Workshop on Providing the traceability of measurements Application practice of provisions GUM concerning calibration of measuring instruments Workshop on Providing the traceability of measurements in the national metrology institutes and accredited calibration and testing laboratories Kyiv, Ukraine – January 2011 Workshop on Providing the traceability of measurements

Uncertainty and Calibration ISO/IEC 17025 5.4.6.1 A calibration laboratory, or a testing laboratory performing its own calibrations, shall have and apply a procedure to estimate the uncertainty of measurement for all calibrations and types of calibration. Workshop on Providing the traceability of measurements

Uncertainty and Calibration 5.4.6.3 When estimating the uncertainty of measurement, all uncertainty components which are of importance in the given situation shall be taken into account using appropriate methods of analysis. Workshop on Providing the traceability of measurements

Uncertainty and Calibration Definitions Uncertainty is a parameter, associated with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed to the measurand. Result of a measurement must always be a measurand and its uncertainty (for a stated interval of confidence). Can be expressed as absolute or relative. Workshop on Providing the traceability of measurements

Uncertainty and Calibration Uncertainty of type A is obtained from the statistics of the measurement results. Uncertainty of type B comes from all other sources. Combined uncertainty combines type A and type B uncertainty. Expanded uncertainty transforms the combined uncertainty for the requested interval of confidence. Workshop on Providing the traceability of measurements

Uncertainty and Calibration Type A is obtained from the statistics of the measurement results. Several (often n = 3) independent observations are needed. Observable scattre in the values obtained. Standard deviation of the mean (68% confidence interval): Xi are the measured values, X is the arithmetic mean of these values. Workshop on Providing the traceability of measurements

Uncertainty and Calibration Type B is obtained from the remaining (other then statistical) sources, need not have normal distribution. Sources uncertainty of the used standard, uncertainty due to the resolution of the calibrated device, uncertainty due to the conditions (temperature) of calibration, … Workshop on Providing the traceability of measurements

Uncertainty and Calibration Inputs can be obtained from: calibration certificates of the used measuring instruments, manufacturer‘s specifications, reference data, previous measurement data and experience, … They always must be renormalized to the normal distribution. The sensitivity coefficients c must be considered. For m non-correlated input uncertainties, we obtain: Workshop on Providing the traceability of measurements

Uncertainty and Calibration Combined uncertainty Combines both types into one value of 68% confidence: Expanded uncertainty Transforms the result to the needed confidence interval by multiplying with a coverage factor: Mostly to 95 % for k = 2, sometimes to 99.7 % for k = 3. Workshop on Providing the traceability of measurements

Uncertainty and Calibration Coverage factor Usually for 95% confidence interval it can be taken from table. veff 1 2 3 4 5 6 7 8 10 20 50 ∞ k 13,97 4,53 3,31 2,87 2,65 2,52 2,43 2,37 2,28 2,13 2,05 2,00 Workshop on Providing the traceability of measurements

Uncertainty and Calibration Welch-Satterthwait equation ui (i = 1, 2, …, N) - uncertainty of determination of error from input xi (considered non correlated), vi = n – 1 - degrees of freedom for standard uncertainty of type A, for standard uncertainty determined by method type B assume vi → ∞. Calculated value of veff is rounded to the nearest smaller value given in table. Workshop on Providing the traceability of measurements

Uncertainty and Calibration Reporting the results Uncertainty should have one or two significant digits. If it begins with the number 1 or 2, then two significant digits. Measurand must be rounded to the same position as the uncertainty The coverage factor must be reported. Workshop on Providing the traceability of measurements

Measurement results and uncertainties calibration of a NAWI Examples Measurement results and uncertainties calibration of a NAWI Workshop on Providing the traceability of measurements

Workshop on Providing the traceability of measurements Examples Errors of indication – subject of the calibration Discrete values Characteristic of the weighing range In addition, or as an alternative to the discrete values a characteristic, or calibration curve may be determined for the weighing range, which allows to estimate the error of indication for any indication I within the weighing range Workshop on Providing the traceability of measurements

Workshop on Providing the traceability of measurements Examples Discrete values For each test load, the error of indication is calculated as Ej = Ij – mref,j Workshop on Providing the traceability of measurements

Examples Characteristic of the weighing range A function E = f(I) may be generated by an appropriate approximation based on the “least squares” approach: vj² = (f(Ij) – Ej)² = minimum with vj = residual f = approximation function Workshop on Providing the traceability of measurements

Examples Uncertainty of the measurement Standard uncertainty for discrete values The basic formula is E = I – mref with the variances u²(E) = u²(I) + u²(mref) Workshop on Providing the traceability of measurements

Examples I = IL + IdigL+ Irep + Iecc - I0 - Idig0 Uncertainty of the measurement Standard uncertainty of the indication I = IL + IdigL+ Irep + Iecc - I0 - Idig0 Workshop on Providing the traceability of measurements

Examples Uncertainty of the measurement Standard uncertainty of the reference mass mref = mN + mc + mB +mD + mconv + m... Workshop on Providing the traceability of measurements

Workshop on Providing the traceability of measurements Examples Uncertainty of the measurement Standard uncertainty of the error u²(E) = d0²/12 + dI²/12 + s²(I) + u²(Iecc) + u²(mc) + u²(mB) + u²(mD) + u²(mconv) Workshop on Providing the traceability of measurements

Workshop on Providing the traceability of measurements Examples Uncertainty of the measurement Expanded uncertainty at calibration U(E) = ku(E) Workshop on Providing the traceability of measurements

Workshop on Providing the traceability of measurements Examples Uncertainty of NAWI in use Standard uncertainty of a weighing result The user of an instrument should be aware of the fact that in normal usage of an instrument that has been calibrated, the situation is different from that at calibration in some if not all of these aspects: the indications obtained for weighed bodies are not the ones at calibration the weighing process may be different from the procedure at calibration: a. certainly only one reading for each load, not several readings to obtain a mean value b. reading to the scale interval d of the instrument, not with higher resolution c. loading up and down, not only upwards – or vice versa d. load kept on load receptor for a longer time, not unloading after each loading step – or vice versa e. eccentric application of the load, f. use of tare balancing device Workshop on Providing the traceability of measurements

Workshop on Providing the traceability of measurements Examples Uncertainty of NAWI in use the environment (temperature, barometric pressure etc.) may be different on instruments which are not readjusted regularly e.g. by use of a built-in device, the adjustment may have changed, due to ageing or to wear and tear. Unlike the items 1 to 3, this effect is usually depending on the time that has elapsed since the calibration, it should therefore be considered in relation to a certain period of time, e.g. for one year or the normal interval between calibrations. It is therefore difficult to conclude from the results of a calibration as given in the calibration certificate: Workshop on Providing the traceability of measurements

Workshop on Providing the traceability of measurements Examples Uncertainty of NAWI in use Uncertainty of the error of a reading Uncertainty from environmental influences Uncertainty from the operation of the instrument Workshop on Providing the traceability of measurements

Workshop on Providing the traceability of measurements Examples Uncertainty of NAWI in use Standard uncertainty of a weighing result Workshop on Providing the traceability of measurements

Workshop on Providing the traceability of measurements Examples Uncertainty of NAWI in use Repeatability (standard deviation of single reading) Workshop on Providing the traceability of measurements

Workshop on Providing the traceability of measurements Examples Uncertainty of NAWI in use Digital rounding Workshop on Providing the traceability of measurements

Workshop on Providing the traceability of measurements Examples Uncertainty of NAWI in use Temperature effect Workshop on Providing the traceability of measurements

Workshop on Providing the traceability of measurements Examples Uncertainty of NAWI in use Temperature effect Workshop on Providing the traceability of measurements

Workshop on Providing the traceability of measurements Examples Uncertainty of NAWI in use Temperature effect (coeffitient C ) Workshop on Providing the traceability of measurements

Workshop on Providing the traceability of measurements Examples Uncertainty of NAWI in use Expanded uncertainty of a weighing result Errors accounted for by correction Errors included in uncertainty Workshop on Providing the traceability of measurements

Calibration of a digital manometer Examples Calibration of a digital manometer Workshop on Providing the traceability of measurements

Workshop on Providing the traceability of measurements Examples Calibrated digital pressure meter Measuring range: (0 to 35) kPa Resolution: 0,01 kPa Thermal error: 0,02 % FS/°C Reference temperature: 20 °C Temperature of enviroment: (20 ± 1) °C Used standard manometer Measuring range: (1,5 to 100) kPa Accuracy: 0,01 % MV Difference of reference levels of calibrated and measuring standart pressure meter is insignificant Workshop on Providing the traceability of measurements

Workshop on Providing the traceability of measurements Examples Measured values Workshop on Providing the traceability of measurements

Workshop on Providing the traceability of measurements Examples Components of uncertainty Repeatability Uncertainty of measuring standart pressure meter Resolution of calibrated pressure meter Thermal error of calibration pressure meter Workshop on Providing the traceability of measurements

Evalution of uncertainty of measurement Standard uncertainty of type A: Uncertainty of the standard: Uncertainty of resolution of calibrated device:

Evaluting of uncertainty of measurement Uncertainty of thermal error of the calibrated device: Standard uncertainty of type B: Combined standard uncertainty:

Expanded uncertainty of measurement Effective degrees of freedom: Coverage factor: Expanded uncertainty of measurement:

Calibration of digital multimeter Examples Calibration of digital multimeter Workshop on Providing the traceability of measurements

Calibration of digital multimeter Definition of measurement Direct measurement Multimeter 6 ½ digit DC Voltage 10 V Workshop on Providing the traceability of measurements

Calibration of digital multimeter Uncertainty type A Reading Meas value [ V ] 1 10.00005 2 10.00013 3 10.00008 4 10.00011 5 10.00006 6 10.00002 7 10.00007 8 10.00009 9 10 10.000077 u A 0.000011 Reading n – number of reading r – average xi – single reading Average Workshop on Providing the traceability of measurements

Calibration of digital multimeter Uncertainty type B Error of indication EX of calibrating multimeter is: EX = UDMM – UE + URES – USPEC where: UE is the output voltage of calibrator (standard) UDMM is average value indicated by calibrated DMM URESL is correction due to finite resolution of DMM USPEC is corection of output voltage of calibrator from his specification (included temp. depence, load depence, drift since last cal, linearity) Workshop on Providing the traceability of measurements

Calibration of digital multimeter Output voltage of calibrator UE Output voltage of calibrator is 10,000 000 V Uncertainty from Calibration certificate of the calibrator for 10 V is ±72 µV for k = 2 and normal distribution

Calibration of digital multimeter Resolution of DMM URES The least significant digit is 10 µV (10,000 00 V) Uncertainty of resolution is ±(5 µV / √3) = ±2,9 µV (rectangular distribution) Workshop on Providing the traceability of measurements

Calibration of digital multimeter Specificatin of calibrator USPEC Range ppm from reading + ppm from range 200 mV 20 + 0,0 + 15 µV 2 V 12 + 4 20 V 7 + 0,5 200 V 20 + 4 1000 V 25 + 4

Calibration of digital multimeter Specification of calibrator USPEC 10 V * 7E-6 + 20 V * 0,5E-6 = 70E-6 V + 10E-6 V = 80E-6 V = 80 µV If not specified, we assume the rectangular distribution: 80 µV / √3 = 46,2 µV Workshop on Providing the traceability of measurements

Calibration of digital multimeter The combine uncertainty Workshop on Providing the traceability of measurements

Calibration of digital multimeter Expanded measurement uncertainty U = k * u(Ex) = 2 * 59,7 µV = 119,4 µV We assume normal distribution, than k = 2 for coverage probability of approximately 95%. The error of multimeter is: (0,000 08 ± 0,000 12) V The reported expanded uncertainty of measurement is stated as the standard uncertainty of measurement multiplied by the coverage factor k = 2, which for a normal distribution corresponds to a coverage probability of approximately 95%. Workshop on Providing the traceability of measurements

Expression of the uncertainty of Measurement in calibration MICROMETR CALLIPER Expression of the uncertainty of Measurement in calibration Workshop on Providing the traceability of measurements

MICROMETER CALLIPER calibration method The error of the micrometer calliper is determined by set of gauge blocks. There are recommended calibration points in the technical standard ISO 3611. The values are as follows: (2,5; 5,1; 7,7; 10,3; 12,9; 15; 20,2; 22,8; 25) mm Workshop on Providing the traceability of measurements

MICROMETER CALLIPER calibration method Workshop on Providing the traceability of measurements

MICROMETER CALLIPER calibration method Preparation (marking, cleaning, visual inspection, rework of slight damage) Preliminary tests and activities (smooth run, function of the locking device, surface, acclimatisation) Calibration (checking of the lower limit of the measuring range, determination of the errors) Workshop on Providing the traceability of measurements

MICROMETER CALLIPER calibration method Workshop on Providing the traceability of measurements

MICROMETER CALLIPER input data Range : (025) mm Reading : analog Division : 0.01 mm Standard : Set of gauge blocks Expanded uncertainty of the standard U = 0,62 k = 2 Temperature conditions : (20 ± 2) °C Workshop on Providing the traceability of measurements

MICROMETER CALLIPER input data Measured values [mm]: 25,000; 25,001; 25,000; 25,002; 25,001 25,001; 25,002; 25,000; 25,000; 25,003 The result of the measurement is (corrected) arithmetical mean of these values Workshop on Providing the traceability of measurements

MICROMETER CALLIPER contributors to resulting uncertainty To resulting uncertainty consider following influences: dispersion of the measured values uncertainty of the standard resolution of the calliper thermal expansion Workshop on Providing the traceability of measurements

MICROMETER CALLIPER basic equation - mathematical model LK = A + E + M + L (a · Dt + Da · Dt20) LK ... corrected result of the maeasurement A ... arithmetic mean of the measured values E ... standard correction M ... reading correction L ... nominal length Workshop on Providing the traceability of measurements

MICROMETER CALLIPER basic equation - mathematical model LK = A + E + M + L (a · Dt + Da · Dt20) a = (a1 + a2) / 2 ... mean value of the expansion coefficient Dt = (t1 - t2) ... difference in material temperatures Da = (a1 - a2) ... difference in expansion coefficients Dt20 = 20 – (t1 + t2) / 2 deviation of the mean temperature from 20 °C Workshop on Providing the traceability of measurements

MICROMETER CALLIPER basic equation - mathematical model Notes 1: For making out the mathematical model not only mathematical knowledge is sufficient Experience with particular measurement task is necessary Not enough experience leads to insufficient mathematical model Workshop on Providing the traceability of measurements

MICROMETER CALLIPER basic equation - mathematical model Notes 2: Equation for resulting uncertainty is derived from the mathematical model. Only mathematical knowledge is needed for this derivation Workshop on Providing the traceability of measurements

MICROMETER CALLIPER resulting uncertainty equation u =  (uA2 + uE2 + uM2 + + (L · a· uDt)2 + (L · uDt 20 · uD)2 ) uA ... type A uncertainty uE ... uncertainty of the standard uM... uncertainty of the reading L ... nominal length Workshop on Providing the traceability of measurements

MICROMETER CALLIPER resulting uncertainty equation u =  (uA2 + uE2 + uM2 + + (L· a· uDt)2 + (L · uDt 20 · uD)2 ) a = (a1 + a2) / 2 ... mean value of the expansion coefficient uDt ... uncertainty of the difference in material temperatures uDt 20 ... uncertainty of the deviation of the mean temperature from 20 °C uD ... uncertainty of the difference in expansion coefficients Workshop on Providing the traceability of measurements

MICROMETER CALLIPER type A uncertainty uA = s/√n s … experimental standard deviation n … number of measurement uA … standard uncertainty associated with the arithmetical mean Workshop on Providing the traceability of measurements

MICROMETER CALLIPER type A uncertainty The measured values are as follows: (25,000; 25,001;25,000;25,002;25,001;25,001; 25,002; 25,000; 25,000; 25,003): Using scientific calculator or Excel sheet, the value of the experimental standard deviation is: s = 0,00105 mm = 1,05 µm Workshop on Providing the traceability of measurements

MICROMETER CALLIPER type A uncertainty uA = s/n = 1,05/10 = 0,332 µm The value of the type A uncertainty is 0,332 µm By increasing of the number of measurement n, the type A uncertainty associated with the arithmetical mean can be decreased Workshop on Providing the traceability of measurements

MICROMETER CALLIPER uncertainty of the standard Reported expanded uncertainty for the length 25 mm and coverage factor from the certificate: U = 0,62 µm coverage factor k = 2 The value of the standard uncertainty of the Standard for the length 25 mm is: uE = U/k = 0,62/2 = 0,31 µm Workshop on Providing the traceability of measurements

MICROMETER CALLIPER uncertainty of the reading The division of the scale Workshop on Providing the traceability of measurements

MICROMETER CALLIPER uncertainty of the reading Experienced operator is able to estimate result of the measurement with resolution 0,001 mm Assume rectangular distribution with bounds ± 1µm uM = 1/3 = 0,577 µm The value of the standard uncertainty of the reading is: 0,577 µm Workshop on Providing the traceability of measurements

MICROMETER CALLIPER thermal expansion there are three facts: There is the difference in material temperatures of the calliper and gauge block There is the difference in expansion coefficients of the calliper and gauge block There is deviation of the mean temperature from the reference temperature (20 °C) Workshop on Providing the traceability of measurements

MICROMETER CALLIPER thermal expansion There is the difference in material temperatures of the calliper and gauge block... Estimate the maximum difference in material temperatures of the calliper and gauge block to ± 2 °C Standard uncertainty of the difference: uDt = 2/3= 1,15 °C Workshop on Providing the traceability of measurements

MICROMETER CALLIPER thermal expansion There is the difference in expansion coefficients of the calliper and gauge block Estimate the maximum difference in expansion coefficients of the calliper and gauge block to ± 2 µm/m°C Standard uncertainty of the difference: uDa = 2/3 = 1,15 µm/m°C Workshop on Providing the traceability of measurements

MICROMETER CALLIPER thermal expansion There is deviation of the mean temperature from the reference temperature (20 °C) The temperature in the laboratory is in kept in limits (20 ± 2) °C Standard uncertainty of the deviation from reference temperature: uDt 20 = 2/3 = 1,15 °C Workshop on Providing the traceability of measurements

MICROMETER CALLIPER thermal expansion Contribution to uncertainty by thermal expansion: uLt =  ((L · a · uDt )2 + (L · uDt 20 · uD)2) = = ((0,025 · 11,5 · 1,15)2 + (0,025· 1,15 · 1,15)2) = 0,332 µm Workshop on Providing the traceability of measurements

MICROMETER CALLIPER resulting standard uncertainty u = √(uA2 + uE2 + uM2 + uLt2 ) = = √(0,312 + 0,3322 + 0,5772 +0,3322)= = 0,806 µm Workshop on Providing the traceability of measurements

MICROMETER CALLIPER resulting expanded uncertainty U = k · u = 2 · 0,806 = 1,61 µm   1,6 µm  2 µm k … coverage factor k = 2 ( 5.1 and 5.2 EA 4/02) u … resulting standard uncertainty Workshop on Providing the traceability of measurements

MICROMETER CALLIPER evaluation in excel Workshop on Providing the traceability of measurements

System of prepackages control Thank you for your attention Ivan Kříž ČMI ikriz@cmi.cz www.cmi.cz Workshop on Providing the traceability of measurements