>
<big><big><font size="3" face="Times New Roman">graph.<br>
<br>
In reality, this is simplifying error analysis. Error
analysis is much more<br>
complicated than this but beyond the scope of this class.<br>
<br>
Finally, if you set the s.f. properly in your Excel table
before making a graph<br>
they will carry over to the graph. This is at least true if
using the PC version<br>
of Excel. I'm not sure if this is true if using a version for
another operating<br>
system or a knockoff (Google Docs, OpenOffice, etc.). You can
set s.f.<br>
on the axes by editing the axes.<br>
<br>
Dr. Zellmer</font></big></big>
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