SOP for Measurement of Uncertainty (MOU]

1.0              Objective:

This SOP describes detailed procedure for estimating Measurement of Uncertainty (MOU] associated with any measurement activity.

2.0       Scope:

            Applicable for use in testing and calibration laboratories.

3.0              Responsibility:

            All Associates are responsible for estimating uncertainty.

            QA Officer/QA Manager: Review the records and governing the document.

4.0  Procedure:

4.1              Information

ISO/IEC 17025 guidelines, Quality Manual, And NABL 141 requirements.

4.2       Frequency

Once for all data analyzed and as and when new method/process/equipment arrives.

4.3              Step by Step Instructions

Measurement of Uncertainty (MOU):

MOU is defined as “A parameter associated with the result of a measurement that characterizes the dispersion of the values that could reasonably be attributed to the measurand”

Process of MOU calculations involves following four steps;

Step 1: Specify Measurand

Clearly mention what is being measured and also its relationship with input quantities on which it depends.

Step 2: Identify source of uncertainty

List all possible sources of uncertainty including sources from chemical assumptions. (Sources of uncertainty may arise from various sources, e.g. Sampling, Matrix effect, Interferences, Environmental conditions, uncertainty due to mass and glass wares, reference values being used, approximation and assumptions, random variations, repeatability and reproducibility of method used, equipment/instruments used, and operator specific.)

Step 3: Quantify uncertainty components

Measure uncertainty component associated with each potential identified source. Check whether available data accounts sufficiently for all sources of uncertainty.

Step 4: Calculate Standard uncertainty

Each component of uncertainty that contributes to measurement of uncertainty is expressed by an estimated standard deviation termed as standard uncertainty U₁ and is equal to the positive square root of variance U₁².

Step 4: Calculate Combined uncertainty

Express the individual uncertainties calculated in step 3 as standard deviation and combined them as per rule. U_c  is calculated square root of the sum of the squares based on the law of propagation of uncertainty.

 

                        U_c = √ U₁²⁺ U₁²⁺ U₁²⁺ U₁^{2+………….+} U_n²    

Step 5: Calculate Expanded uncertainty

Expanded uncertainty U_e is determined by multiplying U_c with coverage factor “k” such that the estimated true value of a measurement result Y may lie within “y - U_c “ and “y + U_c “ values, where “y” is the measured value of the parameter Y = y ± U.

The coverage factor k is generally taken as 2, which is equivalent to a confidence level of 95%.

Methods for estimating Numerical Values of Uncertainty:

There are two methods to estimate numerical values of uncertainty;

Method 1: Type A Evaluation and

Method 2: Type B Evaluation.

Type A Evaluation of standard uncertainty:

ü  Type A evaluation of standard uncertainty is the method of evaluating the uncertainty by statistical analysis of a series of observation.

ü  Here, standard uncertainty is the experimental standard deviation of the mean that follows from averaging or an appropriate regression analysis.

ü  Estimation is based on analysis of a series of observations by a valid statistical method.

ü  Type A evaluation of standard uncertainty applies to situation where several independent observations have been made under the same condition of measurement. If there is sufficient resolution in the measurement process, there will be an observable scatter or spread in the values obtained.

ü  Type A evaluation of standard uncertainty (U_i) generally arise out of random effects and follows a “Gaussian curve” or a normal probability distribution and the same is calculated using following formula;

U_i = SD/N or RSD = SD/ Mean of observations; N is no of observations recorded to get mean and SD.

SD – Standard Deviation;

SD Calculation: It is the amount of variation from the average of a set of a data. It is calculated using following formulae;

 

SD (n) = √ ∑ (X – X) ² / n-1

Where;

X = Individual Measurement

X = Average Measurement (Mean)

n = Number of measurements

Type B Evaluation of Standard Uncertainty:

ü  This is another method of evaluation of uncertainty by means other than the statistical analysis of a series of observations. The standard uncertainty is “ u (Xi) “ is evaluated by scientific judgment based on all available information on the possible variability of Xi (measurand).

ü  Values belonging to this category may be derived from;

·         Previous measurement data

·         Experience with behavior and properties of relevant materials and instruments

·         Manufacturers specification

·         Data provided in calibration and other certificates

·         Uncertainties assigned to reference data taken from handbooks.

ü  Type B evaluation of standard uncertainty can be as reliable as Type A evaluation of standard uncertainty, especially in a measurement situation where a Type A evaluation is based only on a small number of statistical independent observations.

ü  During testing or calibration the maximum error at any point is considered as the accuracy class of the equipment. If A is the accuracy of the measuring equipment, then for rectangular probability distribution standard uncertainty (Ui) is calculated using the formula;

(Ui) = √ A²/k

Where;

K=2 at 95% confidence level (CL) and k = 3, at 99% CL.

ü  The contribution to the standard uncertainties due to different factors are tabulated in the following format; n _1-1

Source of uncertainty	Type	Ui	Value	Divisor	Degree of freedom	Standard Uncertainty
Chemist	A	U1	SD1	1	n 1-1	SD1/N or SD1/M
Sampling	A	U2	SD2	1	n 2-1	SD1/N or SD1/M
Envt. Conditions	A	U3	SD3	1	n 3-1	SD1/N or SD1/M
Reference Material	B	U4	X4	√3 or √6		X4/√3or √6
Instrument Accuracy	B	U5(1)	X5(1)	√3 or ∞		X5/√3or √6
Calibration Uncertainty	B	U5(2)	X5(2)	√3 or ∞		X5/√3or √6

Note1:  n₁ , n₂, n₃….. are number of observations while testing variation in work piece and variation due to instrument respectively.

Note 2:When calculating expanded uncertainty using uncertainty values from external calibration certificates, the divisor for  U₅₍₂₎ divisor is 2.

When to use Normal distribution:

Sometimes quoted uncertainty is an input or output quantity stated along with level of confidence. In such cases, value of coverage factor needs to be defined, so that the quoted uncertainty may be divided by this coverage factor to obtain the value of standard uncertainty. When distribution type is not known, it may be assumed to be normal values of the coverage factor. Various levels of confidence for a normal distribution are as follows;

Confidence Level (CL)	68.27%	90%	95%	95.45%	99%	99.73%
Coverage factor (k)	1.000	1.645	1.960	2.000	2.576	3.000

When to use Rectangular distribution:

In some cases where it is possible to estimate only upper and lower limits of an input quantity and there is no specific knowledge about the concentration of values within interval, one can assume that it has equal probability to lie anywhere within interval ( rectangular distribution). In such cases, quoted uncertainty may be divided by coverage factor √3 to obtain the value of standard uncertainty.

                                                TABLE

Assumed probability distribution	Expression used to obtain the standard uncertainty	Comments or examples
Rectangular	U (Xi) = ai/√3	A digital thermometer has a least significant digit of 0.1°C. The numeric rounding caused by finite resolution will have semi-range limits of 0.05°C. Thus the corresponding standard uncertainty will be U (Xi) = ai/√3 = 0.05/ 1.732 = 0.029°C.
Triangular	U (Xi) = ai/√6	A tensile testing machine is used in a testing laboratory where the air temperature can vary randomly but does not depart from the nominal value by more than 3°C. The machine has a large thermal mass and is therefore most likely to be at the mean air temperature, with no probability of being outside the 3°C limits. It is reasonable to assume a triangular distribution, therefore the standard uncertainty for U (Xi) = ai/√6 = 3/ 2.449 = 1.2°C.
Normal (from repeatability evaluation)	U (X1) =  S (q¯)	A statistical evaluation of repeatability gives the result in terms of one standard deviation; therefore no further processing is required.
Normal (from a calibration certificate)	U (Xi) = U/k	A calibration certificate normally quotes an expanded uncertainty U at a specified, high coverage probability. A coverage factor, k, will have been used to obtain this expanded uncertainty from the combination of standard uncertainties. It is therefore necessary to divide the expanded uncertainty by the same coverage factor to obtain the standard uncertainty.
Normal (from a manufacturer’s specification	U(Xi) = Tolerance limit/k	Some manufacturers’ specifications are quoted at a given coverage probability (sometimes referred to as confidence level), e.g. 95% or 99%. In such cases, a normal distribution can be assumed and the tolerance limit is divided by the coverage factor k for the stated coverage probability. For a coverage probability of 95%, k = 2 and for a coverage probability of 99%, k = 2.58. If a coverage probability is not stated then a rectangular distribution should be assumed.

Steps to summarize an estimate of Uncertainty associated with measurand:

    

Step 1	Specification of Measurand	Write down clear statement about what is to be measured including relationship between measurand and parameters on which it depends.
Step 2	Identification of uncertainty sources	List out all the possible sources of uncertainty associated with the input quantities that contribute to the value of the measurand.
Step 3	Quantify components of uncertainty	Estimate the size of uncertainty associated with each potential source of uncertainty defined.
Step 4	Calculate Total uncertainty	Combine the quantified uncertainty component expressed as standard deviations. As per appropriate rule to give a combined standard uncertainty, Uc.

Evaluation of Combined Uncertainty:

Rule 1- When all the calculated uncertainties are of same units:

After measuring the individual uncertainty due to type A and type B, the combined uncertainty (U_c) is calculated as follows;

 

    U_c = √ (U₁² + U₁² + U₁²+……)

Where;

U_1, U_2, U₃ etc are uncertainties calculated with same units.

Rule 2- When all the calculated uncertainties are of different units:

                                                                                     Uncertainty

Combined Uncertainty = Result Declared ×

                                                                            Expected value or mean value

    U_c = √ (R₁×U₁/V₁) + (R₁×U₁/V₁)+……

Where; U_1, U_2, U₃ etc are uncertainties calculated R₁, R₂ etc are the result declared V₁, V₂ etc are the expected value or mean value.

Combined Uncertainty of Precision:

Up = √  U₁² + U₁² + U₁²

As per rule 1.

Total Combined Uncertainty:

After measuring the individual uncertainty due to Type A and Type B, the combined uncertainty (U_c ) is calculated as;

 

U_c = √  U_p² + U₄² + U₅₍₁₎² + U₅₍₂₎²

Expanded Uncertainty:

The product of combined uncertainty and coverage factor (k) gives the expanded uncertainty. The expanded uncertainty (Ue) calculated as;

Ue = U_c × 2

K is the coverage factor for different effective degree of freedom at 95% CL.

Final Result = Mean Value ±  Ue

5.            References

1.      Clause 5.4.6 of ISO/IEC 17025:2005 guidelines,

2.      Purchase specifications

3.      NABL 141

4.      Eurachem/CITAC guide for “Quantifying uncertainty in Analytical Measurement”

6.0       Documentation:

All the data of uncertainty parameters will be recorded and it will be filed in the respective files.

                   

7.0       History of Revision:

Revision No.	Effective Date	Revision details	Reason for revision