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Uncertainty of Measurement Nihal Gunasekara Sri Lanka

Bangladesh BEST Programme. Uncertainty of Measurement Nihal Gunasekara Sri Lanka. 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

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Uncertainty of Measurement Nihal Gunasekara Sri Lanka

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  1. Bangladesh BEST Programme Uncertainty of MeasurementNihalGunasekaraSri Lanka

  2. 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

  3. Bangladesh BEST Programme • What do you need for a measurement ? • Instrument • Rulers • Stopwatches • Weighing scales • Thermometers

  4. 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

  5. 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

  6. 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 !!!!!

  7. 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

  8. Bangladesh BEST Programme Measurement Uncertainty X U U A range containing the true value

  9. 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

  10. 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

  11. 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 !!!!

  12. 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

  13. 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 !!!!

  14. 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

  15. 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

  16. 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

  17. 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”

  18. 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

  19. Bangladesh BEST Programme Mathematical Equation for Standard Deviation

  20. 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.

  21. Bangladesh BEST Programme Distribution – Shape of Errors The spread of set of values can take different forms Mean or average reading Probability of occupation Value of reading Normal or Gaussian distribution

  22. 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 Full width Range Value of reading Value of reading

  23. Bangladesh BEST Programme Triangular Distribution Probability of occurrence Value of reading

  24. Bangladesh BEST Programme • What is not a Measurement Uncertainty ? • Mistakes made by a operator • Tolerances of a product • Specifications of instruments

  25. 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

  26. How to Calculate Uncertainty of Measurement Specify Measurand STEP 1 STEP 2 Identify Uncertainty sources Simplify by grouping the sources covered by available data STEP 3 Quantify grouped and remaining components Convert components to standard uncertainties

  27. How to calculate Uncertainty of measurement Calculate the combined standard Uncertainty STEP 4 Review and if required re-evaluate large components Calculate the Expanded Uncertainty

  28. 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.

  29. 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

  30. 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 =

  31. 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 lower limits

  32. Rectangular Distribution Lower limit 2a a a f(x) Area enclosed by rectangle = 1 x Upper limit Best estimate

  33. There are simple mathematical expressions to evaluate the standard deviation for this. Another such distribution we normally encounter is the triangular distribution Area enclosed by Triangle=1

  34. Confidence Level Gaussian probability distribution -ks +ks 68% Within 1sof mean k = 1 95% Within 2s of mean k = 2 99% Within 3sof mean k = 3

  35. 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=

  36. 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)

  37. 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

  38. 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”

  39. 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

  40. 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

  41. 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 Value U95(mg) k u (mg) 1 20.000088 g 0.019 2 0.0095 2 19.999995 g 0.019 2 0.0095 3 5.000030 g 0.0043 2 0.0045 Total 45.000113 g 0.0235

  42. 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

  43. 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)

  44. Bangladesh BEST Programme Uncertainty Budget ss

  45. Bangladesh BEST Programme Comparison of magnitudes of Standard Uncertainty Components mg

  46. 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

  47. 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

  48. 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%”

  49. 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”

  50. Bangladesh BEST Programme Calibration and Measurement Capability (CMC) Example: SWEDAC

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