Re: [Phys-l] season's greetings

["Jeffrey Schnick" <JSchnick@Anselm.Edu>]

I set up a spreadsheet (Excel 2003 SP2--I haven't tested it with old versions of Excel) for doing uncertainty propagation in the case of a function of up to six variables when each of the variables is characterized by a gaussian distribution of known mean and standard deviation.  The spreadsheet uses Monte Carlo to determine a distribution for the function.
The user enters the number of variables, the mean and standard deviation for each variable, and the function (and copies the last item to a column).  The spreadsheet produces a set of 1000 randomly-determined values for each variable.  Each set conforms to a gaussian distribution consistent with the given mean and standard deviation for that set.  The function uses one value from each set to determine a value for the function. It does so 1000 times, thus creating a set of 1000 values for the function.

The spreadsheet displays:
1) A bar graph of the distribution for each variable with a curve corresponding to the gaussian distribution with the given mean and standard deviation.  It analyzes the set of randomly-chosen values plotted on the bar graph and displays the mean and standard deviation for that set.  (They turn out to be close to but not the same as the given values.)
2) A bar graph of the distribution for the variable with a curve corresponding to the gaussian distribution with the calculated mean and standard deviation.

The spreadsheet is the file gaussEPv5.xls.zip at
http://www.anselm.edu/internet/physics/cbphysics/math.html

The spreadsheet may be of use to folks who want to go beyond sig figs but don't want to bog students down with algebraic uncertainty propagation.  It can also be used as a check on the latter.  It provides more information than algebraic methods in that one can see what the calculated function's distribution looks like, in particular, whether or not it is reasonably close to being gaussian.


-----Original Message-----
From: phys-l-bounces@carnot.physics.buffalo.edu [mailto:phys-l-bounces@carnot.physics.buffalo.edu] On Behalf Of John Denker
Sent: Monday, August 28, 2006 2:13 PM
To: Forum for Physics Educators
Subject: [Phys-l] season's greetings

    /It's that time of year again:  the start of school.   Chapter 1 of
    the textbook goes on and on about significant digits.  A few
    students sit there, fidgeting, wondering whether the teacher is
    really crazy enough to believe this stuff.  The more they think
    about it, the less sense it makes.  But// only a few of them bother
    to think about it at all; most of them learned long ago that
    thinking about classwork is a lose/lose proposition./


Maybe I'm overly optimistic, but I think we can do better than that.  We
can start by eradicating the sigfig nonsense.   I recently updated my
web page on the subject.


      How Many Digits Should Be Used?

Use many enough digits to avoid unintended loss of significance. Use few enough digits to be reasonably convenient.


      How Should Uncertainty Be Expressed?

Express the uncertainty separately and explicitly. For example,
1.234(55) or equivalently 1.234±0.055.

State the form of the distribution, unless this is obvious from context.
Examples include Gaussian, square, triangular, et cetera.


      What About Significant Digits?

The whole notion of significant digits is heavily flawed; see section 8 <http://www.av8n.com/physics/uncertainty.htm#sec-abomination> for more on this. Anything that can be done by means of significant digits can be done much better by other means. People who care about their data don't use significant digits.


    Table of Contents

    * 1  Introduction <http://www.av8n.com/physics/uncertainty.htm#htoc1>
          o 1.1  How Many Digits Should Be Used?
            <http://www.av8n.com/physics/uncertainty.htm#htoc2>
          o 1.2  How Should Uncertainty Be Expressed?
            <http://www.av8n.com/physics/uncertainty.htm#htoc3>
          o 1.3  What About Significant Digits?
            <http://www.av8n.com/physics/uncertainty.htm#htoc4>
    * 2  Fundamental Notions of Uncertainty
      <http://www.av8n.com/physics/uncertainty.htm#htoc5>
    * 3  Propagation of Uncertainty
      <http://www.av8n.com/physics/uncertainty.htm#htoc6>
          o 3.1  Example: Extracting Signal from Noise
            <http://www.av8n.com/physics/uncertainty.htm#htoc7>
          o 3.2  Another Example: Magnesium Mass. Monte Carlo Solution
            <http://www.av8n.com/physics/uncertainty.htm#htoc8>
          o 3.3  Example: Interpolating Between Graduations
            <http://www.av8n.com/physics/uncertainty.htm#htoc9>
          o 3.4  Crank Three Times
            <http://www.av8n.com/physics/uncertainty.htm#htoc10>
          o 3.5  Manual Propagation of Uncertainty
            <http://www.av8n.com/physics/uncertainty.htm#htoc11>
          o 3.6  Example: Manual Propagation
            <http://www.av8n.com/physics/uncertainty.htm#htoc12>
          o 3.7  Amplification of Uncertainty
            <http://www.av8n.com/physics/uncertainty.htm#htoc13>
    * 4  How Much Accuracy Is Enough? How Much Data Is Enough?
      <http://www.av8n.com/physics/uncertainty.htm#htoc14>
          o 4.1  Why is this hard?
            <http://www.av8n.com/physics/uncertainty.htm#htoc15>
          o 4.2  Recommendations -- How To Do It Right
            <http://www.av8n.com/physics/uncertainty.htm#htoc16>
          o 4.3  Report the Form of the Distribution
            <http://www.av8n.com/physics/uncertainty.htm#htoc17>
          o 4.4  Too Many Digits, Oh My!
            <http://www.av8n.com/physics/uncertainty.htm#htoc18>
          o 4.5  How To Survive an Attack of Sig Figs
            <http://www.av8n.com/physics/uncertainty.htm#htoc19>
          o 4.6  Sensitivity Analysis, On-Line Analysis, and
            Cross-Checking
            <http://www.av8n.com/physics/uncertainty.htm#htoc20>
    * 5  "Correctness" versus Agreement
      <http://www.av8n.com/physics/uncertainty.htm#htoc21>
    * 6  Measurement Errors versus Modeling Errors
      <http://www.av8n.com/physics/uncertainty.htm#htoc22>
    * 7  Contributions to the Uncertainty -- and Related Notions
      <http://www.av8n.com/physics/uncertainty.htm#htoc23>
          o 7.1  Tolerance
            <http://www.av8n.com/physics/uncertainty.htm#htoc24>
          o 7.2  Significance
            <http://www.av8n.com/physics/uncertainty.htm#htoc25>
          o 7.3  Precision
            <http://www.av8n.com/physics/uncertainty.htm#htoc26>
          o 7.4  Accuracy
            <http://www.av8n.com/physics/uncertainty.htm#htoc27>
          o 7.5  Readability and Reproducibility
            <http://www.av8n.com/physics/uncertainty.htm#htoc28>
          o 7.6  Empirical Uncertainty
            <http://www.av8n.com/physics/uncertainty.htm#htoc29>
          o 7.7  Other Limits to the Notion of Uncertainty
            <http://www.av8n.com/physics/uncertainty.htm#htoc30>
    * 8  Significant Figures are an Abomination
      <http://www.av8n.com/physics/uncertainty.htm#htoc31>
          o 8.1  If You Mean Place Value, Say Place Value
            <http://www.av8n.com/physics/uncertainty.htm#htoc32>
          o 8.2  Uncertainty, Insignificance, and Guard Digits
            <http://www.av8n.com/physics/uncertainty.htm#htoc33>
          o 8.3  Rough -- Very Rough -- Uncertainties
            <http://www.av8n.com/physics/uncertainty.htm#htoc34>
          o 8.4  Sectarian Differences
            <http://www.av8n.com/physics/uncertainty.htm#htoc35>
          o 8.5  Some Answers Not Expressible Using Sig Digs
            <http://www.av8n.com/physics/uncertainty.htm#htoc36>
          o 8.6  Some Numbers are Exact
            <http://www.av8n.com/physics/uncertainty.htm#htoc37>
          o 8.7  Units Won't Solve the Problem
            <http://www.av8n.com/physics/uncertainty.htm#htoc38>
          o 8.8  Relative versus Absolute Uncertainty
            <http://www.av8n.com/physics/uncertainty.htm#htoc39>
    * 9  Appendix: Mean, Standard Deviation, Gaussian, etc.
      <http://www.av8n.com/physics/uncertainty.htm#htoc40>
    * 10  References <http://www.av8n.com/physics/uncertainty.htm#htoc41>

 
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