Estimation of Uncertainties in Chemical Measurement Guide to the Expression of Uncertainty in Measurement (GUM)- An Overview Angelique Botha
Method of evaluation: Analytical measurement Step 1: Specification and modeling Step 2: Identify the uncertainty sources Step 3: Quantify the uncertainty sources Step 4: Calculate the total uncertainty (combined standard uncertainty) Step 5: Calculate the expanded uncertainty Step 6: Reporting the uncertainty
Step 1: Specification and modeling A clear and unambiguous statement of what is being measured. Measurand Matrix Method: Rational methods, empirical methods Model: A quantitative expression relating the value of the measurand to the parameters on which it depends (SOP).
Modeling Derive a functional relationship between the measurand and input quantities from the method and procedure of the measurement It may be an analytical expression ( X X ) Y = f 1,,..., 2 X N Should include every quantity, including all corrections/correction factors that contribute to the uncertainty
Modeling (cont.) Or a simple expression plus corrections/correction factors for systematic effects Y = Ystd + ( corrections) Y determined directly by comparison with the value of a standard Or determined experimentally or exist only as a computer algorithm that must be evaluated numerically Or a combination of all of these
Step 2: Identify uncertainty sources Incomplete definition of the measurand Imperfect realisation of the definition Imperfect mathematical model Sampling method Uncertainties in values of measurement standards and reference materials Uncertainties in constant or other parameters obtained from other sources Environmental factors Random variation in repeated observations Instrument resolution, etc.
Step 2: Identifying uncertainty sources Sampling Storage conditions Instrument effects Reagent purity Measurement conditions Sample effects Computational effects Blank correction Operator effects Random effects
Cause and effect analysis The strategy has two stages: Identify the effects on a result using a cause and effect diagram i.e. Ishikawa or fishbone diagram. Simplify and resolve duplication.
Cause and effect analysis (cont.) Step 1: Write the complete equation for the result.
Cause and effect analysis (example) ( EtOH ) = d = M gross M tare V
Initial list d(etoh)
Cause and effect analysis (cont.) Step 1: Write the complete equation for the result. Step 2: Consider each step of the method or parameter in the equation and add additional main branches.
Initial list: Main branches M(gross) M(tare) d(etoh) Volume
Cause and effect analysis (cont.) Step 1: Write the complete equation for the result. Step 2: Consider each step of the method and add additional main branches. Step 3: For each branch, add factors until their effect on the result becomes negligible.
Initial list Temperature Precision M(gross) Linearity Bias Temperature Calibration Precision M(tare) Linearity Bias Calibration d(etoh) Precision Calibration Volume
Cause and effect analysis (cont.) Step 1: Write the complete equation for the result. Step 2: Consider each step of the method and add additional main branches. Step 3: For each branch, add factors until their effect on the result becomes negligible. Step 4: Resolve duplications, re-arrange, group related causes.
Cause and effect analysis (cont.) RULE 1: Cancelling effects: remove both. RULE 2: Similar effects, same time: combine into a single input. RULE 3: Different instances: re-label.
Combination of similar effects Temperature M(gross) Linearity Bias Temperature M(tare) Linearity Bias Calibration Calibration Precision Precision d(etoh) Precision Calibration Volume Temperature Precision
Cancellation M(gross) Linearity Bias Calibration M(tare) Linearity Bias Calibration d(etoh) Calibration Volume Same balance: bias cancels Temperature Precision
Step 3: Quantifying uncertainty Two categories Those whose estimate and associated uncertainty are directly determined by the current measurement Those whose estimate and associated sources uncertainty are brought into the measurement from external sources Their uncertainties require different ways of evaluation
Step 3: Quantifying uncertainty sources (cont.) Classification of uncertainty components according to the method of evaluation Type A components: those that are evaluated by statistical analysis of a series of observations Type B components: those that are evaluated by other means Both are based on probability distributions Standard uncertainty of each input estimate is obtained from a distribution of possible values for the input quantity: based on the state or our knowledge Type A founded on frequency distributions Type B founded on a priori distributions
Type A evaluation For component of uncertainty arising from random effect Applied when multiple independent observations are made under the same (repeatability) conditions Data can be from repeated measurements, control charts, curve fit by least-squares method, etc. Obtained from a probability density function derived from an observed frequency distribution (usually Gaussian)
Type A evaluation (cont.) Best estimate of the expected value of an input quantity arithmetic mean q = 1 n n k = 1 q k Distribution of the quantity experimental standard deviation Spread of the distribution of the means experimental standard deviation of the mean Type A standard uncertainty Degrees of freedom s 1 n 1 n ( ) ( ) 2 q = q q k s ( q ) = k = 1 s k ( q ) k n ( x ) s( q ) u i = ν i = n 1
Type B evaluation Evaluated by scientific/professional judgment Based on all available information Previous measurement data Experience of the behaviour of instruments or materials Calibration certificates Manufacturer s specifications Reference data from textbooks Can be as reliable as type A components Can sometimes be verified by experiment
Type B evaluation If the value is obtained from a calibration certificate, a normal distribution is assumed and the standard uncertainty and degrees of freedom must be retrieved from the certificate Standard uncertainty : ( x ) u i = Degrees of freedom from the Student s t- distribution tables U k
Type B evaluation It may be stated that the quoted uncertainty defines an interval having a 90, 95 or 99% level of confidence (normal distribution) Level of confidence Coverage factor k 90 1,64 95 1,96 99 2,58
Type B evaluation If the quoted uncertainty is stated to be a particular multiple of a standard deviation the standard uncertainty is simply the quoted value divided by the multiplier.
Type B evaluation Rectangular a a Triangular a a U-shaped a a a - Ч a + a - Ч a + a - Ч a + Best estimate µ = a + + 2 a Standard uncertainty of the best estimate a u( x i ) = u( x i ) = ( x ) 3 a 6 u i = a 2 Degrees of freedom = infinity if the reliability is 100%
Step 4: Calculating the combined uncertainty Before combining, convert all uncertainty contributions to standard uncertainties. Rules: Standard deviations from repeated observations in an experiment. For results and data from previous studies use the standard deviation or calculate the standard deviation from the probability distribution that applies.
Step 4: Calculating the combined uncertainty All uncertainty components must be converted to be in the same unit of measurement (speak the same language ). To combine standard uncertainties, they must all be converted to standard variances (squared). They can then be combined linearly (summed). To obtain a combined standard uncertainty, the resulting variance must be converted to a deviation (positive square root).
Step 4: Calculating the combined uncertainty Sensitivity coefficients are used to rewrite standard uncertainties in the same unit of measurement Found by obtaining partial derivatives, or a numerical estimation method can be used For simple mathematical models, these sensitivity coefficients are normally 1
Step 4: Calculating the combined uncertainty The combined standard uncertainty is the positive square root of the combined variance, given by u 2 c ( y) = 2 n f u 2 ( x = x 1 i i where f is the function describing the estimation of the measurand (First order Taylor series) i )
Step 4: Calculating the combined uncertainty These partial derivatives are often called sensitivity coefficients c i c i = f xx i It describes how the output estimate y vary with changes in the input quantities x 1, x 2,, x N The change in y produced by a small change in x i is given by ( y) = ( x ) i f x i i
Step 4: Calculating the combined uncertainty The combined variance can now be written as u 2 c ( y) = n [ c u( x )] i i i = 1 2 n i = 1 u 2 i ( y) where u i (y) is called the uncertainty contribution The sensitivity coefficients can be evaluated numerically by changing the value of an input quantity by a small amount and determining the effect it has on the estimate of the measurand.
Step 5: Determine the expanded uncertainty U = k u c ( y) k = coverage factor chosen from the t-distribution table, depending on the desired level of confidence and the effective degrees of freedom
Expanded uncertainty Issues to consider when choosing a coverage factor k: The level of confidence required Knowledge of the sample distribution Knowledge of the number of values used to estimate random effects Most cal labs adopt an approx. 95% level of confidence with k=2 for effective degrees of freedom > 30.
Calculating the expanded uncertainty Obtain the measurement result y and the combined standard uncertainty u c (y) ν Compute eff from the Welch-Satterthwaite formula: 4 uc ( y) ν eff = N 4 u i i= 1 ν i ( y) Obtain the coverage factor for the desired level of confidence from the t-distribution tables. Calculate the expanded uncertainty
Degrees of freedom Uncertainty Type A repeated observations Type A linear least square regression Type A: least-squares fit of m parameters to n data points Type B: 100% reliability Degrees of freedom n 1 n 2 n m Type B: reliability < 100% ν 1 2 100 100 R 2
Should include Result of measurement Reporting Expanded uncertainty with coverage factor and level of confidence Example of uncertainty statement The expanded uncertainty of measurement is ±.., estimated at a level of confidence of approximately 95% with a coverage factor k =..
Concluding remarks GUM: provides a framework for assessing uncertainty, it cannot substitute for critical thinking, intellectual honesty and professional skill Correct practice of the GUM helps to identify possible uncertainty sources and quantify their contribution to the total uncertainty The GUM cannot eliminate any unknown significant systematic errors in the measurement.