Significant Figures
& Rounding
Using Significant Figures and Rounding 
with Respect to
Laboratory Notebook
Maintenance
and Final Reports

Introduction:
What is a significant figure?
Measured significant figures
General rules for determining the number of significant figures in a number
Exact numbers
Significant figures used in calculation
Rounding significant figures
Conclusion:

Introduction:

The purpose of this web page is to set forth in some manner the method with which we treat numbers at IonSource.Com.  We are aware that there are many great significant figure tutorials presented on the internet and we agree that this presentation does not necessarily add anything new to those discussions, for a list of some of these other web pages see the links at the bottom of this page.  The sole purpose of this presentation is to describe to the reader how we deal with significant figures and rounding exclusively at IonSource.Com. 

Scientists routinely attempt to describe the world with numbers and if you are a mass spectroscopist you had better love numbers because in many instances they are all you have, except perhaps for the occasional flaming turbo.  As a good friend once told me, "Every credible scientific study should be reducible to a table filled with meaningful significant numbers."

It is important to establish a a policy with which you treat numbers.  Some companies go so far as to establish a document called an SOP, standard operating procedure.  Then when a regulatory agency comes to call and when they need show them how they derived an assay result without bias, they can point to the SOP and say, "See, we passed this release test by 0.00001 glicks because our SOP tells us to always round up in this situation."  The situation you do not want to be in is the one where you barley pass a test because the analyst always rounds up but the regulatory agency finds instances where another analyst, or worse the same analyst, did something else in a different situation. This can lead the agency to the conclusion that you only round up when you need to pass a test.

Even if you are not answerable to a regulatory agency and the world does not rest on your shoulders you will gain respect from your peers by treating numbers with respect and by reporting only significant figures and by rounding properly.

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What is a significant figure?

There are two types of significant figures, measured and exact

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Measured significant figures

As scientists we get a large amount of the figures we report and use in calculation from  measured observation. Whether a digit is determined to be significant or not is determined by the capability of the measuring device.  In a number derived from a measurement the last significant digit to the right inherently expresses an uncertainty.  For example if you are sure that your low resolution quadrupole type mass spectrometer can deliver accurate measurements to a tenth of a mass unit then you would be justified in reporting masses to a tenth of a mass unit.  For example if one measured a mass of 110.1 u this number would contain four significant figures with the last digit expressing the uncertainty.  The uncertainty would be plus or minus 0.1 u.  Even if the instrument is capable of reporting 10 digits passed the decimal point one should only report the significant digits.  Errors can arise in calculations if insignificant figures are used in a calculation.  If a number resulting from a measurement is used in a calculation that involves multiplication or division all significant figures should be carried through the calculation and then the result should be rounded at the end of the calculation to reflect the term used in the calculation with the fewest significant figures. For example 10.4 X 5.0 should be reported as 52 and not 52.0.  If the calculation involves addition and subtraction a different rule applies, one should preserve common decimal places of the numbers involved.  For example if two numbers obtained from a measurement are used in an addition, 10.1 + 1000.234 the reported number should be 1010.3. Notice that 10.1 has 3 significant figures and 1000.234 has 7 significant figures and the result of the addition has 5 significant figures.

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General rules for determining the number of significant figures in a number:
 

A) All non-zero numbers are significant. 
B) All zeros between significant numbers are significant, for example the number1002  has 4 significant figures.
C) A zero after the decimal point is significant when bounded by significant figures to the left, for example the number 1002.0  has 5 significant figures.
D) Zeros to the left of a significant figure and not bounded to the left by another significant figure are not significant. For example the number 0.01 only has one significant figure.
E) Numbers ending with zero(s) written without a decimal place posses an inherent ambiguity. To remove the ambiguity write the number in scientific notation. For example the number 1600000 is ambiguous as to the number of significant figures it contains, the same number written 1.600 X 106 obviously has four significant figures.
Several Notes:

1)  It is important to know the accuracy and precision of the measuring device one is using and it is important to report only those digits that have significance. To reiterate, your electrospray mass spectrometer may be able to spit out 10 numbers past the decimal place but you should only use the digits that have significance in reporting or in a calculation.

2) It is generally accepted that the uncertainty is plus or minus 1 unit at the level of the uncertainty, for example the "true value" for the number 0.003 can be described as being bounded by the numbers 0.002 and 0.004.  It is important to note that in some instances scientists will sometimes want to express an uncertainty that exceeds 1 at the level of the uncertainty and this should be noted explicitly in the following fashion, 0.003 ± 0.002

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Exact Numbers

Exact values are those that are counted without ambiguity, for example the number of mass spectrometers in the lab is exactly three, or the number of cars in the parking lot is exactly four.  These numbers carry no ambiguity and can be considered to have an infinite number of significant figures.  When using these numbers in a calculation the restriction on reporting is borne by the measured number if any.

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Rounding significant figures
(now it gets personal)

As far as I can tell rounding of significant figures carries a certain degree of controversy and people will argue with you based on what they were taught at some point in their education. For example I learned from my "Biostatistics" course in college that when rounding a number that is followed by a 5, for example 1.1150, one should round up to the even number, 1.12 or not round up if the number was already even.  The explanation that the professor gave was that even numbers are easier to deal with in a calculation, which now seems to me like an odd reason. More recently I have been told from statisticians that I respect that this procedure removes the rounding bias.  They explain that without bias half of the time the number is rounded up, to me this makes a lot of sense, after all as scientists we want to be as unbiased as humanly possible.  Others always round up in this situation regardless of whether the number is even or odd.  Our position on this subject is we don't care what you do, but you should establish your own policy and follow it absolutely consistently, but of course just so you will understand, the method we have adopted is correct (a little joke). Another painful detail that can cause controversy is that if the number following the 5 is not a zero, for example 1.1151, the number should be rounded up. This is the policy that we follow.  Again set your own policy or if you are working with a larger group follow that policy. Be consistent.

Rounding policies that everyone agrees with:

If you are rounding a number to a certain degree of significant digits if the number following that degree is less than five the last significant figure is not rounded up, if it is greater than 5 it is rounded up.

Examples: 

A) 10.5660 rounded to four significant figures is 10.57

B) 10.5640 rounded to four significant figures is 10.56

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Conclusion:

We agree that we have not addressed every controversy on this subject but we hope that you understand how we deal with numbers at IonSource.Com.   For a quality easy to follow tutorial on rounding and significant figures visit Dr. Stephan Morgan at the University of South Carolina.  If you need to find a consultant to teach a course on statistics at your company we suggest Statistical Designs , they also have several tutorials on-line. The people at Statistical Desings teach statistics and experimental design for the American Chemical Society.  For an interesting paper on significant figures and rounding visit Dr.Christopher Mulliss at his web site.

Other significant figure and rounding sites we have found:

http://dbhs.wvusd.k12.ca.us/SigFigRules.html
http://www.chem.ufl.edu/~chm2040/Notes/Chapter_1/figures.html#nist
http://www.scimedia.com/chem-ed/data/sig-figs.htm
http://edie.cprost.sfu.ca/~rhlogan/sig_fig.html
http://www.angelfire.com/oh/cmulliss/index.html
http://www.bae.ncsu.edu/bae/courses/bae578/sigfig.html
 
 

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Last updated:  Monday, March 06, 2000 04:04:36 AM

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