1. 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

2. Uncertainty and Calibration
ISO/IEC 17025
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

3. Uncertainty and Calibration
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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. Measurement results and uncertainties calibration of a NAWI
Examples
Measurement results and uncertainties calibration of a NAWI
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14. 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

15. 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

16. 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

17. 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)
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18. Examples I = IL + IdigL+ Irep + Iecc - I0 - Idig0
Uncertainty of the measurement
Standard uncertainty of the indication
I = IL + IdigL+ Irep + Iecc - I0 - Idig0
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19. Examples Uncertainty of the measurement
Standard uncertainty of the reference mass
mref = mN + mc + mB +mD + mconv + m...
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20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

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

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

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

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

31. 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

32. Calibration of a digital manometer
Examples
Calibration of a digital manometer
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33. 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

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

35. 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

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

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

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

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

40. Calibration of digital multimeter
Definition of measurement
Direct measurement
Multimeter 6 ½ digit
DC Voltage 10 V
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41. Calibration of digital multimeter
Uncertainty type A
Reading
Meas value
[ V ] 1 2 3 4 5 6 7 8 9 10 u A
Reading
n – number of reading
r – average
xi – single reading
Average
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42. 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

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

44. Calibration of digital multimeter
Resolution of DMM URES
The least significant digit is 10 µV (10, V)
Uncertainty of resolution is ±(5 µV / √3) = ±2,9 µV (rectangular distribution)
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45. Calibration of digital multimeter
Specificatin of calibrator USPEC
Range
ppm from reading + ppm from range
200 mV
20 + 0, µV
2 V
12 + 4
20 V
7 + 0,5
200 V
20 + 4
1000 V
25 + 4

46. Calibration of digital multimeter
Specification of calibrator USPEC
10 V * 7E 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
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47. Calibration of digital multimeter
The combine uncertainty
Workshop on Providing the traceability of measurements

48. 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, ± 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

49. Expression of the uncertainty of Measurement in calibration
MICROMETR CALLIPER
Expression of the uncertainty of Measurement in calibration
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50. 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
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51. MICROMETER CALLIPER calibration method
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52. 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)
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53. MICROMETER CALLIPER calibration method
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54. 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, k = 2
Temperature conditions : (20 ± 2) °C
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55. 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
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56. 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
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57. 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
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58. MICROMETER CALLIPER basic equation - mathematical model
LK = A + E + M + L (a · Dt + Da · Dt20)
a = (a1 + a2) / 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
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59. 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
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60. 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
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61. 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

62. MICROMETER CALLIPER resulting uncertainty equation
u =  (uA2 + uE2 + uM2 +
+ (L· a· uDt)2 + (L · uDt 20 · uD)2 )
a = (a1 + a2) / mean value of the expansion coefficient
uDt uncertainty of the difference in material temperatures
uDt uncertainty of the deviation of the mean temperature from 20 °C
uD ... uncertainty of the difference in expansion coefficients
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63. MICROMETER CALLIPER type A uncertainty
uA = s/√n
s … experimental standard deviation
n … number of measurement
uA … standard uncertainty associated with the arithmetical mean
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64. 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
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65. 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

66. 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
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67. MICROMETER CALLIPER uncertainty of the reading
The division of the scale
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68. 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
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69. 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)
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70. 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
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71. 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
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72. 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
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73. 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
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74. MICROMETER CALLIPER resulting standard uncertainty
u = √(uA2 + uE2 + uM2 + uLt2 ) =
= √(0, , , ,3322)=
= 0,806 µm
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75. 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
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76. MICROMETER CALLIPER evaluation in excel
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77. System of prepackages control
Thank you for your attention
Ivan Kříž
ČMI
Workshop on Providing the traceability of measurements