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1. MEASUREMENT UNCERTAINTY IN TESTING

2. Sr. No Topics 1 Measurment of uncertainty 2 Z-score 3 Measurment of uncertainty questioner

3. We measure things to get data that helps us make good decisions. Measurements are important in many areas Trade – To make sure buying and selling is fair. Health Care – To know if a person is healthy or sick. Law & Order – To help find the truth in a case. Industry – To check if products are made the right way. Testing Labs – To see if products follow the specifications. Calibration Labs – To check if measuring equipment's are correct. Why Do We Measure?

4. National and international measurement system

5. What is uncertainty ? In simple words, uncertainty is the doubt about the exactness of the reading / measurement. Measurement uncertainty provides an idea about the quality of the particular measurement system ​. It provides an insight about the parameters/ factors contributing to erro r. ​ It provides opportunities for improvement in the existing measuring system ​.

6. Important terms Accuracy - Closeness of agreement between a measured quantity value and a true quantity value of a measurand. Precision - The  precision  of a measurement system  refers to  how close the agreement is between repeated measurements. Measurand - Quantity intended to be measured. Error -Measured quantity value minus a reference quantity value. Standard Deviation - The positive square root of the variance. Variance - It tells you how much the values in a group differ from the average (mean) . Degree of freedom - The number of terms in a sum minus the number of constraints on the terms of the sum. Resolution / Least count – Minimum measurable value through a particular instrument or tool.

7. Measurement of uncertainty Measurment of Uncertainty It shows how close or far a measurement could be from the true value Type A Repeating the measurement and seeing how much the results change Type B Data from things like tools, manuals, or past data — not from repeating the test Uncertainty Type A Type B

8. 2 3 4 1 Type A uncertainty deals with random errors , the kind that: Error Experimental data the actual readings taken with measuring instrument Evaluated using statistical methods — basically, math based on data Careless in making reading Wrong techniques Information Evaluation Example What is Type A Uncertainty? Measurement uncertainty-Type A

9. • Random error presumably arises from unpredictable or stochastic temporal and spatial variations of influence quantities. • The effects of such variations, hereafter termed random effects, give rise to variations in repeated observations of the measurand. • Although it is not possible to compensate for the random error of a measurement result, it can usually be reduced by increasing the number of observations ; its expectation or expected value is zero. You only include random errors that cause variation between repeated measurements. Random Error

10. Example based on Type A (repeated measurements) Measuring the same cable sample’s resistance multiple times using the same CR meter. Repeating tensile tests on multiple pieces of the same wire batch under identical conditions. Measuring the insulation wall thickness at several points on the same wire, then calculating the standard deviation. 1 2 3 Calculation for Type A Take Multiple measurements Calculate the mean (average) Find the standard deviation 1 2 3 4 Use it to calculate standard uncertainty Type A

11. CALCULATING MEASUREMENT UNCERTAINTY TYPE A UNCERTAINTY – Take a set of readings / repeated measurements taken in a relatively short time span Calculate the mean of the readings, (x̅) Degree of freedom, V = (No. of readings – 1) Standard Deviation, σ = x̅)² Standard Uncertainty, Ur = σ / √n where, n= No. of readings, σ = Standard deviation - Standard Uncertainty, Ur, % = (Ur*100) / (x̅)  

12. Identify uncertainty range Select probability distribution Apply divisor Calculate standard uncertainty How to Calculate Type B Uncertainty All known sources of error (not just repeatability) are included in the uncertainty budget. Type B important in accredited labs Manufacturer’s accuracy, calibration reports, reference material values, environmental effects, operator influence, and resolution limits. Type B uncertainty is estimated using information other than repeated measurements , such as specifications, calibration certificates, or expert judgment. Common Sources of Type B Uncertainty What is Type B Uncertainty? Measurement uncertainty-Type B

13. • Systematic error, like random error, cannot be eliminated but it too can often be reduced. • If a systematic error arises from a recognized effect of an influence quantity on a measurement result, hereafter termed a systematic effect, the effect can be quantified. • If it is significant in size relative to the required accuracy of the measurement, a correction or correction factor can be applied to compensate for the effect.  • It is assumed that, after correction, the expectation or expected value of the error arising from a systematic effect is zero. • It is assumed that the result of a measurement has been corrected for all recognized significant systematic effects and that every effort has been made to identify such effects. Systematic Error

14. You have only one measurement , so you can’t calculate variability from repeats. you rely on published data or literature for uncertainty values You use reference materials with stated values You're dealing with systematic effects (e.g., instrument bias) You're preparing an uncertainty budget (e.g., for NABL or ISO/IEC 17025 compliance) Environmental conditions fluctuate unpredictably WHEN should I use Type B

15. You choose Type B when the uncertainty cannot be determined by repeating measurements . Instead, you rely on: Scenario: A thermometer is calibrated by an accredited lab, and the certificate says: Uncertainty = ±0.3°C. Why Type B? You didn’t repeat any measurements — you're relying on the lab's data. This is non-statistical info from a trusted source. Instrument specifications Calibration certificates Manufacturer's manuals Historical data Handbooks or reference standards Professional judgment/expert opinion

16. Types of Probability Distribution 1) Normal Distribution (Bell Curve / Gaussian Curve) 2) Rectangular Distribution (Uniform Distribution) 3) Triangular Distribution (Symmetrical Trapezoidal Distribution) 4) U-Shaped Distribution

17. Normal Distribution In some situation, the quoted uncertainty in an input or output quantity is stated along with level of confidence. In such cases, one has to find the value of coverage factor so that the quoted uncertainty may be divided by this coverage factor to obtain the value of standard uncertainty In the absence of any specific knowledge about the distribution, one may assume it to be normal. Values of the coverage factor for various level of confidence for a normal distribution are as follows: Confidence Level 68.27 % 90 % 95 % 95.45 % 99 % 99.73 % Coverage Factor 1.000 1.645 1.960 2.000 2.576 3.000

18. Rectangular Distribution In cases, where it is possible to estimate only the upper and lower limits of an input quantity and there is no specific knowledge about the concentration of the values within the interval, one can only assume that it is equally probable.  To calculate standard uncertainty in case of rectangular distribution, the absolute input value should be divided by the PDF (Probability Distribution Factor) √3. Rectangular Distribution is also called Uniform distribution. Examples – Accuracy, Resolution, etc ….

19. Define the instruments affecting the systematic error of the measurement. Define the parameters of those instruments to be included in the calculation. (Note : Generally, the 3 parameters used for calculation are Std. Uncertainty, Accuracy & Resolution/Least  Count) 1) Uncertainty from Standard Uncertainty in Calibration Report, U1 U1 = Standard Uncertainty / Coverage factor Standard Uncertainty, U1 = Std. Uncertainty / K where, K = Coverage factor as per the Cal. Report Standard Uncertainty, U1, % = (U1*100) / Full Scale Deflection (full Range) Degree of freedom, V1 = ∞ (Infinite) TYPE B UNCERTAINTY 

20. 2) Uncertainty from Accuracy of the instrument, U2 U2 = Accuracy / Probability Distribution Factor Standard Uncertainty, U2 = Accuracy / √3 Standard Uncertainty, U2 % = (Ur*100) / FSD (Range) Degree of freedom, V2 = ∞ (Infinite) 3) Uncertainty from Resolution of the instrument, U3 U3 = Resolution / 2 * Probability Distribution Factor Standard Uncertainty, U3 = Resolution / 2√3 Standard Uncertainty, U3 % = (Ur*100) / FSD (Range) Degree of freedom, V3 = ∞ (Infinite)

21. Combined Standard Uncertainty DEFINITION : The standard measurement uncertainty that is obtained using the individual standard measurement uncertainties associated with the input quantities in a measurement model is known as Combined Standard Uncertainty The combine standard uncertainty is calculated by using the RSS method Combined Uncertainty, Uc = …. Example of Combined Standard Uncertainty Combined Uncertainty, Uc = = = = 1.079 %  

22. Effective Degree of Freedom The effective degree of freedom ( V eff ) is estimated from the Welch-Satterthwaite formula. It is calculated for determining the appropriate value of coverage factor (k) which is obtained from the student ‘t’ table by comparing the value of effective degree of freedom (if >30 then consider ∞)

23. Student ‘t’ Table

24. Expanded Uncertainty DEFINITION - Product of a combined standard measurement uncertainty and a factor larger than the number one The combined uncertainty is a good measure of the uncertainty of a measurement result. However, in certain regulatory situations, the requirements need to stipulate that the quantity be within a specific interval. It has been termed as expanded uncertainty, and it is denoted by Ue . The expanded uncertainty is obtained by multiplying Combined Standard Uncertainty (Uc) and the Coverage factor (K) Expanded Uncertainty, Ue = Uc X K

25. Uncertainty Budget Uncertainty budget is a tabular format which represents that while quoting a measurement result & its uncertainty, all the uncertainty components are listed and documented fully as to how they are evaluated. It gives an overview of the whole measurement system at one place

26. Reporting of result After going through the whole uncertainty calculation, it is very important to summarise its output in a manner in which it is very easy to implement it without going through the calculations The reporting of result should include all the important output parameters Below mentioned is an example of the same, The measured result is measurement of uncertainty. The reported uncertainty is expanded using a coverage factor k for a level of confidence assuming a normal distribution.

27. Measured from data Repeated measurements Statistical calculation Type A Observed value Mean (𝑥ˉxˉ ) Standard Deviation (s) Standard Uncertainty (u) Expanded Uncertainty (U) Calculation Estimated from experience or documents Other info (specs, calibration) Type B List of uncertainty sources Choose distribution type Find divisor Calculate standard uncertainty Calculation RSS ( Root Sum Square) Method Combine Uncertainties Report the result with uncertainty Report Measurement of uncertainty

28. Z- Score Skills Performance Knowledge NABL ILC Ability PT Evaluation

29. What is Z-Score       04 03 02 01 ILC and PT  Calculation of the Z score Key takeaway Agenda

30. standard score, is a statistical measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations. Definition of Z-Score The value how much exact to the mean (above, below or equal to the mean) Z-score provides a quantitative measure of how close a lab's result is to the assigned or consensus value What It Tells You it's a powerful performance indicator that helps a laboratory continuously improve accuracy, reliability, and compliance, especially in NABL or ISO/IEC 17025-accredited environments. How Z-Score Helps a Laboratory Improve It Helps the engineer validate that the test setup, method, and instrument are producing correct results. How Z-Score Helps a Testing Engineer Z-score

31. Inter-Laboratory Comparison (ILC) Definition : Comparison of test/calibration results among multiple labs. Use : When PT is not available . Organized by : Labs or third party. Standard : ISO/IEC 17025:2017, Clause 7.7.2(b) Tools : Z-score, En-score Purpose : Check consistency, detect bias, validate methods. Proficiency Testing (PT) Definition : Formal evaluation of lab performance by approved PT provider. Mandatory : If available – at least 1 per year per major discipline . Organized by : NABL-approved PT providers (e.g., NPTP, ERA). Standard : ISO/IEC 17043, Clause 7.7.2(a) ILC & PT – Key Overview for NABL Compliance

32. ζ (Zeta) Score – For Checking Your Own Uncertainty What is it? Checks if your result matches the assigned value within your claimed uncertainty . When to use: When labs report their own uncertainties Useful to find if: Your result is wrong Or your uncertainty is underestimated

33. What is it? Checks if your result is okay within your expanded uncertainty (U) — used often in calibration. Where: U: Expanded uncertainty En Score – For Calibration Labs (Expanded Uncertainty)

34. 01 03 02 05 04 Sample collection from the lot ILC Participation - External laboratories  Testing the sample Comparision of the results Z-Score Interlaboratory comparison Process Cycle

35. Proficiency Test Process Cycle   PT Schedule Comparison of Assigned Value with Independent  reference value Distribution of PT Item Performance Evaluation of Participants    Homogeneity    and Stability Assessment Metrological Traceability Receipt of Test Results from Participant Labs

36. Homogeneity Assessment: The process of checking whether a material or sample is uniform in composition and characteristics throughout . In simple terms: It means making sure that all parts of a sample or batch are the same , so test results are reliable and not affected by variation in the sample itself. Example (for lab/testing context): Before sending samples for proficiency testing or inter-laboratory comparison, labs perform a homogeneity assessment to ensure each sample given to different labs is identical in quality and properties . Ten (10) samples were randomly collected for Homogeneity check and transferred to testing labs. The Results received from laboratory were analyzed for homogeneity as per ISO 13528-2015 . PT Item was found sufficiently Homogeneous. Thereafter PT samples will be dispatched to participant labs.

37. Inter Laboratory Comparison Proficiency Test Certified Reference Material (CRM) Production Round Robin Testing (RR Testing) Accreditation Audits (NABL/ISO 17025) Z-SCORE

38. To describe the exact location of the score within a distribution Why do we use z-scores? To find the exact position of a score in a group of scores (distribution) To describe and compare scores more clearly To calculate a z-score, we use: The mean (average) The standard deviation (spread) In short, a z-score helps us locate and understand any score in a set of data. Z-score changes a raw score (X) into a positive or negative number: ​ A positive z-score means the score is a bove the average (mean)​ A negative z-score means the score is below the average​ The number of the z-score tells us how far the score is from the mean, measured in standard deviations. Z ≤ 2:  Acceptable 2 < Z < 3: Warning Z ≥ 3:  Unsatisfactory

39. The relationship between Z-scores and locations in a distribution Z-Score Formula: ​ Where: Z = z-score X = raw score (the observed value) μ = mean of the dataset σ\sigma = standard deviation of the dataset

40. Two distributions, to describe the position of X=76. Z= +2.00 The score is located above the mean by exactly 2 standard deviation. Z= +0.50 The score is located above the mean by 1/2 standard deviation. Example:

41. Learning Check = 50 10 40 30 For a population with µ = 50 and σ\sigma  = 10, find the z-score foreach of the following scores: X = 55 X = 40 X = 30 Z Score will be Z= +0.50 Z= - 1.00 Z = - 2.00

42. Calculation of the Z score Obtain the Test Result (X) Get the Assigned/Reference Value (μ) Get the Standard Deviation (σ) Use the Z-Score Formula Calculate the Z-Score Interpret the Z-Score Get your lab’s measured value from the test or analysis. This is the mean or consensus value provided by the PT/ILC provider. Use the standard deviation provided by the PT provider or calculated from all participating labs. Subtract the mean from your result, then divide by the standard deviation. Your Z-score result performance Interpretation

43. Z-Score: A Lab’s Mirror How accurate your test result is compared to a reference value How consistent your lab's performance is over time How to detect and correct errors in testing methods or equipment How your lab compares to others in proficiency testing (PT/ILC) How to improve quality through data-driven decisions How to build confidence in your testing process

44. A testing engineer has performed the test for the flammability test at the testing laboratory  R R Kabel. He wants to calculate the measurement of uncertainty for it . The below mentioned values are the measured value during testing of the sample. Sr. No Measured value 1 0.812 2 0.823 3 0.805 4 0.851 5 0.834

45. TYPE A  Sr. No. No. of Readings (xi) Unit Mean (x̅) xi - x̅ (xi - x̅)2 0.812 0.823 0.805 0.851 0.834 min min min min min 1 2 3 4 5 0.825 -0.013 -0.002 -0.02 0.026 0.009 0.00016900 0.00000400 0.00040000 0.00067600 0.00008100

46. Step 1 Mean  (x̅) = Avg of all the measured value Step 2  xi - x̅ Measured value minus mean Step 3 ∑(xi - x̅)2 square of the value ∑(xi - x̅)2 Summation of values No. of Readings (n) = 5 Total measured value Step 4  Degree of freedom = (n-1) 4 Measured value minus 1  Step 5 ∑(xi - x̅)2/Degree of freedom = (n-1) ∑(xi - x̅)2  divided by  degree of freedom

47. Step 6 Standard Deviation ( σ) √∑(xi - x̅)2/Degree of freedom = (n-1) √n = Standard Uncertainty (Ur) = σ/√ n Standard Uncertainty  (Ur) % = ( Ur*100)/(x̅) Square root of total number of readings σ/√ n Standard Uncertainty divided by mean 2.236067977 0.003741657 0.047136022 Ans= 0.00837

48. TYPE B Types of distribution 1) Normal (Bell) Uncertainty of instrument (from Calibration report) 2) Rectangular (Uniform) Accuracy, resolution Total range of the instrument CR Meter=20 m Ohm, Temp. sensor= 200 °C and Measuring tape= 1 m

49. Step 7 SR. NO Source Value Unit  Distribution  "K" Factor According to Distribution standard Uncertainty(u1) standard Uncertainty (%) DOF Uncertainty from the certificate divided by k factor standard Uncertainty(u1) divided by FSD 0.28 0.05 0.05 0.59 0.5 0.5 0.000125 0.0005 0.0005 % % micro ohm °C % °C m m m Normal Distribution Rectangular Distribution Rectangular Distribution Normal Distribution Rectangular Distribution Rectangular Distribution Normal Distribution Rectangular Distribution Rectangular Distribution 2 1.732 1.732 2 1.732 1.732 2 1.732 1.732 0.14 0.02886836 0.02886836 0.295 0.288683603 0.288683603 0.0000625 0.000288684 0.000288684 0.007 0.0000000000 0.0000000003 0.268181818 0.002624396 0.262439639 0.00625 0.02886836 0.02886836 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ Uncertainty as per calibration certificate Accuracy Resolution Uncertainty as per calibration certificate Accuracy Resolution Uncertainty as per calibration certificate Accuracy Resolution CR Meter 20 20000000000 Temp sensor with indicator 110 Measuring tap (scale) 1 1 FSD 2 FSD 3 FSD

50. Step 8 Combined Uncertainty  Uc = Uc =  (𝑈𝑟^2+𝑈1^2+𝑈2^2+𝑈3²) …. 0.1447795706 Squar root of standard uncertainty Uc = Uc =  √(𝑈𝑟^2+𝑈1^2+𝑈2^2+𝑈3²) …. 0.380499107 Squar root of combined uncertainty

51. Step 9 Effective degree of freedom Veff = Uc^4 0.020961124 Combine uncertainty raised to 4 Ur^4 0.000004936 Standard Uncertainty raised to 4 Ur^4/V 0.000001234 Standard Uncertainty raised to 4 divided by degree of freedom Veff = 16984.89501 Combine uncertainty raised to 4 divided by  Standard Uncertainty raised to 4 divided by degree of freedom

52. Step 11 Step 10 So as per student table  K = 2 From the student table at 95.45% confidence level Expanded Uncertainty ( Ue )  in percentage 0.060408038 =D17*E73% Mean value plus expanded uncertainty  in percentage Expanded Uncertainty  ( Ue ) = Ue  = Uc * K  0.760998214 =F59*D71 Expanded uncertainty multiply with k factor taken from student table

53. Assessment Quiz: Measurement Uncertainty 1 9 7​ 2​ 4​ 8​ 3 11 9​ 10 5 ​ 6​ 12

54. Q1. Type A evaluation of uncertainty is based on: a) Calibration certificate values b) Statistical analysis of repeated measurements c) Manufacturer’s specification d) Environmental conditions Answer:  b    Question 1

55. Q2. Which of the following is NOT a valid source of Type B uncertainty? a) Resolution of the instrument b) Drift over time c) Repeatability of measurement d) Manufacturer’s calibration certificate   Question 2 Answer: c (repeatability is Type A).​

56. Q3. If the calibration certificate of a reference gauge block states uncertainty = ±0.3 µm (k=2), what is the standard uncertainty ? a) 0.15 µm b) 0.3 µm c) 0.6 µm d) 0.075 µm   Question 3 Answer: a (because U=k×uU = k \times uU=k×u → u=U/ku = U/ku=U/k).

57. Q4. Type A evaluation can be done even if we only have one measurement. True / False?   Question 4 → False

58. Q5. Type B evaluation relies on professional judgment and other information without statistical analysis. True / False?   Question 5 → True

59. Q6. Expanded uncertainty is obtained by: a) Dividing combined standard uncertainty by k b) Multiplying combined standard uncertainty by k c) Adding Type A and Type B directly d) Taking square root of Type A only   Question 6 Answer: b

60. Q7. You measure the weight of a sample on a balance with a least count of 0.001 g. You take 30 readings, but all values fluctuate randomly within ±0.002 g. Which contributes more to your uncertainty? a) Type A b) Type B c) Both equally d) Cannot decide without coverage factor   Question 7 Answer: a (random fluctuation dominates).

61. Q8. The uncertainty budget combines different sources of uncertainties by: a) Simple addition b) Subtraction c) Root sum square (RSS) method d) Multiplication   Question 8 Answer:  c

62. Q9. Expanded uncertainty is calculated by multiplying the combined standard uncertainty with: a) A correction factor b) A coverage factor (k) c) A calibration coefficient d) A linear regression constant   Question 9 Answer:  b

63. Q10. You have measured the length of a steel rod 10 times and calculated the standard deviation as 0.02 mm . This uncertainty falls under: a) Type A b) Type B   Question 10 Answer: a

64. Q11. Your caliper has a least count of 0.01 mm , as per manufacturer’s specification. This uncertainty falls under: a) Type A b) Type B   Question 11 Answer: b

65. Q12. Can an uncertainty budget have only Type B sources ? Explain with an example.   Question 12 → Yes, if no repeated measurements are taken (e.g., using a certified gauge block directly).

66. Feedback and Questions