Report

Bangladesh BEST Programme Uncertainty of Measurement Nihal Gunasekara Sri Lanka 1 Bangladesh BEST Programme What is a measurement ? Property of something How heavy of an object is How hot of an object is How long it is A measurement gives a number of that property 2 Bangladesh BEST Programme What do you need for a measurement ? Instrument Rulers Stopwatches Weighing scales Thermometers 3 Bangladesh BEST Programme How do you report a measurement ? The length of table is 20 m The weight of the object is 3 kg The temperature of the sample is 50 °C The volume of liquid is 50 ml Use SI units for all measurements 4 Bangladesh BEST Programme What is not a measurement ? Comparing two pieces of strings to see which is longer Comparing two liquids to see which is hotter Comparing height of two persons to see who is taller 5 Bangladesh BEST Programme What is uncertainty of measurement ? The uncertainty of measurement tells us something about its quality Uncertainty of measurement is the doubt that exists about the result of any measurement Can we expect accurate results from all measuring instruments ? A margin of doubt !!!!! 6 Bangladesh BEST Programme Definition of Uncertainty of Measurement “ Non-negative parameter characterizing the dispersion quantity values being attributed to a measurand, based on the information used” JCGM 200: 2012 BIPM 3rd Edition 7 Bangladesh BEST Programme Measurement Uncertainty X U U A range containing the true value 8 Bangladesh BEST Programme Expressing Uncertainty of Measurement Margin of doubt about any measurement !!!! How big is the margin ? How bad is the doubt ? Two numbers are needed to quantify an uncertainty Width of the margin or interval Confidence level 9 Bangladesh BEST Programme Error Versus Uncertainty Error : is the difference between the “measured value” and the” true value” of the thing being measured Error = measured value - true value (reference value) Uncertainty : is a qualification of the doubt about the measurement result 10 Bangladesh BEST Programme Error Versus Uncertainty Error can be corrected !!!!! How ? Apply correction form calibration certificates But any error whose value we do not know is a source of uncertainty !!!! 11 Bangladesh BEST Programme Why is uncertainty of measurement important? “ We wish to make good quality measurement and to understand the result” ISO 17025 requirements Calibrations & Testing laboratories shall have a procedure for calculation of MU Where not possible for some test methods of testing labs, the contributing factors need to be identified and a reasonable estimation be made When estimating MU all components that contribute to MU should be taken into account 12 Bangladesh BEST Programme Basic Statistics on Sets of Numbers “Measure thrice, cut once- operator error” Risk can be reduced by checking the measurement several times !!!! Take several measurements to obtain a value !!!! 13 Bangladesh BEST Programme Basic Statistical Calculations To increase the amount of information of your measurement : take several readings !!!! Two most important statistical calculations : Average or arithmetic mean Standard deviation - s 14 Bangladesh BEST Programme Getting the Best Estimate Repeated measurements give different answers If there is variation in readings when they are repeated Take many readings Get the average Best estimate for the “true” value Mean or average value Value of reading 15 Bangladesh BEST Programme How Many Readings Should you Average ? More measurements : better estimate of true value What is a good number ? 10 20 would give slightly better estimate than 10 16 Bangladesh BEST Programme Standard Deviation – Spread of Readings Repeated measurements : different readings How widely spread the readings are ? Usual way to quantify spread is “Standard Deviation” The standard deviation of a set of numbers tells us “about how different the individual readings typically are from the average of the set” 17 Bangladesh BEST Programme Calculating an Estimated Standard Deviation Example : Let the readings are 16, 19,18, 16, 17, 19,20,15,17, and 13 Average is 17 Find the difference between each reading and the average ie. -1 +2 +1 -1 0 +2 +3 -2 0 -4 And square each of those ie 1 4 1 1 0 4 9 4 0 16 Find the total and divide by n-1 (in this case n is 10) ie. 1+4+1+1+0+4+9+4+0+16 = 40 = 4.44 9 9 Standard deviation s = = 2.1 18 Bangladesh BEST Programme Mathematical Equation for Standard Deviation s r i r 2 n 1 19 Bangladesh BEST Programme Where do Errors and Uncertainties come from ? Measuring instrument - ageing effect, drift, poor readability etc Item being measured - ice cube in a warm room Measurement process - measurement itself may be difficult Imported uncertainties – instrument uncertainty Environment – temperature, air pressure, humidity vibration etc. 20 Bangladesh BEST Programme Distribution – Shape of Errors The spread of set of values can take different forms Probability of occupation Normal or Gaussian distribution Mean or average reading Value of reading 21 Bangladesh BEST Programme Uniform or Rectangular Distribution When measurements are quite evenly spread between the highest and lowest values a rectangular or uniform distribution is produced Probability of occurrence Range Value of reading Full width Value of reading 22 Bangladesh BEST Programme Triangular Distribution Probability of occurrence Value of reading 23 Bangladesh BEST Programme What is not a Measurement Uncertainty ? Mistakes made by a operator Tolerances of a product Specifications of instruments 24 Bangladesh BEST Programme How to Calculate Uncertainty of Measurement Identify the sources of uncertainty in the measurement Estimate the size of the uncertainty from each source Combine individual uncertainties to give an overall figure 25 How to Calculate Uncertainty of Measurement Specify Measurand Identify Uncertainty sources STEP 1 STEP 2 Simplify by grouping the sources covered by available data STEP 3 Quantify grouped and remaining components Convert components to standard uncertainties 26 How to calculate Uncertainty of measurement Calculate the combined standard Uncertainty Review and if required re-evaluate large components STEP 4 Calculate the Expanded Uncertainty 27 Bangladesh BEST Programme Estimation of Total Uncertainty Type A evaluation – method of evaluating the uncertainty by the statistical analysis of a series of observations Type B evaluation - uncertainty estimates by means other than the statistical analysis of a series of observations. 28 Bangladesh BEST Programme Type B Evaluation Category may be derived from: Previous measurement data Experience with or general knowledge of the behaviour and properties of relevant materials and instruments Manufacture’s specifications Data provided in calibration and other certificates Uncertainties assigned to reference data taken from handbooks 29 Bangladesh BEST Programme Standard Uncertainty for a Type A Evaluation “When a set of several repeated readings has been taken the mean and estimated standard deviation, s, can be calculated for the set” Fro these , the estimated standard uncertainty , u of the mean is calculated from : U= 30 Bangladesh BEST Programme Standard Uncertainty for Type B Evaluation “Where the information is more scarce (in some Type B estimates), you might be able to estimate the upper and lower limits of uncertainty. You may then have to assume the value is equally likely to fall anywhere in between ie. rectangular or uniform distribution “ The standard uncertainty for rectangular distribution is found from: U = “a “ is the semi range or half width between upper and 31 lower limits Rectangular Distribution 2a a a f(x) Area enclosed by 1 2a rectangle = 1 a Lower limit a a 2 a x Upper limit Best estimate 32 There are simple mathematical expressions to evaluate the standard deviation for this. Another such distribution we normally encounter is the triangular distribution a a Area enclosed by f x Triangle=1 1 a a a a 2 a x 33 Confidence Level Gaussian probability distribution 68% 95% 99% -ks +ks Within 1s of mean k = 1 Within 2s of mean k = 2 Within 3s of mean k = 3 34 Bangladesh BEST Programme Combining Standard Uncertainties Individual standard uncertainties calculated by Type A and Type B evaluations can be combined validly by “root sum of the squares” The result is the “combined standard uncertainty” This is represented by uc If the Type A and Type B uncertainties are a, b, c & d, then combined standard uncertainty is : uc = 35 Bangladesh BEST Programme Coverage Factor The overall uncertainty is stated at the confidence level of 95% with the coverage factor k=2 Multiplying the combined standard uncertainty uc by the coverage factor gives the result which is called “ expanded uncertainty “ usually shown by the symbol “Uc “ Uc = kuc (y) 36 Bangladesh BEST Programme Reporting Uncertainty State the result of the measurement as : Y = y ± U and give the units of y and U where the uncertainty U is given with no more than two significant digits and y is correspondingly rounded to the same number of digits The nominal value of 100 g mass is 100.02147 g The expanded uncertainty is 0.00079 g The result of measurement is expressed as 100.02147 g ± 0.00079 g and the coverage factor k = 2 37 Bangladesh BEST Programme Statement of Uncertainty in Measurement Calibration Certificate : “The reported expanded uncertainty in measurement is stated as the standard uncertainty in measurement multiplied by the coverage factor k = 2, which for a normal distribution corresponds to a coverage probability of approximately 95 %. The standard uncertainty of measurement has been determined in accordance with Guide to expression of uncertainty in measurement (GUM) JCGM 100:2008” 38 Bangladesh BEST Programme How to Reduce Uncertainty in Measurement Calibrate measuring instruments Use calibration corrections given in the certificate Make your measurements traceable to International system of units (SI) Confidence in measurement traceability from an accredited laboratory (UKAS, SWEADC, NABL etc.) Choose the best measuring instruments for smallest uncertainty Check measurements by repeating them Check all calculations when transferring data Use an uncertainty budget to identify the worst uncertainties and address them 39 Bangladesh BEST Programme Some Good Measurement Practices Follow the manufacture’s instruction for using and maintaining instruments Use experienced staff and provide training Validate software Check raw data by a third party Keep good records of your measurements and calculations 40 Bangladesh BEST Programme Preparation of Uncertainty Budgets Example: Calculation of uncertainty of a balance calibration Capacity of balance : 50 g Resolution of balance : 0.1 mg Measured max. Std. deviation : 0.0939 mg Number of measurements :10 Task : Calibration of scale value of 45 g Method : A combination of three masses are required Mass 1 2 3 Total Value 20.000088 g 19.999995 g 5.000030 g 45.000113 g U95 (mg) 0.019 0.019 0.0043 k 2 2 2 u (mg) 0.0095 0.0095 0.0045 0.0235 41 Bangladesh BEST Programme Preparation of Uncertainty Budget Observations: 1st zero reading : 0.0000 g 1st reading of standard mass : 45.0003 g 2nd reading of standard mass : 45.0003 g 2nd zero reading : 0.0001 g Calculations: Mean zero reading ( zi ): 0.00005 g Mean reading on standard mass ( ri ) : 45.00030 g 42 Bangladesh BEST Programme Preparation of Uncertainty Budget The basic measurement model is: Ci = Mi – (ri - zi ) Where C is the calculated correction Mi is the calibrated value of standard mass ri is the mean of two repeated readings zi is the mean of two no-load (zero) readings Correction : Ci = Mi – ( ri – zi ) = 45.000113 g – (45.00030 – 0.00005 ) g = -0.000137 g = - 0.1 mg (rounded to least count of balance) 43 Bangladesh BEST Programme Uncertainty Budget Source of uncer. (quant..) Units Type Prob. Dis. of evalu. ss Uncer. (U or s) Divisor Stand. Uncer. uc Cal. Uncer. umass mg B Normal 0.0235 Resolution uresolution mg B Rect. Repeatabili ty urepeatability mg A Normal 1 0.0235 0.00055 0.1/2 0.02887 0.00083 0.0939 0.02972 0.00088 Sum 0.00226 Comb. std uncer. 0.0475 mg Cov. Fac. k Expan.uncr 2 0.09544mg Bangladesh BEST Programme Comparison of magnitudes of Standard Uncertainty Components mg 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 Std. mass Scale res. Bal. repeat. Expa. Uncer. 45 Bangladesh BEST Programme Calibration and Measurement Capability (CMC) History In order to enhance the harmonization in expression of uncertainty on calibration certificates and on scope of accreditation of calibration laboratories, ILAC approved a resolution at its third General Assembly meeting in 1999. ILAC and BIPM have signed a MOU to harmonize the terminology, namely the “Best Measurement Capability (BMC)” used on the scope of accreditation of calibration laboratories with the “Calibration and Measurement Capability (CMC)” of CIPM MRA This document was effective November 2011 46 Bangladesh BEST Programme Calibration and Measurement Capability (CMC) The scope of accreditation of an accredited laboratory shall include CMC expressed in terms of: Measurand Calibration/measurement/performance method Measurement range Uncertainty of measurement 47 Bangladesh BEST Programme Calibration and Measurement Capability (CMC) In the formulation of CMC: “The smallest uncertainty of measurement that can be expected to be achieved by a laboratory during a calibration or measurement” “The uncertainty covered by the CMC shall be expressed as the expanded uncertainty having a specific coverage probability of approximately 95%” 48 Bangladesh BEST Programme Calibration and Measurement Capability (CMC) In the formulation of CMC : “ Take the notice of the performance of the “best existing device” which is available for a specific category of calibrations” Consideration should also be given to “repeatability of measurement” 49 Bangladesh BEST Programme Calibration and Measurement Capability (CMC) Example: SWEDAC Measured Quantity Method of Calibration Range Readability Calibration and Measurement Capability( ±) Calibration of weighing balance MM/MA/01 0 to 200 g 0.01 mg 0.10 mg Performance test of laboratory oven MM/TE/01 50 to 250 °C 1 °C 0.2 °C One mark pipette MM/VO/01 0 to 200 ml 0.001 ml 50 Bangladesh BEST Programme Examples Example 1 : Determination of uncertainty of the mass 1000 g Reference mass standard used : uncertainty given in the calibration certificate is 0.005 g at 95% confidence level Resolution of the balance : 0.001 g 51 Bangladesh BEST Programme Example of uncertainty calculation Determine the weight of 1kg Observation Value of test mass 1 2 3 4 5 6 7 8 9 10 1000.143 1000.144 1000.144 1000.146 1000.146 1000.146 1000.144 1000.143 1000.145 1000.145 Mean Value : 1000.1446 g Standard deviation : 0.0011 g Estimated Standard deviation of mean : 0.0011/√10=0.00035 g 52 Bangladesh BEST Programme Uncertainty Budget Source of uncer. (quant.) Units Type of Eval. Pro. Dist. Uncer. (U or s) Divisor Stand. Uncer. uc Cal. Uncer. umass mg B Normal 5 Resolution uresolution mg B Rect. Repeatabili ty urepeatability mg A Normal 2 2.5 6.25 0.5x 0.4082 0.1666 1.1 0.35 0.1225 Sum 6.539 Comb.std uncer. 2.55 mg Cov. Fac. k Exp. uncer. 2 5.1 mg 53 Bangladesh BEST Programme Presentation of Results The result is reported as: The value of the test mass = 1000.145 g Expanded uncertainty = ± 0.005 g with k=2 at 95% confidence level or The value of test mass is 1000.145 g ± 0.005 g with k=2 at 95% confidence level 54 Bangladesh BEST Programme Preparation of Uncertainty Budget Example 2: Calibration of an oven at 100 °C Reference thermometer : Calibrated set of TC, uncertainty given in the calibration certificate is 0.5 °C at 95% confidence level Digital thermometer with a resolution of 0.1 °C Test oven used with a resolution of 1 °C The standard deviation of 10 readings obtained at 100 °C is 0.6 °C 55 Bangladesh BEST Programme Uncertainty Budget Source of uncer. (quant.) Units Type of Eval. Pro. Dist. Uncer. (U or s) Divisor Stand. Uncer. uc Cal. Uncer. utc °C B Normal 0.5 Dig. Ther. uresolution °C B Rect. Dig. Ther. urepeatability °C A Dig. Ther. U cjc °C Test Oven uresolution °C 2 0.25 0.0625 0.1/2 0.0289 0.00084 Normal 0.6 0.190 0.0361 B Rect. 0.2 0.1156 0.0134 B Rect. 1/2 0.289 Sum Co. Std. u 0.0835 0.1963 0.44 °C Cov. Fac. k 2 Exp. uncer. 0.9 °C 56 Bangladesh BEST Programme Comparison of Magnitudes of Standard Uncertainty Components °c 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Dig. Th. Dig. Th. Dig.Th. Dig. Th. Tes. Ov. Exp. Un. U tc Ures Urep U cjc Ures 57 Bangladesh BEST Programme Sensitivity Coefficients Sensitivity coefficient converts all uncertainty components to the same unit as the measurand Ex. The standard uncertainty due temperature( u1 ):0.05 °C The standard uncertainty in the bridge (u2 ) : 0.001 Ω The standard uncertainty in diameter ( u3 ) : 0.01mm Combined standard uncertainty Uc = 58 Bangladesh BEST Programme Sensitivity Coefficients The general formula for the sensitivity coefficient is: Where : ci is the sensitivity coefficient for component xi y the measurand is a function of xi is the partial derivative of yi with respect to xi “The partial derivative gives the slope of the curve that results when the function yi, the measurand, is plotted for the appropriate range of xi values” 59 Bangladesh BEST Programme Preparation of Uncertainty Budget Example 3: Measurement of resistivity of a rod using the following equation Where : R l A d is the rod resistance in ohms is the length of the rod in meters is the cross sectional area of the rod in m is the diameter of the rod in m 60 Bangladesh BEST Programme Uncertainty Budget Input Data : Distance between knife degrees : 1.00003 m , unce. ± 0.01 mm, 95% CL Measured diameter of the rod : 6.001 mm No. of measurements of diameter : 10 Estimated std. dev. Of diameter : 0.25 µm Micrometer uncertainty : ±3 µm at 95% CL Measurement Data : Mean resistance : 604.44 µΩ No. of resistance measurements : 5 Estimated std. dev. : 0.3 µΩ Bridge reading uncertainty : ±1 µΩ Rod temperature : 20 ± 0.05 °C 61 Bangladesh BEST Programme Uncertainty Components and their Evaluation Rod diameter uncertainty ud Type A evaluation: The sensitivity coefficient c is obtained by differentiating the model equation for ρ with respect to d, thus 62 Bangladesh BEST Programme Uncertainty Components and their Evaluation Micrometer uncertainty u m Micrometer uncertainty is 3 µm, um = U/k = 3.0/2 µm 63 Bangladesh BEST Programme Uncertainty Components and their Evaluation Rod length uncertainty u l Uncertainty value supplied is 0.01 mm Standard uncertainty ul is calculated as : ul = U/k = 0.01/2 mm The sensitivity coefficient ci is calculates as : 64 Bangladesh BEST Programme Uncertainty Components and their Evaluation Resistance uncertainty u R Uncertainty of resistance includes several terms a.Repeatability uncertainty urdg Type A evaluation is Sensitivity coefficient crdg is given by 65 Bangladesh BEST Programme Uncertainty Components and their Evaluation b.Bridge reading uncertainty ub is given by : ( Assume rectangular distribution) Sensitivity coefficient is as in the previous case : 66 Bangladesh BEST Programme Uncertainty Component and their Evaluation C. Resistance temperature uncertainty u T The model equation has not included a term for temperature but the resistance varies with temperature as: The model equation can be written as : Differentiate this equation with respect to t then we get: 67 Bangladesh BEST Programme Uncertainty Components and their Calculations As per data supplied the possible temperature variation is 0.05 °C Uncertainty due to temperature variation is : Sensitivity coefficient is given by : cT = 68 Bangladesh BEST Programme Uncertainty Budget Source of uncer. (quant.) U Typ. Pro. nit of Dist. Ev. Uncer. (U or s) Div. Stand. Uncer. uc Sen. Coff. ci x uc = ui (y) Rod dia. ud m B Nor. 7. 91e-8 1 7.1e-8 5.7xe-6 4.5e-13 2.03e-25 Mi. Ca. um m A Nor. 3.0e-6 2 1.5e-6 5.7xe-6 8.7e-12 7.61e-23 Length ul m A Nor. 1e-5 2 0.5e-5 -1.7e-8 8e-14 6.45e-27 Res. U rdg Ω B Nor. 1.34e-7 1 1.34e-7 2.83e-5 3.8e-12 1.44e-23 Bri. Ca. Ub Ω A Rec. 1e-6 5.77e-7 -2.83e-5 1.6e-11 2.67e-22 R. Tem. ut ° C B Rec. 5e-2 2.89e-2 6.7e-11 1.9e-12 3.71e-24 Sum Std. Un. 3.62e-22 1.9e-11 Ωm k Exp. Un. 2 3.8e-11 Ωm 69 Bangladesh BEST Programme Comparisons of Magnitudes of Standard Uncertainty Components 40 35 30 pΩm 25 20 15 10 5 0 Rod. Dia.Ud Mic. Cal. Um Len. Ul Res. Urdg Brg. Ub R.tem. Ut Exp. Un U 70 Bangladesh BEST Programme Preparation of Uncertainty Budget Example 4: Temperature measurement using a TC A digital thermometer with a Type K TC was used to measure the temperature inside a chamber at 500 °C Specification of digital thermometer: Resolution :0 .1 °C Measurement accuracy : ±0.6 °C TC calibration certificate provides : Uncertainty is ± 2.0 °C at 95% confidence level Correction at 500 °C is 0.5 °C 71 Bangladesh BEST Programme Preparation of Uncertainty Budget Measure temp. (T) = Displayed temp. + Correction Calculation of uncertainty components Urept - standard uncertainty in the repeatability of the measured results Udig -standard uncertainty in the digital thermometer Utc - standard uncertainty in the thermocouple 72 Bangladesh BEST Programme Preparation of Uncertainty Budget Measurement record: Measurement 1 2 3 4 5 6 7 8 9 10 Temperature °C 500.1 500.0 501.1 499.9 4 99.9 500.0 500.1 500.2 499.9 500.0 Mean value is 500.02 Standard deviation s is 0.103 °C Standard deviation of mean SDOM is 0.03 °C (Type A) 73 Bangladesh BEST Programme Uncertainty Budget Source of uncer. (quant..) Units Type Prob. Dis. of evalu. ss Uncer. (U or s) Divisor Stand. Uncer. uc Cal. Uncer. utc °C B Normal 1 Cal. Uncer. udig °C B Rect. Repeat. Urep. °C A Normal 2 0.5 0.25 0.6 0.349 0.1223 0.103 0.326 0.0011 Sum 0.3734 Comb. std uncer. Cov. Fac. k Expan.uncr 0.61 °C 2 1.2 °C 74 Bangladesh BEST Programme 1.4 1.2 1 °C 0.8 0.6 0.4 0.2 0 U (TC) U (dig. Ther.) U (Rept.) Exp. Uncer. 75 Bangladesh BEST Programme Preparation of Uncertainty Budget Example 5: Calibration of 250 ml volumetric flask A balance with a resolution of 1 mg is used for the calibration Uncertainty of balance is ± 1 mg Weight of volumetric flask is 200.001g Three readings are obtained: First measurement : 449.822 g Measured temperature : 20.2 °C Second measurement : 450.055 g Measured temperature : 20.1 °C Third measurement : 449.892 g Measured temperature : 20.2 °C 76 Bangladesh BEST Programme Preparation of Uncertainty Budget The volume at 20 °C is given by : 1 V20 ( RL RE * w a a * 1 * 1 t 20 b Z values are given in Tables B6, B7 and B8 in ISO 4787 : 2010 for different types of glass at common air pressure Vs temperature 77 Bangladesh BEST Programme Preparation of Uncertainty Budget Measurement First Weight of water (g) 249.821 Second 250.054 Third 249.891 Mean value 249.922 Std. deviation 0.1195 SDOM (Type A ) 0.06899 g 78 Bangladesh BEST Programme Preparation of Uncertainty Budget First measurement Second measurement Third measurement Volume at 20 °C ml 250.55 250.78 250.65 Average volume is 250.66 ml at 20 °C Uncertainties : Std. uncertainty of weighing process U1 = 0.06899 g Weighing uncertainty U2 = cer. Value/2 = 0.0005 g Balance resolution U3 = half inet./1.7321 = 0.00029 g 79 Bangladesh BEST Programme Preparation of Uncertainty Budget Sensitivity coefficient: 80 Bangladesh BEST Programme Uncertainty Budget Source of uncer. (quant.) Uni t Typ. Pro. of Dist. Ev. Uncer. (U or s) Repeatabi lity U1 g B Nor. 0.1195 Calibr. U2 g A Nor. 0.001 Resolu. U3 g A Rec. 0.0005 Div. Stand. Uncer. uc 2 Sen. Coff. ci x uc = ui (y) 0.06899 1.003 0.0692 0.4789e-2 0.0005 1.003 0.0005 0.2e-6 0.00028 1.003 0.00028 7.84e-4 Sum Std. Un. 0.004868 0.0697 k Exp. Un. 2 0.14 ml 81 Bangladesh BEST Programme Uncertainty Budget 0.16 0.14 0.12 ml 0.1 0.08 0.06 0.04 0.02 0 Rep.U1 Cal. U2 Res. U3 Exp. Un. 82 Bangladesh BEST Programme Estimation of Standard Uncertainty Modeling of the measurement process The measurands are the particular quantities subject to a measurement Only one mesurand or output quantity Y that depends upon number of input quantities Xi Y= f(X1,X2,X3,……Xn) Y- measurement result X1,X2,X3,……Xn - input values f - functional relationship 83 Bangladesh BEST Programme Estimation of Standard Uncertainty An estimation of the measurand Y, the output estimate denoted by y, is obtained from the previous equation using input estimates xi for the values of input quantities Xi as y = f ( x1, x2, x3,………xn ) The uncertainty of measurement of input estimates are determined by : Type A evaluation Type B evaluation 84 Bangladesh BEST Programme Type A Evaluation __ Mean q Standard Deviation 1 m sq m q k 1 k m __ 2 ( ) q q (m 1) k 1 k 85 Bangladesh BEST Programme Type A Evaluation Standard Deviation of the Mean (SDOM) sq __ s q m Standard Uncertainty u s A __ q s q m 86 Bangladesh BEST Programme Combined Standard Uncertainty y = f(x1,x2,x3,……xn) Law of Propagation of Uncertainties 2 f f U 2 ( x1 ) U c2 ( y ) x1 x2 2 U 2 ( x2 ) ............ 2 f 2 U ( xn ) ...... xn 87 Bangladesh BEST Programme Expanded Uncertainty & Coverage Factor U = k .uc (y) U- Expanded Uncertainty Uc (y)- Combined Standard Uncertainty k- Coverage factor , obtained from the t-distribution corresponding to the level of confidence desired (95 %) 88 Bangladesh BEST Programme Reporting Results Results are reported in a “ Calibration Certificate” or “Test Report” Information to be included: Name and address of laboratory, and the location Unique identification of test report or calibration certificate Identification of each page Name and address of the customer Description of item, including capacity or range, resolution, serial number, manufacture and model number, any identification number etc. Condition of received 89 Bangladesh BEST Programme Reporting of Results Request received date Date of performance of test or calibration Identification of method used Environmental conditions Uncertainty of measurement Traceability of measurement including reference standards used eg. “Set of accuracy class E2 traceable to Primary standards maintained at Bangladesh Standards and Testing Institution (BSTI) – certificate number……….” Name (s), function(s) and signature(s) or equivalent identification of person(s) authorizing the test or calibration certificate Recommendation of re-calibration should not be included 90 Bangladesh BEST Programme Presentation of Results Example : Calibration of Volumetric Glassware METHOD OF CALIBRATION The volumetric flask was calibrated generally in accordance with the method manual Ref. No MM/VO/01 – Calibration of volumetric glassware by the gravimetric method, TEST EQUIPMENT USED Description Model Precision Balance BP 221 S Liquid in Glass Thermometer Manufacture Sartorius - - Digital Pressure Gauge Model 370 Setra Liquid : Deionised Water Capacity 220g -10 to 52C 600 to 1100mbar Resolution 0.1 mg 0.1C 0.01 mbar 91 Bangladesh BEST Programme Presentation of Results Calibration of Results Nominal capacity (ml) Volume at reference temperature of 20oC (ml) Expanded Uncertainty U (ml) 100 99.87 0.08 The measurement results can be varied U The reported expanded uncertainty in measurement is stated as the standard uncertainty in measurement multiplied by the coverage factor k = 2, which for a normal distribution corresponds to a coverage probability of approximately 95 %. The standard uncertainty of measurement has been determined in accordance with Guide to expression of uncertainty in measurement (GUM) JCGM 100:2008 Note: The user is obliged to have the flask re-calibrated at appropriate intervals Authorized by Authorized Signatory Designation Test Performed by Name Designation 92 page ( ) of ( ) Bangladesh BEST Programme Presentation of Results NOTE : Temperature effect When the temperature at which the glassware is used (t2) differs from the reference temperature (t1=200C), the corresponding volume change can be calculated via the following equation. Where : is the volume change due to temperature change is the cubical thermal expansion coefficient of the material by which the glassware is made is the temperature change Material Fused Silica (Quarts) Borosilicate Glass Soda-Lime Glass Coefficient of Cubical Thermal Expansion OC-1 *10-6 1.6 9.9 27 93 Bangladesh BEST Programme 94