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Re: [Phys-L] Sig Figs homework from my 7th grader



On 10/09/2013 01:53 PM, Dan Beeker wrote:

In general and in my humble opinion, using significant figures to
determine an uncertainty is a bad practice.

Agreed!

That said one can beat a dead horse and proceed as follows:

That dead horse likes to bite people.

If one assumes, or has been told, that the least significant figure
truly indicates the uncertainty then we have

Before we look at how the numbers might be used, let's think
about where they came from. The guy who gave us those number
is telling us
"Hey, I measured something and then rounded it off
so much that roundoff error is now the dominant
contribution to the uncertainty, because I'm being
held prisoner in a moldy dungeon where they force
me to do stupid things all day."

Why should we believe any of the numbers this guy gives us?
My suggestion: We should take a break from doing homework
and call the hostage rescue team.

Also, we should call Buffy to come deal with that horse.

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

Suppose we are given two numbers but are /not/ explicitly
told how much uncertainty is associated with them:

For starters: 2.54 ... how much uncertainty?

Next: 299792.458 ... how much uncertainty?

I claim that in both cases, the uncertainty is zero. For
one thing, as a matter of principle, there is no such thing
as an uncertain number. You can have a /distribution/ over
numbers, and the /distribution/ can have a mean and a std
deviation ... but a distribution is not a number ... in
much the same way that vectors are not scalars.

Secondly, you should have learned in 3rd grade that 2.54 is
*exactly* equal to 254/100, with no uncertainty whatsoever,
in accordance with the axioms of the numeral system. Evidently
we are given the choice between
a) Forgetting the axoims of arithmetic and learning sig
figs; or
b) Remembering the axioms of arithmetic and forgetting
everything you've ever heard about sig figs.

I strongly recommend plan (b).

Thirdly, a hint: Write down the following numbers:
-- The number of cm per inch: ______________
-- The speed of light: ______________ km/s.

Now, what is the uncertainty "associated" with the two numbers
you just wrote down?

===============
Here's an issue of a qualitatively different kind:

.... ± sqrt(.005^2 + 0.0005^2)

What makes anybody think it is OK to add those two uncertainties
in quadrature?

Suppose we have some large number x (with a fairly large
absolute uncertainty) plus a smaller number e (with very
small uncertainty) and we know
y = x + e
This type of problem shows up in innumerable applications.
Geodesy (aka surveying) is a familiar practical example.
The quadratic formula often provides additional examples.

Now the uncertainty on x is large and the uncertainty on is
large ... but when we subtract them we get back e with very
small uncertainty. The uncertainties on x and y drop out,
because they are /correlated/. If you round off y according
to the uncertainty on y by itself, you are going to get
final answer wrong ... wrong by several orders of magnitude.

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

What did these 7th graders do to deserve an attack of the
sig figs?

It's just completely age-inappropriate. In all likelihood,
they don't know the first thing about probability, so what's
the point of pestering them about "uncertainty" or "signficance"?
I guess the policy is
"They're too young to understand the right answer now,
so let's teach them a bunch of completely wrong stuff
that they won't understand /ever/. The only way they
can do the assignment is to oh-so-carefully avoid looking
for any meaning in what they are doing, and just follow
the rules, because that's what's important."

My suggestion: Forget about sig figs. Have the students play
some coin-tossing games or something else that will teach them
some actual factual facts about probability.