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UNCERTAINTY OF MEASUREMENTUncertainty in Measurement:
Uncertainty in Measurement Why do we need Uncertainties? No measurement is perfect Accreditation requirements ISO/IEC 17025 Section 5.4.6, Estimation of Uncertainty of Measurement Section 5.9, Assuring the quality of Results Section 5.10, Reporting the Results Accreditation Body Requirements ILAC/APLAC RequirementsMEASUREMENT UNCERTAINTY:
MEASUREMENT UNCERTAINTY Uncertainty evaluation is best done by personnel who are thoroughly familiar with the Test & calibration and understand the limitations of the measuring equipment and the influences of external factor The lower uncertainty is usually attained by using better equipment, better control of environment and ensuring consistent performance of the test. Measurement gives best estimate of true value Even after applying corrections to the measurand …….. True value cannot be determined True value of measurand is indeterminate Error is unknown & unknowable Error : Uncertainty = Cause : Effect UNCERTAINTY ALWAYS REMAINSRelevance of measurement uncertainty (clause 7.6 of ISO 9001:2000):
Relevance of measurement uncertainty (c lause 7.6 of ISO 9001:2000) Measurements provide evidence of conformity of products to specified requirements . Specified requirements are related to tolerance of product characteristics. Two factors to be considered Tolerance of product characteristics. Variability/uncertainty of measurement system. Uncertainty << product tolerance. Auditors to ensure Uncertainty is known. It should be in accordance with measurement objectives.Slide 5:
NABL Policy NABL’s Policy is that all laboratories shall follow the requirements of ISO/ IEC 17025 (Cl. 5.1.2, 5.4.6, 5.9, 5.10) on Uncertainty of MeasurementSlide 6:
BASIC TERMS TRUE VALUE BIAS ERROR OF MEASUREMENT MEASUREMENT UNCERTAINTY SENSITIVITY COEFFICIENT CORRELATION COEFFICIENT COVERAGE FACTOR CONFIDENCE INTERVALSlide 7:
BASIC TERMS True Value True value has been defined as the value that is perfectly consistent with the definition of a given specific quantity. It can be obtained by ideal measurement system. Conventional True Value It is defined as the value accepted by consensus among people knowledgeable on the subject. It is also referred to as assigned value, best estimate of the value, conventional value, accepted value, and reference value. Precision:
Precision “ The closeness of agreement between repeated measurements of the same quantity under the same conditions” Precision depends only on the distribution of random errors and does not relate to the true value or accepted reference value. The precision of a measurement process is linked with the dispersion of repeat observations among themselves. The most widely used measure of dispersion of statistical data is the standard deviation of the data. The lower the standard deviation, the more the measurement data are clustered among themselves and more precise the measurement process is. So standard deviation of the measurement data is a measure of its precision.Precision of Measurement:
Precision of Measurement 1. Repeatability (Equipment variation): variation in measurements under exact conditions.(same time frame,condition) 2. Reproducibility (Appraiser variation): variation in the average of measurements when different operators measure the same part.(diff.timeframe,condition) Accuracy:
Accuracy “ The closeness of agreement between the result of a measurement and the true value of the measurand.” As true value is unknown, accuracy of the measurement is also unknown as error hence qualitative . The following factors influence the accuracy of a measurement process: The precision of the measurement process, characterized by the dispersion of repeated observations due to the influence of sources of random errors. The standard deviation of repeat measurement observations is an index of precision. The bias of the measurement process, which is due to the presence of known and unknown systematic error components. In order for a measurement process to be accurate, it should be precise as well as unbiased. As measurement accuracy is indeterminate in nature, for better understanding of the quality of the measurement result, the concept of measurement uncertainty was evolved.Accuracy of Measurement:
Accuracy of Measurement Broken down into three components: 1. Stability: the consistency of measurements over time. 2. Accuracy: a measure of the amount of bias in the system. 3. Linearity: a measure of the bias values through the expected range of measurements.UNCERTAINTY OF MEASUREMENT:
UNCERTAINTY OF MEASUREMENT Standard Uncertainty: component of uncertainty that contributes to the uncertainty of a measurement result by an estimated standard deviation is termed ‘ standard uncertainty’ Ex.: Type A & Type B Standard uncertainty Evaluation of Standard Uncertainty :
Evaluation of Standard Uncertainty Type-A evaluation of standard uncertainty Type-A evaluation of uncertainty is based on statistical analysis of repeat observations of the measurand obtained under the same conditions of the measurement. Standard uncertainty is expressed as standard deviation. Type-B evaluation of standard uncertainty Type-B uncertainty is evaluated by scientific judgement based on all the available information about the variability of the variables contributing to the uncertainty. Type-B evaluation is based on probability distribution. The knowledge base, on which type-B evaluation can be carried out may consist of the following : Previous measurement data Manufacturer’s specification Data provided in calibration certificates Uncertainties assigned to reference data etc.Combined Uncertainty :
Combined standard Uncertainty uc(y) Standard Uncertainty of the result y of a measurement when the result is obtained from the values of a number of other quantities, equal to the positive square root of a sum of terms, the terms being the variances or covariance of these other quantities weighted according to how the measurement result varies with these quantities Combined UncertaintyExpanded Uncertainty :
Quantity defining an interval about the result of a measurement that may be expected to encompass a large fraction of the distribution of values that could reasonably be attributed to the measurand An expanded Uncertainty U is calculated from a combined standard Uncertainty uc and a coverage factor k using U = k × u c Relative standard uncertainty: standard Uncertainty divided by the value Expanded UncertaintySlide 16:
Coverage Factor: A numerical factor used as a multiplier of the standard uncertainty of measurement in order to obtain an expanded uncertainty of measurement Confidence Level :68.27 90 95 95.5 99 99.73 Coverage Factor :1.0 1.645 1.960 2.00 2.562 3.00Important uncertainty contributors:
Important uncertainty contributors Environment The reference element of the measuring equipment The measuring equipment and setup Software and calculations The metrologist The measuring object The definition of the measurand (standard or perfect / ideal measurement) The measuring procedure Physical constants Intentional bias out of cost or resource concerns. The "operator model" is a relatively recent concept developed by ISO Technical Committee 213 and accounts for intentional uncertainty. This is probably the most overlooked uncertainty contributorSlide 18:
CLASSIFICATION OF UNCERTAINTY COMPONENTS TYPE A (those evaluated by statistical methods) (Random error) & TYPE B (those evaluated by other means) (Systematic error)Slide 19:
Evaluated by means other than statistical analysis of a series of observations. Type – B uncertainty evaluation is based on : Previous measurement data. : Manufacturer specification : Data provided in calibration certificates. :Uncertainty assigned to reference, data & so on. Calculation of Type B UncertaintySlide 20:
Uncertainty is obtained by dividing -supplied data by stated specific multiple for the applicable probability distributions like - : Normal or Gaussian distribution : Triangular distribution : Rectangular distribution Calculation of Type B UncertaintyMEASUREMENT OF UNCERTAINTY:
21 TYPES OF DISTRIBUTION 1)Normal MEASUREMENT OF UNCERTAINTYMEASUREMENT OF UNCERTAINTY:
22 • An estimate is made from repeated observations of a randomly varying process. u(x) = s • An Uncertainty is given in the form of a standard deviation s, a relative standard deviation s / x , or a coefficient of variance CV% without specifying the distribution. u(x) = s u(x)=x⋅(s/x(mean)), cv%.x/100 MEASUREMENT OF UNCERTAINTY An Uncertainty is given in the form of a 95% (or other) confidence interval x±c u(x) = c /2 (for c at 95%)MEASUREMENT OF UNCERTAINTY :
MEASUREMENT OF UNCERTAINTY 2. TriangularMEASUREMENT OF UNCERTAINTY :
MEASUREMENT OF UNCERTAINTY The available information concerning x is less limited than for a rectangular distribution. Values close to x are more likely than near the bounds. • An estimate is made in the form of maximum range(±a) described by symmetric distribution u(x)= a/sqrt6 Example : Glass ware calibration accuracy is triangularMEASUREMENT OF UNCERTAINTY :
25 MEASUREMENT OF UNCERTAINTY 3. RectangularMEASUREMENT OF UNCERTAINTY :
26 MEASUREMENT OF UNCERTAINTY Values close to x are more likely near the bounds and confidence level not specified. An estimate is made in the form of maximum range(±a) with no knowledge of space of distribution • An estimate is made in the form of maximum range(±a) described by distribution u(x)= a/sqrt3 Example : Accuracy,Resolution,Tolerance limits etc.,Slide 27:
Combined uncertainty is given by U C = √u²A+u²B at 95% confidence level. Expanded uncertainty is given by U E = Uc x k at 95% confidence level Measurement Result = x ± U E units Calculation of Expanded UncertaintyMEASUREMENT OF UNCERTAINTY :
28 MEASUREMENT OF UNCERTAINTY Step 1 : Write the formula for the measurand Step 2: Check in the procedure for any tolerances that has been mentioned to study the tolerance effect on measurand Step 3:From above two steps list the sources of Uncertainty for the measurand Step 4:Where ever required calculated Uncertainty in recovery (in case of instrumental methods) Step 5: Write logical /Scientific assumptions wherever Uncertainty contribution is negligibleMEASUREMENT OF UNCERTAINTY :
29 MEASUREMENT OF UNCERTAINTY Step 6:Then calculate the Standard Uncertainty Step 7:If measurement units are different ( like ml and mg) calculate Relative Standard Uncertainty Step 8:Then calculate Combined Uncertainty and then Expanded Uncertainty using necessary coverage factor obtained from effective degree of freedomMEASUREMENT OF UNCERTAINTY :
30 MEASUREMENT OF UNCERTAINTY Common Parameters ● Mass ● Volume ● Formula Weight ● Purity of Reference MaterialsMEASUREMENT OF UNCERTAINTY :
17 MEASUREMENT OF UNCERTAINTY 2)Based on Calibration Uncertainty If Uncertainty mentioned by the Calibration agency in the calibration certificate used for Uncertainty calculation then Uncertainty to be calculated as Example : Accuracy of the balance 0.0002g Resolution of the balance 0.0001g Uncertainty from calibration certificate 0.0006g Accuracy is rectangular hence U a =0.0002/√3 = 0.0001154g Resolution is rectangular hence U r = 0.0001/ √3 = 0.0000577g Uncertainty from calibration certificate =0.0006 k factor =2 hence U c = 0.0006/2 =0.0003gMEASUREMENT OF UNCERTAINTY :
17 MEASUREMENT OF UNCERTAINTY Then Uncertainty is √ (U a ) 2 + (U r ) 2 +(U c ) 2 Then Uncertainty is √(0.0001154) 2 + (0.0000577) 2 + (0.0003) 2 = 0.00033 g If mass is taken by difference then Uncertainty is multiplied by 2 timesMEASUREMENT OF UNCERTAINTY :
33 MEASUREMENT OF UNCERTAINTY To calculate Uncertainty associated with volume Pipette Pipette details : 5 ml pipette Uncertainty Component Value Distribution Uncertainty Calibration accuracy ± 0.015 triangular 0.015/sq6= 0.00061 Repeatability σ√ 6 0.0004 Temperature Variation 5x 3 x 2.1 x 10 -4 = 0.0032 rectangular 0.0032/Sq3 =0.0018 Combined Uncertainty = Sqrt(0.00061 2 + + 0.0004 2 + 0.0018 2 ) = 0.017ml Note : Temperature Variation is ±3 co-efficient of expansion of water is 2.1 x 10 -4 / deg C. Hence formula for TV = Volume capacity x 2.1 x 10 -4 x 3MEASUREMENT OF UNCERTAINTY :
34 MEASUREMENT OF UNCERTAINTY Volumetric Flask, 100 ml Volumetric Flask details : Uncertainty Component Value Distribution Uncertainty Calibration accuracy ± 0.1Max Triangular 0.1/sqrt6=0.041 Repeatability σ√ 6 0.002 Temperature Variation 100x 3 x 2.1 x 10 -4 = 0.064 Rectangular 0.064/sqrt3 =0.037 Combined Uncertainty = Sqrt (0.041 2 + + 0.002 2 + 0.037 2 ) = 0.055ml Note : Temperature Variation is ±3 co-efficient of expansion of water is 2.1 x 10 -4 / deg C. Hence formula for TV = Volume capacity x 2.1 x 10 -4 x 3MEASUREMENT OF UNCERTAINTY :
35 MEASUREMENT OF UNCERTAINTY Purity reference material : Purity Uncertainty Distribution Standard Uncertainty a) 99.9 ± 0.1 Rectangular 0.1/sq3=0.058 b) 99.9min 100 max ± 0.1 Triangular 0.1/sq6=0.041 c) 99.9 ± 0.1,95% confident level ± 0.1 Normal 0.1/2 = 0.05MEASUREMENT OF UNCERTAINTY GRAVIMETRIC ANALYSIS:
36 MEASUREMENT OF UNCERTAINTY GRAVIMETRIC ANALYSIS Ash content Tno 1 2 3 4 5 6 Sample Weight ,g 2.2119 2.0826 2.1212 2.0602 2.0292 2.0144 Crucible + Residue Weight ,g 30.4582 23.2393 23.2468 30.183 29.2323 31.131 Crucible Weight , g 30.2946 23.0846 23.0896 30.032 29.0821 30.9816 Residue Weight , g 0.1636 0.1547 0.1572 0.152 0.1502 0.1494 Ash%= Residue weightX100/ Sample weight 7.40 7.43 7.41 7.38 7.40 7.42 Average Ash% 7.41 Standard Deviation σ 0.0187 σ about mean= σ/√6 0.00763 Type A Uncertainty U AMEASUREMENT OF UNCERTAINTY GRAVIMETRIC ANALYSIS:
37 MEASUREMENT OF UNCERTAINTY GRAVIMETRIC ANALYSIS Sources of Uncertainty 1) Due to Sample weight 2) Due to Residue weight 3) Due to Temperature Tolerance of ±20 degree centigrade 4) Due to Muffle Furnace Temperature ToleranceMEASUREMENT OF UNCERTAINTY GRAVIMETRIC ANALYSIS:
38 MEASUREMENT OF UNCERTAINTY GRAVIMETRIC ANALYSISMEASUREMENT OF UNCERTAINTY GRAVIMETRIC ANALYSIS:
39 MEASUREMENT OF UNCERTAINTY GRAVIMETRIC ANALYSIS Calculation of Uncertainty Type B 1)Sample weight Balance ABSFCT used with details as below Particulars Value Standard uncertainity Distribution Accuracy , g 0.0002 0.0002÷√3 Rectangular Resolution, g 0.0001 0.0001÷√3 Rectangular Calibration uncertainty , g 0.0006 0.0006÷2 Normal K=2 Hence uncertainty due to Mass ,g √(0.0001154) 2 +(0.0000577) 2 +(0.0003) 2 = 0.00033 U M Since Sample Weight is taken by taring hence uncertainty is 0.00033, g 2)Residue weight Same Balance is used uncertainty as above for Balance is 0.00033,g but Residue weight is taken by difference(Final weight- Initial weight)hence uncertainty is multiplied 2 times Therefore uncertainty in residue weight is 0.00066 , g U RMEASUREMENT OF UNCERTAINTY GRAVIMETRIC ANALYSIS:
40 MEASUREMENT OF UNCERTAINTY GRAVIMETRIC ANALYSIS Ash at 580 o C Ash at 620 o C Tno 1 2 Tno 1 2 Sample Weight ,g 2.2119 2.0004 Sample Weight ,g 2.0602 2.0001 Cruicible + Residue Weight ,g 30.4582 32.4590 Crucible + Residue Weight ,g 30.1838 30.1768 Cruicible Weight,g 30.2946 30.3116 Crucible Weight ,g 30.0318 30.0297 Residue Weight,g 0.1636 0.1474 Residue Weight ,g 0.152 0.1472 Ash %= Residue WeightX100/ Sample Weight 7.40 7.37 Ash%= Residue WeightX100/ Sample Weight 7.38 7.36 Ash, % 7.385 Ash, % 7.375 3)Due to Temperature Tolerance of ±20 degree centigrade Procedure specifies ash to be done at 600±20 degree centigrade uncertainity due to temperature difference is defined as below. To find the uncertainity due to temperature difference two sets of results done one at 580 degree centigrade and one at 620 degree centigradeMEASUREMENT OF UNCERTAINTY GRAVIMETRIC ANALYSIS:
41 MEASUREMENT OF UNCERTAINTY GRAVIMETRIC ANALYSIS So difference of 20 degree centigrade change is 1)0.025% (2)0.035% SINCE 0.035% is Maximum it is taken for calculation Temperature range for Ash is 580-620 degree centigrade i.e., 40 degree difference therefore 0.035% is multiplied by 2 i.e.,0.07% 0.07% is divided by square root of 3 because this change is rectangular distribution Therefore uncertainty due to temperature tolerance is 0.07%÷√3 =0.040% U T 4)Muffle furnace Temperature Tolerance Details of the muffle furnace is as below Particulars Value Standard uncertainity Distribution Accuracy, o C 5 5÷√3=2.8868 Rectangular Resolution, o C 1 1÷√3=0.5774 Rectangular Calibration uncertainity, o C 1.3 1.3÷2=0.65 Normal K=2 Hence uncertainty in muffle furnace temperature is √2.887 2 + 0.577 2 +0.65 2 = 3.01 o C So ±3 o C uncertainty in muffle furnace temperature changes the ash content by 0.035*3/20 i.e., 0.00525 also it is rectangular hence 0.00525÷√3 is uncertainty= 0.0030 % U FMEASUREMENT OF UNCERTAINTY GRAVIMETRIC ANALYSIS:
42 MEASUREMENT OF UNCERTAINTY GRAVIMETRIC ANALYSIS Calculation of Uncertainty Parameter Value V Standard Uncertainty U Relative Standard Uncertainty, U/V Sample Weight , g 2.0866 0.00033 0.00016 Residue Weight ,g 0.1546 0.00066 0.00427 Temperature difference o C(Change in Ash value %) 7.41 0.04 0.00540 Muffle Furnace Temperature Tolerance o C(Change in Ash Value%) 7.41 0.003 0.00040 Repeatability as σ about mean 7.41 0.00763 0.00103 Uncertainty in Ash content = Mean Ash Value(7.41%)*√0.00158 2 +0.00427 2 +0.00540 2 +0.00040 2 +0.00103 2 = 0.052% U Effective Degrees of Freedom 10479 ≈ ∞ So Coverage Factor is 2.0 Hence Expanded Uncertainty in Ash content is 7.41 ± 2*0.052 =7.41 ± 0.104 % Uncertainity is ±1.4MEASUREMENT OF UNCERTAINTY GRAVIMETRIC ANALYSIS:
43 MEASUREMENT OF UNCERTAINTY GRAVIMETRIC ANALYSIS ( U) 4 ------------------------------------------------------------------------------------------------------------ (U A ) 4 /5 + (U M ) 4 /∞+ (U R ) 4 /∞+ (U T ) 4 /∞+ (U F ) 4 / ∞ Effective Degrees of Freedom Effective Degrees of Freedom 10479 ≈ ∞ So Coverage Factor is 2.0 Hence Expanded Uncertainty in Ash content is 7.41 ± 2*0.052 =7.41 ± 0.104 % Uncertainity is ±1.4Slide 44:
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