[ale] [OT] Help with Significant Figures Explaination

tom tfreeman at intel.digichem.net
Fri Oct 17 16:44:52 EDT 2008


I'll hang on to that for next year. Not quite where I was hoping
to head, but decidedly interesting.

On Fri, 17 Oct 2008, Greg Freemyer wrote:

> I think it is relatively easy to explain via an experiment / lab:
>
> You have a random stack of paper about 6 centimeters tall (2 1/2
> inches).  Have lab partner 1 measure it with a standard ruler that can
> do millimeters.
>
> Say it is 63mm tall.
>
> Now have lab partner 1 take a single piece of paper.  Measure its
> thickness with a micrometer or some other more accurate tool.  I'll
> guess it is about .1mm.
>
> (If you don't have a fine grained measurement tool, measure the height
> of a stack of exactly 100 sheets and divide the measurement by 100 to
> get each sheets thickness.  That also shows significant digits
> relative to division.)
>
> Add the single sheet to the stack and have lab partner 2 measure the
> stack and write down the thinkness on a piece of paper and turn the
> paper upside down.  (Hint: It will still measure 63mm.)
>
> Now have lap partner 1 guess what is written down on lab partner 2's
> piece of paper.
>
> The odds are very high that the two disagree.
>
> Now have them repeat the process, but have them follow rules about
> rounding and significant digits.  Hopefully they can agree on how tall
> the stack is.
>
> ===> Now for multiplication
>
> Repeat the above, but now add 37 sheets of paper to the stack not one.
> (But don't measure the height of the 37 sheets, just of one sheet.)
>
> Does that help.
>
> Greg
>
> On Fri, Oct 17, 2008 at 11:33 AM, Thompson Freeman
> <tfreeman at intel.digichem.net> wrote:
>> To start, thanks to both you and Robert for responses.
>>
>> As an aside, I had a friend in the banking business of the
>> early 1990's who reported something similar to your
>> accountant charge. He sat down with his opposite number in
>> another trust division one day, and some eight hours later
>> they still could not reconcile their common "numbers"
>> closer than hundreds of millions of dollars. Admittedly the
>> story is hearsay, so take it for what it is worth.
>>
>> To the main topic of tonight's symposium... (with
>> appologies to Tom Leher)
>>
>> I've gotten students to accept the need for, and to
>> (largely) use significant figures. This isn't the challenge
>> any more, thankfully.
>>
>> They have a challenge justifying the two computational
>> rules, as in "Why does this work this way?" Dumb these
>> students are not. Just not sophisticated enough to tromp
>> through a statistically oriented exposition (which I would
>> need time to relearn myself). Hence my interest in a less
>> sophisticated development of why significant figures
>> calculations work the way they do.
>>
>> Example time: 752 x 1256 = 944512 in arithmetic, or 944000
>> when _reporting_ the result in a lab exercise. The reported
>> value is rounded to have the same number of sigfig as the
>> smallest number of sigfigs in the multiplicands.
>>
>> The other piece, which everybody remembers, is addition.
>> 23.45 - 19.4578 = 3.9922 as pure arithmetic, but 3.99 when
>> reporting the results computation from data. For addition,
>> you ignore all the digits to the right of the larger least
>> significant digit.
>>
>> Of course, the point of both rules and the determination of
>> significant figures in the first place is to indicate the
>> level of uncertainty without explicitly calculating and
>> displaying it. Many of the items I have located on the web
>> are suggesting that we should completely ditch significant
>> figures and routinely compute the statistical uncertainty -
>> a noble goal but not happening in the near term.
>>
>>
>>
>> On 10/17/2008 08:32:28 AM, Jeff Lightner wrote:
>>> You might talk about how "significant" the figures become
>>> in space.
>>> What appears to be "insignificant" at the start of a
>>> launch to send
>>> something to Jupiter for example becomes greatly
>>> "significant" in error
>>> by the end of the trip due to magnification over distance.
>>>   It would be
>>> embarrassing to set your multi-billion dollar satellite to
>>> do a drive by
>>> on Jupiter only to see it instead miss it by nearly as
>>> many miles as the
>>> dollars that were spent.
>>>
>>> Also "significant" in numbers doesn't always have to be
>>> that far right
>>> of the decimal.  Most accountants spend more time worrying
>>> about why
>>> they're off a penny (0.01 dollars) than why they're off
>>> $3,000,000.00
>>> simply because it usually a lot harder to find that penny.
>>>  Why worry
>>> about a penny?  Because if you're off a penny it indicates
>>> there is an
>>> error and you have no way of knowing whether the error is
>>> just a penny
>>> one way or if it is instead a variance of $3,000,000.00
>>> one way and
>>> $3,000,000.01 the other way.
>>>
>>> -----Original Message-----
>>> From: ale-bounces at ale.org [mailto:ale-bounces at ale.org] On
>>> Behalf Of
>>> Robert Reese~
>>> Sent: Friday, October 17, 2008 12:07 AM
>>> To: ale at ale.org
>>> Subject: Re: [ale] [OT] Help with Significant Figures
>>> Explaination
>>>
>>> Why not contrast them to "insignificant figures"?
>>> Sometimes teaching
>>> the opposite works just as well, or even better, than
>>> teaching the
>>> topic.
>>>
>>> Cheers,
>>> Robert~
>>>
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