1. Measurement Uncertainty and Traceability
CLEEN MMEA Certainty Seminar
MIKES, Espoo
Maija Ojanen

2. Introduction Why should we estimate measurement uncertainty?
To estimate the reliability of the results
To compare different results and measurement methods
To see if the device meets its specifications
To be cost efficient
To know in which applications the results can be used !!!
Maija Ojanen

3. Identification of uncertainty components
Measurement device
Stability
Needs to be studied based on successive calibrations
Linearity
Hysteresis
Resolution
Drift…
Deviation of the results, sampling
Measurement procedure
Environmental conditions
Temperature, vibration, relative humidity, air pressure, gravitation, air flow, …
Change in the above mentioned can result in significant effect in the measurement result
Calibrations typically done in well-controlled laboratory conditions!
Maija Ojanen

4. Uncertainty classification
Type A uncertainty
Can be evaluated by statistical methods
Type B uncertainty
Evaluated by other means: calibration certificate, specifications, literature, earlier experience etc.
Cannot be improved by repeating the measurements!
Standard uncertainty
Combined standard uncertainty
Expanded uncertainty
Maija Ojanen

5. Uncertainty analysis
Express in mathematical terms the dependence of the measurand (output quantity) on the input quantities. In the case of a direct comparison of two standards the equation may be very simple, e.g. Y = X1+X2.
Identify and apply all significant corrections.
List all sources of uncertainty in the form of an uncertainty analysis.
Calculate the standard uncertainty for repeatedly measured quantities (type A uncertainty).
Evaluate type B uncertainty by other means.
Calculate for each input quantity the contribution to the uncertainty associated with the output estimate resulting from the input estimate and sum their squares to obtain the square of the combined standard uncertainty of the measurand. If input quantities are known to be correlated, the associated covariances must be included.
Calculate the expanded uncertainty by multiplying the standard uncertainty associated with the output estimate by a coverage factor k (typically k = 2 -> 95 % confidence level).
Maija Ojanen

6. Uncertainty distributions
Gaussian distribution
Uncertainty from calibration certificate
Repeatability
Standard uncertainty = standard deviation (1 σ)
Rectangular distribution
Resolution of digital display
Manufacturer specifications
Triangular distribution
Difference of two rounded values
±1 σ
±2 σ
Maija Ojanen

7. Calculation example The measurand Y depends on input quantities Xi
Using relative uncertainties (xi /Xi) multiplied and divided input quantities produce an equal relative uncertainty in the measurand
If the input quantity is raised to nth power , the resulting relative uncertainty in the measurand is n-tuple
Maija Ojanen

8. Calculation example
Combined standard uncertainty is calculated as a square sum of the uncertainty components
Expanded uncertainty is calculated by multiplying the combined standard uncertainty by a chosen coverage factor k (typically k = 2)
Note: in this example all the input quantities are non-correlated!
Maija Ojanen

9. Correlation Two input quantities dependent on each other
Common calibration reference
E.g. measurement of a 5-m distance with a 1-m ruler: measured distance is a sum of five separate measurements with same error/uncertainty
Effect of correlation must be studied case by case; no general model
Can either increase or decrease the combined standard uncertainty
Maija Ojanen

10. Example of an uncertainty budget
“Any detailed report of the uncertainty should consist of a complete list of the components, specifying for each the method used to obtain its numerical value”
P. Kärhä, P. Toivanen, F. Manoochehri, and E. Ikonen, “Development of a detector-based absolute spectral irradiance scale in the nm spectral range,” Appl. Opt. 36, (1997).
Maija Ojanen

11. Precision vs. accuracy
Maija Ojanen

12. Traceability
Maija Ojanen

13. Literature
BIPM, IEC, IFCC, ISO, IUPAC, UIPAP, OIML. JCGM 100:2008 Evaluation of measurement data - Guide to the Expression of Uncertainty in Measurement. (GUM 1995 with minor corrections), 2008
European cooperation for Accreditation, EA-4/02 Expression of the Uncertainty of Measurement in Calibration, December 1999.
Maija Ojanen