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Fyyr Lu'Storm

01-16-2004, 07:27 PM

Ok, Sobe's post made me have to post this.

Does anyone know about Significant Figures?

I am trying to find the rules that explain the huge errors in the Significant Figures rules.

Which are:

Each non-zero number is a significant figure

All zeroes between two non-zero numbers are significant figures

All zeroes at the end of a number after a decimal point are significant figures

The number of significant figures is then totaled

For example, this is the conversion formula for changing Celsius to Fahrenheit.

a°C =[32 + (9/5)a] °F

Ok, now let us say the thermometer shows that it is 8°C.

Using the Significant Figures rules, what is the temperature in °F? Google up some examples of SigFig computations if you don't understand what I am getting at, ton of stuff out there.

Just nothing to account for the error that SigFig introduces to the problem.

Here is another example:

(9.0012 - 9.0011)12,345=x

I start out with 5 significant figures. But because the subtraction is supposed to be performed first, it knocks my sig fig down to 1 digit.

1.2345=x is not equal to 1=x.

Are there additional rules to account for these introduced errors(that I can't find)? When are you not suppose to use the rules?

Drake09

01-16-2004, 10:42 PM

I didnt read the rest of your thread, but all your rules are right-on, and I will come back to this thread when I am more sober, sir.

Palarran

01-17-2004, 12:46 AM

Well, using significant digits to determine margin of error is just an approximation anyway. A better (but more time consuming) method might be to explicitly carry the maximum error with the numbers as you perform operations. So:

(9.0012 - 9.0011)12,345=x

would be ((9.0012 +/- 0.00005) - (9.0011 +/- 0.00005))(12345 +/- 0.5)

(0.0001 +/- 0.0001)(12345 +/- 0.5)

0...2.4691

If you knew the probability distribution of error (like, if you performed the same measurement, the results might fit a bell curve centered around 9.0012 for example) you could go even further. http://mathworld.wolfram.com/ErrorPropagation.html

(That goes into a branch of math I only looked at briefly, and it was several years ago. :P )

Jinjre

01-17-2004, 01:11 AM

The quick and dirty way I had always determined error was to take my measurement error and simply plug it into the equation, do all the math and see what number gets spit out.

In your temperature example, let's say you can read the thermometer to within +/- 0.5 degrees. Plug 0.5 degrees C into your equation, and see what number it spits out, then follow the normal sigfig rules.

My biggest gripe about MSExcel is that it has no built in formula for setting sigfigs in the format/number menu option dialog box. You can only set decimal places. Kinda stinks when doing large numbers of calculations all of which require 3 sigfigs, only to have to set decimal places to 3, then go back manually and set any number with more than 3 sigfigs to the correct number of decimal places.

Aldarion_Shard

01-17-2004, 12:23 PM

And this is the part where Im happy I work in a field (biology) where the error in measurement techniques is far greater than any errors due to significant figures, and so we cna just completely ignore them. :)

Panamah

01-17-2004, 12:38 PM

Does anyone know about Significant Figures?

There's a joke in here somewhere.

Fyyr Lu'Storm

01-17-2004, 12:45 PM

Biology huh?

I have a fever.

My thermometer shows 40 degrees C.

What is my temperature in Fahrenheit? Using sigfigs.

Zyphyr

01-17-2004, 05:55 PM

Here is another example:

(9.0012 - 9.0011)12,345=x

I start out with 5 significant figures. But because the subtraction is supposed to be performed first, it knocks my sig fig down to 1 digit.

1.2345=x is not equal to 1=x.

Are there additional rules to account for these introduced errors(that I can't find)? When are you not suppose to use the rules?

There is one rule you are missing, which specificaly addresses this sort of situation :

- Only the final answer has the significant figure rules applied.

As such, the subtraction doesn't produce a number with only 1 significant figure, it produces an intermediate result with 5 sig figs.

That gives you the answer of x=1.2345, not the erroneous answer of X=1.

Part of the reason for that rule is that you can, if you wish, restate the equation from

x=(a-b)*c

to

x=ac-bc

Since both equations are the same, the number of sig figs absolutely must be the same as well and applying the rules to intermediate values can potentially result in differing numbers of sig figs.

Fyyr Lu'Storm

01-17-2004, 06:23 PM

From Chemistry Sixth Edition, Zumdahl Zumdahl.

"Here, apply the addition/subtraction rule first; then the multiplication/division rule to arrive at the four[edit: referring to a different but similar problem] significant figure answer...However, you should round off at the end of all the mathematical operation in order to avoid round-off error. Make sure you keep track of the correct number of significant figures during intermediate steps, but round off at the end."

((8.925-8.904)/8.925) x 100 is another problem in the text book. And continues...

=".24; The difference of 8.925-8.904=.21 has only 2 significant figures. When a two significant figure number is divided by a four significant figure number, the result is reported to two significant figures(division rule)."

If you have a credible resource to contradict this textbook and the professor, I would be grateful.

Jinjre

01-18-2004, 10:20 AM

Fyyr, your example above is a prime example of what I think is the biggest fallacy of significant figures.

Let's say my instrument has a detection limit of 0.001. Following the sigfig guidelines, when I do the subtraction, I must report to only 2 sigfigs, yet this is misleading, as by dropping the final zero, it looks as though my instrument can only see to 0.01, forcing me to report less accurate data.

The more accurate answer would be 0.240, sigifgs be damned. When working with very small numbers (one of my instruments has a detection limit of 0.000002), this can really take away from the accuracy of the numbers reported.

edited to add: Bill Nye is really a cool guy. I love doing some of his stuff in the lab. He did a great demo of condensing water vapor by soaking marshmallows in liquid nitrogen, putting them in his mouth, then exhaling through his nose. As the water vapor in the lungs passes by the very cold marshmallows, it condenses and as it exits the nose it does so in the form of fog. Liquid nitrgoen is fun! (so is 'dry ice' for that matter)

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