From: Ronald Wong (ronwong@inreach.com)
Date: Sat Oct 07 2000 - 06:49:28 PDT
Message-Id: <l03102802b5f98b0a5a43@[209.209.19.28]> Date: Sat, 7 Oct 2000 14:49:28 +0100 From: Ronald Wong <ronwong@inreach.com> Subject: re: Sig figs & averaging
Geoff said:
>I've been thinking about significant figures a lot recently, and
>contemplating when scientists do and don't use them. Here's a simple
>example.
>
>If you had two objects with masses of 2 g and 3 g, and you wanted to find
>their average, I think that most chemists would report the average mass as
>2.5 g. But according to sig fig rules, their average mass should be 3 g.
>...
>Advice?
The first thing to do is remember that
A) All the figures used by scientist in their work are significant.
B) Scientists do NOT give the mass of an object as 2 g or 3 g.
Because of the inherent limitation of the tools that are used, or the
undiscovered shortcomings of the procedures being followed, or just the
fact that randomness enters into all of our activities in one form or
another, ALL measurements end up being uncertain. Because of this,
scientists have to specify how uncertain their measurements are if their
measurements are to have any real value. By doing this, we are able to see
where the last significant figure is in their results and how much its
value varied in their experiment or the calculations made thereafter.
The fact that uncertainty is am important part of every scientific
investigation is one of the fundamental lessons that we should be teaching
our students. How this is done is another topic all by itself.
To respond to Geoff's question:
In his example, there is only one significant figure given for each of the
two values. So both the "2" and the "3" are "unsure" (as Gary Horne said).
For a scientist, the 2 g or the 3 g would represent the average value of a
series of measurements where the readings varied in the units place. Since
Goeff's measurements did not specify how much the values varied with
respect to their average, they lack some important information.
If you don't know the uncertainty of these two measurements then you have
two problems:
A. You don't know how precise these values are (2 +/- 1 g is not the same
as 2 +/- 2 g) and
B. You cannot specify the uncertainty that arises from any mathematical
operation involving these two measurements (their average, for instance).
Any mathematical operation will tend to produce results that have a
greater percentage of uncertainty. Keeping track of these uncertainties
is important because the final result can end up being totally
meaningless due to the way the uncertainties add up.
Since Geoff's example involves two numbers of only one significant figure,
the highest level of precision possible would be +/- 1 g for each of the
measurements.
Assuming that this was indeed the case (BIG assumption here), then the
LARGEST average of the two numbers would be the average of two largest
possible values for these numbers (3 g & 4 g). This comes to 3.5 g. In a
similar fashion, the SMALLEST average, the average of the two lowest
possible values for these two numbers (1 g & 2 g) is 1.5 g. The number in
the middle of these two results is the average value (i.e. AVERAGE = 2.5
g). It differs from the maximum and minimum averages by 1 g. So we can
summarize our results by saying the Average of 2 +/- 1 g and 3 +/- 1 g is
2.5 +/- 1 g.
You could do this graphically by just drawing a number line on the board
and placing two points on the number line. One at 1.5 and another at 3.5.
When asked, "If the region between these two points represent all the
possible average values (and it's important to point out that ANY point in
this region could be the actual average value), which point within this
region would seem to be the most representative one for the AVERAGE of the
average?", most students would opt for the middle, 2.5 (although some of
your students may come up with some interesting reasons why other points
might be considered). The question then arises, "What number can be given
that will reflect the fact that 2.5 g represents one point in the middle of
a range of points extending from 1.5 g to 3.5 g?". From this comes the
uncertainty of 1 g.
Actually, the final result is only good to the units place.
The uncertainty in the final value is in the unit place so the answer is
only good to the unit place. The 0.5 must be dropped. In statistics, when
the digit being dropped is a 5 and the digit to it's left is even, it
remains unchanged (many teachers ignore this rule and probably with good
reason). Thus 2.5 becomes 2 and the final value for the AVERAGE is 2 +/- 1
g.
Astute students will probably notice that this final result covers a range
from
1 g to 3 g. The low end goes past the lowest value of the original range
(1.5 g) and the highest value falls short of the highest value of the
original range (3.5 g). Keeping this in mind, one can see that Geoff's
choice of 3 g for the overall average is equally acceptable. With an
uncertainty of +/- 1 g, it has the a complementary deficiency of extending
beyond the upper limit and not quite reaching the lower.
The fact that one answer can't be given that will cover the entire range of
possible values and that the final average can be one of the two original
measurements - and thus seems "...very weird" - is a consequence of the
very low precision in his example.
Like its uncertainty, the ambiguity in the final answer brings out an
important fact about science. At it's very foundations, science is NOT
exact. That's probably as important a result as the answer itself.
Cheers - ron
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