I am working on error analysis of some of my measurements and have stumbled upon an issue with the propagation of error. According to this reading,
- "The uncertainty of the average acidity (ΔacidHavg) was calculated as the root sum square of the random and systematic errors. For example, for the A3CSH system, the random error was treated as the averaged uncertainty of the reference acids (±2.2 kcal/mol) divided by the square root of the number of the reference acids, (2.2/√6) = 0.9 kcal/mol, and the systematic error was assigned as √2.2 = 1.5kcal/mol. The root sum square of the random and systematic errors yielded √(0.92+1.52) = 1.7kcal/mol."
Why is σsys=√2.2 instead of σsys=2.2?
It was my understanding that if the reported uncertainty of the reference acids are ±2.2kcal/mol, shouldn't the root-sum-square be
σRSS=√(σ2sys+σ2rand)=√(2.22+(2.2/√6)2)= 2.4 kcal/mol instead?
If my method is wrong, can I have an explanation pr directed some kind of reading explaining this concept?
Thanks!
One mistake is to use the standard uncertainty of the mean for the random uncertainty (NIST Type A). Use the standard uncertainty directly, not the correction for the population size. Imagine you would measure an infinite number of measurements for a true population standard deviation. The value used your example equation would incorrectly go to zero.
Also, the systematic uncertainty (NIST Type B) should be determined from the uncertainties of the devices. It should certainly not be calculated from the random uncertainty. For example, when the volumetric buret has ticks at every 1 mL, the device (measurement) uncertainty is ±0.5 mL. This becomes the systematic uncertainty. I've seen chemistry texts say to use 1/10 of the scale markings, but that is absolute rubbish in practice (try reading from a millimeter ruler with 100 micron precision!!!).
The total uncertainty is from the root-sum equation as
ST2 = SR2 + SS2
The use of S rather than sigma would be recommended. The latter is understood to be the true population standard deviation, whereas what is report is the standard uncertainty of a finite population size.
I recommend the NIST Website on uncertainties. Also search for references to the GUM (Guide to Expression of Uncertainties in Measurements).
I followed up on your suggested readings to get more insight into this matter. Thanks for the help!
You might also consider any one of these books ...
http://www.amazon.com/gp/product/0072472278?colid=1EN2R2ZWLC448&coliid=I1WE5D29D11I9D&ref_=wl_it_dp_o_pC_nS_ttl
http://www.amazon.com/gp/product/0521605792?colid=1EN2R2ZWLC448&coliid=I3J3RY59SJ8523&ref_=wl_it_dp_o_pd_nS_ttl
http://www.amazon.com/gp/product/0521134927?colid=1EN2R2ZWLC448&coliid=I2T9JD469E37HW&ref_=wl_it_dp_o_pd_nS_ttl
http://www.amazon.com/gp/product/019956633X?colid=1EN2R2ZWLC448&coliid=I2A1ADCVIIBZ2A&ref_=wl_it_dp_o_pC_nS_ttl
http://www.amazon.com/gp/product/1469985861?colid=1EN2R2ZWLC448&coliid=IIRSMBWCG7O4&ref_=wl_it_dp_o_pC_nS_ttl
http://www.amazon.com/gp/product/331912062X?colid=1EN2R2ZWLC448&coliid=ITIMFY18PJZDZ&ref_=wl_it_dp_o_pC_S_ttl
http://www.amazon.com/Introduction-Error-Analysis-Uncertainties-Measurements/dp/0935702423/ref=mt_hardcover?_encoding=UTF8&me=
First let’s distinguish between error and uncertainty. Error is the difference between the “correct value" and the measured value. The “correct value” here is the value which we would have obtained if we had a perfect instrument (which we don’t).
There are two types of error, systematic and error. Systematic error is one whose value is consistent, whereas random error changes unpredictably.
Uncertainty of measurement is defined as "non-negative parameter characterizing the dispersion of the quantity values being attributed to a measurand, based on the information used”.
Then there are two types of uncertainty sources: Type A and Type B. Type A refers to uncertainty sources evaluated by statistical analysis of the measured values, whereas Type B is evaluated by other methods.
There are no such things as “systematic uncertainty” and “random uncertainty”.
Reference:
[1] International vocabulary of metrology (VIM), JCGM 200:2008
[2] Evaluation of measurement data — Guide to the expression of uncertainty in measurement (JCGM 100:2008, also known as the "ISO GUM").
These references are used as canon by national metrology institutes around the world (including NIST), although they might not be well known by the larger scientific community yet. Both can be obtained at http://www.bipm.org/en/publications/guides/
> There are no such things as “systematic uncertainty” and “random uncertainty”.
Thank you for the clarification of the syntax. Note however at section 2.26 of the JCGM that you reference we find this ...
Measurement uncertainty includes components arising from systematic effects ...
It leaves room for confusion no doubt. The fine tuning that this reference gives for the language and your insight to post it are greatly appreciated!
Thanks for the clarification, I noticed that in my writing I've been uses these terms as synonyms but as I suspected, it was not the case. Most of my labs error analysis come from techniques other people have applied but was never explained correctly. I want to really understand these concepts to be as correct as possible.
I would suggest the following article by Richard A Davis:
SIMPLIFY UNCERTAINTY ANALYSIS WITH EXCEL MACROS
- Badges
- Science method