Significance, was Re: TI-92, HELP! Significative numbers!!

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Significance, was Re: TI-92, HELP! Significative numbers!!! Thx Ray!

Seems like apples and oranges to me:

First significant figures for scientific, engineering, and laboratory related
data in the "real world"

In the real world, you usually have a set of experimental data that has a
certain degree of accuracy which you represent as x plus or minus delta x,
x+/-dx.  So you know that the value is somewhere in the interval [x-dx,x+dx],
but you don't have enough information to determine with scientific accuracy
where in the interval it might be.  Now you want to combine it in one or more
formulae, with other similar such data that also may have a range of values.
At the end of the calculation, you want to know that the result, call it r is
accurate to a certain level of precision, say r+/-dr.  [Sometimes this error
term is specified in number of significant figures such as this esult is only
accurate to 3 significant digits, so if 123 is only accurate to 3 significant
digits than it must be accurate to a tolerance of less than +/- 0.5]

This is a very difficult mathematical/computational exercise for a variety of
reasons. As a simple example, say we want to multiply x+/-dx by y+/-dy,
assume 0 <= x, dx <= x, 0 <= y,dy <= y, then you can pretty much be assured
that the product xy is in the interval [(x-dx)*(y-dy),(x+dx)*(y+dy)] or
[xy-x*dy-y*dx-dx*dy,xy+x*dy+y*dx+dx*dy] or xy+/-(x*dy+y*dx+dx*dy).

Now that's just for a simple multiplication, it gets more difficult as the
computation and the number of variables in the computation gets more involved.

You can get a good estimate by using the calculus formula:

delta f(x1, ... ,xN) ~= (Partial Derivative of f with respect to x1)*(delta
x1) + ... + (Partial Derivative of f with respect to xN)*(delta xN)

where "~=" means "approximately equals"

In the case of simple multiplication you get:

x * (delta y) + y * (delta x) which is close to the actual result of
x * (delta Y) + Y * (delta x) + (delta x)*(delta y)

(delta x)*(delta y) is a very small number.

So if there are many variables, it is very difficult [even though it may be
extremely important to determine in a scientific experiment] to calculate the
range of accuracy for the result.  It would be extremely difficult to place
the necessary logic in the calculator, to carry out calculations on such
"fuzzy numbers".

There are also statistical methods and sampling methods for estimating error
propagation in calculations, once again rather difficult to implement in a
calculator.  Frequently, you just want to put a maximum bound on the level of
error, rather than actually determining the precise level of error, itself.

So that's one issue.

The next 2 issues have been pretty much addressed:

Fixing the number of decimal places using the mode settings of the FIX
command, merely alters the display presentation of the actual result.  [It
also affects the accuracy of certain built-in operations such as numerical
integration, but that's an entirely different matter.]

Using the ROUND command actually changes the result, assuming that it is more
accurate than the rounding parameters chosen.

The next issue is round off and truncation error:

The calculator, itself has a certain level of accuracy (in approx mode or
with floating point numbers in auto mode) so floating point numbers may only
be accurate to the significance level (around 12 digits I guess) of the
calculator. Also, the algorithms employed in the calculator may not be
absolutely mathematically precise in all cases, constituting truncation
error.  These errors can combine and propagate, resulting in loss of accuracy
in involved calculations and/or programs.  For many, probably most
calculations, this is probably not too important, but sometimes it can ruin
the accuracy of the entire calculation, reducing the level of significance of
the result.  Exact mode MAY be able to eliminate or substantially reduce
round-off and truncation error.

I assume "significant" and "significative" are the same thing.  If
significative is something different, you may want to ignore this.  Of
course, you may want to ignore this anyway. :)


Just my comments on the subject.  Not intending any criticism toward anybody
posting on this rather involved subject.

-----

In article <357972A0.6365@esoterica.pt>,
  Tiago Carvalho <carvalho@ESOTERICA.PT> wrote:
>
> Hi guys,
>
> Firstly I want to thank to Ray Kremer for having done what I should have
> done when I started this discussion, and this is explaining to everybody
> what significative figures mean. Sorry, but I thought you all know what
> they were. I was wrong (not pretending to offend you, but you have such
> powerful calculators and don't know the basis of Maths or Statistics?!
> You use them for what? Only for gaming?!!!).
>
> Secondly, now you have all understand what significative figures are, do
> you know if there is any calculator that knows how to handle with them?
> I think it should be a standard for any scientific calculator, anyway...
>
> > The TI-92 is not programmed to do sig figs,
> Absolutely sure about that?  :-(((
>
> > and such a thing would be very hard to program,
> :-(((((((((((((((( Anyway if anyone manage to program it could you send
> me the code, please?
>
> > while it's not so hard to just do it by yourself, as long as you
> > actually know how.
> Sorry, Ray. From all you have said I agree in everything excepting this
> point. Imagine you have defined a formula that requires lots of
> parameters, as weights, volumes, densities, etc., if you had the
> significative figures option you would only type f(5.0,4.356,3.32...),
> for instance, and get the proper result, but if you don't, you have to
> do all the calculus by handing... Am I right?
>
> Best Regards
> Tiago Carvalho
>
> P.S: Sorry for my bad English...
>


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