IonSource Significant Figures & 
Rounding

Our policy with respect to significant figures and rounding at the IonSource.Com web site.

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 tell you how we treat numbers at IonSource.Com.  We are aware that there are many great significant figure tutorials presented on the internet. We agree that this presentation doesn't necessarily add anything new to those discussions. For a list of other significant figure web pages please see the list of links at the bottom of this page, or do a web search.  The sole purpose of this presentation is to describe to the reader how we deal with significant figures, and rounding at IonSource.Com. This is our significant figure, and rounding SOP, our standard operating procedure for numbers.  

Scientists routinely attempt to describe the world with numbers. If you are a mass spectroscopist you had better love numbers, because in many instances they are all you have.   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 create a document called an SOP, standard operating procedure.  Then when a regulatory agency comes to call, the officers at the company can show the investigators the policy. You do not want to be in a situation where you barely pass a test because the analyst always rounds up, but the regulatory agency finds an instances where another analyst, or worse the same analyst rounded down 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 you will gain respect from your peers by treating numbers with respect, and by reporting only significant figures and by rounding properly.

 

What is a significant figure? 

There are two types of significant figures, measured and exact.



Measured Observations

As scientists we get a large amount of the numbers we report, and use in our calculations from  measured observations. In this instance a number is determined to be significant or not by the accuracy and precision of the measuring device.  With a number derived from a measurement the last digit to the right  expresses the uncertainty.  For example if you are sure that your low resolution quadrupole mass spectrometer can deliver an 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 0.05 u.  Even if the instrument could spit out 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.

 

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 number 1002  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 0.5 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 0.0025 and 0.0035.  It is important to note that in some instances scientists will 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

 

Exact Numbers

Exact numbers 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.

 

Rounding significant figures
(now it gets personal)

As far as we 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 a a bad 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 more 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 that we don't care what you do, but be consistent. 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, and 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

 

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 Designs 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://www.angelfire.com/oh/cmulliss/index.html

 

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Last updated:  Saturday, September 01, 2012 03:26:01 PM