Uncertainties of Type A

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…from CheLabWiki, an online resource for chemical-engineering laboratories located at www.chelabwiki.org; Site Revision #423; 6 January 2009.

Community Portal → Data Analysis → Uncertainties of Type A


Although we cannot quantify errors, we can quantify uncertainties: an uncertainty is a range of values that is likely to contain the (unknown) exact value. Following the ANSI guide,[1] we divide uncertainties into two kinds: type A and type B. Type A uncertainties are those in which we use repeated sampling to quantify the distribution of values. We then obtain the uncertainty from a statistical analyses of the repeated measurements. We caution that type A uncertainties are not necessarily confined to statistical errors because uncertainties are independent of errors.


Evaluating Type A Uncertainties

Table 1. Ten measured values for flow rate of water through a pipe.
Run # Flow Rate Run # Flow Rate
1 5.5 6 5.6
2 5.85 7 5.75
3 5.55 8 5.65
4 5.8 9 5.4
5 5.9 10 5.7

We illustrate evaluation of type A uncertainties by analyzing the flow-rate measurements in Table 1. In those measurements, the sources of uncertainty include any fluctuations in the behavior of the meter, in the way we read it, and in the flow itself. We treat these fluctuations as type A uncertainties and estimate their combined effects by repeating the measurement ten times. We find the mean of those to be 5.67 gpm and their standard deviation to be s = 0.16 gpm.

This standard deviation s applies to any one measurement of y. It means that if we were to make additional measurements, then we expect that, assuming a Gaussian distribution, 68.3% of them would fall within ±1s of our mean and 95.5% would fall within ±2s. The values 68.3% and 95.5% indicate how confident we would be about a particular measurement; hence, they are called confidence levels.

But rather than report any one measured value of y, the mean ym would be more reliable. And as a measure of the quality of the mean, computed from N measured values of y, we use its standard deviation sm. From the propagation of uncertainties procedure, we can show that sm is related to s by[2]


(1)
s_m \ = \ \frac{s}{\sqrt{N}}


For our N = 10 measurements, (1) gives


(2)
s_m \ = \ \frac{0.16}{\sqrt{10}} \ = \ 0.051 \mbox{gpm}


We use this standard deviation as the type A uncertainty,[3]


(3)
u_A \ = \ s_m


Once we have estimates for both type A and type B uncertainties, then we must combine them to obtain the total. That procedure is describe on the page entitled Combined Uncertainty.

References

  1. ↑ U. S. Guide to the Expression of Uncertainty in Measurement, ANSI/NCSL Z540-2-1997, American National Standards Institute, NCSL International, Boulder, CO, 1997.
  2. ↑ J. R. Taylor, An Introduction to Error Analysis, University Science Books, Mill Valley, CA, 1982.ISBN 0-935702-10-5
  3. ↑ J. M. Haile, Analysis of Data, Macatea Productions, Central, SC, 2003. ISBN 0-9728602-0-7.
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