Presentation to: EUROMET workshop TEMPMEKO 2004. Uncertainty in measurements and calibrations - an introduction. Stephanie Bell UK National Physical Laboratory. Some facts about measurement uncertainty (“the Four Noble Truths”!). Every measurement is subject to some uncertainty.
- an introduction
UK National Physical Laboratory
(“the Four Noble Truths”!)
uncertainty of measurement
Rigorous definition: [ISO GUM]
parameter associated with the result of a measurement, that characterises the dispersion of the values that could reasonably be attributed to the measurand.
1. The parameter may be, for example, a standard deviation, or the half-width of an interval having a stated level of confidence
2. Uncertainty of measurement can comprise many components
3. It is understood that the result of the measurement is the best estimate of the value of the measurand and that all components of uncertainty contribute to the dispersion
quantified doubt about the result of a measurement
It is important not to confuse the terms error and uncertainty
Error is the difference between the measured value and the “true value” of the thing being measured
Uncertainty is a quantification of the doubt about the measurement result
In principle errors can be known and corrected
But any error whose value we do not know is a source of uncertainty.
Input – formulation – evaluation
Eight main steps to evaluating uncertainty (homage to the GUM)
1 Decide what you need to find out from your measurements. What actual measurements and calculations are needed to produce the final result. (The Model)
2 Carry out the measurements needed.
3 Estimate the uncertainty of each input quantity that feeds into the final result. Express all uncertainties in consistent terms.
4 Decide whether errors of the input quantities would be independent of each other. If you think not, then some extra calculations or information are needed (correlation).
5 Calculate the result of your measurement (including any known corrections for things such as calibration).
6 Find the combined standard uncertainty from all the individual parts.
7 Express the uncertainty in terms of a size of the uncertainty interval, together with a coverage factor, and state a level of confidence.
8 Record the measurement result and the uncertainty, and state how you got both of these.
From repeated measurements you can:
The two ways to evaluateindividual uncertainty contributions:
“Normal” or Gaussian distribution
Uniform or rectangular distribution
(NB probability uniform, not data - small set!)
Uncertainties are expressed in terms of equivalent probability*
Uncertainty in a mean value - “standard uncertainty” u
… for a Type A uncertainty evaluation
… for a Type B evaluation of a rectangular distributed uncertainty
(Summation in quadrature)
This rule applies where the result is found using addition / subtraction.
Other versions of this rule apply
… for multiplication or division
… for more complicated functions
All uncertainties must be in the same units and same level of confidence
Difference (Correction Ref-UUT)
Difference (Correction Ref-UUT)
Difference = Ref - UUT
Simple quadrature summation of independent standard deviations would suggest
Sdiff2 =Sref2 +SUUT2
But clearly Sdiff is far smaller than the combination of the two others
Where and are the arithmetic means of the observed values xi and yi, respectively and n is the number of measurements.
The columns are:
Symbol or reference
Source of uncertainty –
brief text description of each uncertainty
Value (±) – a number which is the estimate of the uncertainty. The estimate comes from whatever information you have, such as “worst case limits” or “standard deviation”. Units should be shown, e.g. °C, or %rh.
Probablility distribution - either rectangular, normal, (or rarely others)
Divisor - factor to normalise a value to a standard uncertainty - depends on the probability distribution.
ci- sensitivity coefficient
to convert to consistent units, e.g. measurement in volts, effect in terms of relative humidity. (But sometimes ci = 1)
u– standard uncertainty a “standardised measure of uncertainty” calculated from previous columns: u = value ÷ divisorci .
νi– effective number of degrees of freedom – an indication of the reliability of the uncertainty estimate (the uncertainty in the uncertainty!) - sometimes ignored! Sometimes infinte! ()
The rows are:
One row for each uncertainty
One row for combined standard uncertainty, uc, by summing “in quadrature” and taking square root, i.e., uc =
Final row showing expanded uncertainty, U = k uc. Normally a coverage factor k = 2 is used (giving a level of confidence of 95 percent, as long as the number of degrees of freedom is high). The expanded uncertainty is what finally should be reported.
Uncertainty in a mean value “standard uncertainty” u
… for a Type A evaluation
… for a Type B evaluation of a rectangular distribution
Divisors respectively 1 and 1/(23)
Coverage factor k may be a divisor
1σ, 68% confidence, k=1
2σ,95% confidence, k=2
Other things you should know … (cont.)
How to express the answer
Write down the measurement result and the uncertainty, and state how you got both of these.
Express the uncertainty in terms of a coverage factor together with a size of the uncertainty interval, and state a level of confidence.
Example - How long is a piece of string
‘The length of the string was 5.027 m ± 0.013 m. The reported expanded uncertainty is based on a standard uncertainty multiplied by a coverage factor k = 2, providing a level of confidence of approximately 95%.
‘The reported length is the mean of 10 repeated measurements of the string laid horizontally. The result is corrected for the estimated effect of the string not lying completely straight when measured. The uncertainty was estimated according to the method in …[a reference]
k = 1 for a confidence level of approximately 68 %
k = 2 for a confidence level of approximately 95 %
k = 3 for a confidence level of approximately 99.7 %
u(u) 2(eff) -1/2 [ISO GUM]
Mistakes are not uncertainties
Tolerances are not uncertainties
Accuracy (or rather inaccuracy) is not the same as uncertainty
Statistical analysis is not the same as uncertainty analysis
“To follow the Noble Eightfold Path is a matter of practice rather than intellectual knowledge, but to apply the path correctly it has to be properly understood”