Transcript

NABL 141

NABL

NATIONAL ACCREDITATION BOARD FOR TESTING AND CALIBRATION LABORATORIES

GUIDELINES FOR ESTIMATION
and

EXPRESSION OF UNCERTAINTY IN MEASUREMENT

ISSUE NO : 02 ISSUE DATE: 02.04.2000

AMENDMENT NO : 03 AMENDMENT DATE: 18.08.2000

GOVT. OF INDIA MINISTRY OF SCIENCE & TECHNOLOGY Technology Bhavan, New Mehrauli Road, New Delhi - 110016

PROFESSOR V.S. RAMARMURTHY SECRETARY

FOREWARD
The expression of “Uncertainty in Measurements” is an integral component of the accreditation certificate being issued to the calibration laboratories. Globalization of trade and technology implies the need for interchangeability of components, which must be produced with a high degree of exactness in measurement system. This concept is equally true for all other fundamental units of measurement. The International Bureau of Weights and Measures (BIPM), in consultation with various international bodies, have arrived at a new ISO standard on Expression of Uncertainty in Measurements, in 1995. I am glad to dedicate the document of NABL on Guidelines for Estimation and Expression of Uncertainty in Measurement to the cause of calibration laboratories in the country. I take this opportunity to congratulate the scientists who have made handsome contributions in bringing out this document based on the latest ISO standard.

New Delhi 2nd April, 2000 V. S. Ramamurthy, Chairman, NABL and Secretary, DST

AMENDMENT SHEET
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National Accreditation Board for Testing and Calibration Laboratories
Doc. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.04.00 Amend No: 03 Amend Date: 18.08.00 Page No: ii

00 Amend No: 00 Amend Date: Page No: ii . No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02. sources and measures Definitions of related terms and phrases Evaluation of standard uncertainty in input estimates Evaluation of standard uncertainty in output estimates Expanded uncertainty in measurement Statement of uncertainty in measurement Apportionment of standard uncertainty Step by step procedure for calculating the uncertainty in measurement Appendix – A : Use of relevant probability distribution Appendix – B : Coverage factor and effective degrees of freedom Appendix – C : Solved Examples Page No.04. 1 3 5 10 16 18 19 20 21 22 27 32 National Accreditation Board for Testing and Calibration Laboratories Doc. 1 2 3 4 5 6 7 8 9 10 11 12 Section Introduction Uncertainty – concept.Contents Sl.

2 Scope Provisions of this document apply to measurements of all sorts as are carried out in calibration laboratories. 1. National Accreditation Board for Testing and Calibration Laboratories Doc. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02. 1. Measurements which can be treated as outputs of several correlated inputs have been excluded from the scope of this document. The present document will replace NABL’s document 141 (1992). The document will apprise calibration laboratories of the current requirements for evaluating and reporting uncertainty and will assist accreditation bodies with a coherent assignment of test measurement capability to calibration laboratories accredited by them. For specialized measurements. sources and measures Definitions of related terms and phrases Evaluation of standard uncertainty in input estimates Evaluation of standard uncertainty in output estimates Expanded uncertainty in measurement Statement of uncertainty in measurement Apportionment of standard uncertainty Step by step procedure for calculating the uncertainty in measurement Appendix – A: Use of relevant probability distribution Appendix – B: Coverage factor and effective degrees of freedom Appendix – C: Solved examples showing the application of the method outlined here to eight specific problems in different fields.1. appropriately modified forms of the concerned formulae. these may have to be supplemented by more specific details and. The document will also provide broad guidelines to all those who are concerned with measurements about uncertainty in measurement. The document covers the following topics: Uncertainty – concept.00 Amend No: 00 Amend Date: Page No: 1/ 70 . the purpose is to provide guidelines to users about contemporary requirements for global acceptance of various kinds of measurements. in some cases. estimation and apportionment of uncertainty and interpretation of uncertainty. Attempts have been made to make the provisions of this document easy to understand and ready for implementation.04.1 Introduction Purpose The purpose of the document is to harmonize procedures for evaluating uncertainty in measurements and for stating the same in calibration certificates as are being followed by the NABL with the contemporary international approach. In fact.

04.00 Amend No: 00 Amend Date: Page No: 2/ 70 . first edition. One should refer to ISO 3534-I (1993) part – I probability and general statistical terms. 1993. New Delhi (India). National Accreditation Board for Testing and Calibration Laboratories Doc. Department of Science and Technology. statistics – vocabulary and symbols – Part I. European Cooperation for Accreditation of laboratories (EAL – R-2). ISO and OIML for definition of various terms and phrases. International vocabulary of basic and general terms in metrology. International Organization for Standardization (ISO) et. IEC. Switzerland. Switzerland . 2. Singapore Institute of Standards and Industrial Research.1993. 3. al. International Organization for Standardization (ISO) et. Expression of the uncertainty of measurement in calibration. 6. 1997 5. Guidelines for estimation and statement of overall uncertainty in measurement results. Guidelines on the evaluation and expression of the measurement uncertainty. 1995. Probability and general statistical terms. Switzerland . International Bureau of Weights and Measures (BIPM). International standard ISO 3534 – I.1. (1992)..3 Normative References : This document is based primarily on the Guide to the expression of uncertainty in measurement (1993) jointly prepared by BIPM. al. 1. NABL – 141. 4. Singapore 1995. . International Bureau of Weights and Measures (BIPM).. Guide to the expression of uncertainty in measurement. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02. International Organization for Standardization (ISO) .

etc. Without such an indication.2 2. National Accreditation Board for Testing and Calibration Laboratories Doc.1 Uncertainty – Concept.1 2. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02. measured results can not be compared.04. but can be reduced by exercising appropriate controls. What we obtain from the concerned measurement process is at best an estimate of or approximation to the true value. except when known in terms of theory. 2. either among themselves or with reference values given in a specification or standard.2 These cannot be eliminated totally.2.1. A statement of results of measurement (as a process) is complete only if it contains both the values attributed to the measurand and the uncertainty in measurement associated with that value.2. for example. It is widely recognized that the true value of a measurand (or a duly specified quantity to be measured) is indeterminate.1. for example: the way connections are made or the measurement method employed uncontrolled environmental conditions or their influences inherent instability of the measuring equipment personal judgement of the observer or operator.3 2. Both systematic and random errors affecting the observed results (measurements) contribute to this uncertainty. which could reasonably be attributed to the measurand. a doubt about how well the result of measurement represents the true value of the quantity being measured.1 Source Errors in the observed results of a measurement (process) give rise to uncertainty about the true value of the measurand as is obtained (estimated) from those results. the standard deviation (or a given multiple of it). that is. 2. associated with the result of a measurement.2 2. 2.1. These contributions have been sometimes referred to as systematic and random components of uncertainty respectively.4 2. The uncertainty of measurement is a parameter. The parameter may be. Random errors presumably arise from unpredictable and spatial variations of influence quantities.00 Amend No: 00 Amend Date: Page No: 3/ 70 . Incidentally.2. or the half-width of an interval having a stated level of confidence. there still remains an uncertainty.1. Sources and Measures Concept Quality of measurements has assumed great significance in view of the fact that measurements (in a broad sense) provide the very basis of all control actions. that characterizes the dispersion of the true values. Even when appropriate corrections for known or suspected components of error have been applied. the word measurement should be understood to mean both a process and the output of that process.

4 The standard uncertainty of the result of a measurement.3.1 The model function f represents the procedure of the measurement and the method of evaluation. may simply pass of as random in another case. yields an interval that is likely to cover the true value of the measurand with a stated high level of confidence.5 National Accreditation Board for Testing and Calibration Laboratories Doc. measurements) etc. 2. when that result is obtained from the values of a number of other quantities is termed combined standard uncertainty.x2.2. The standard uncertainty associated with estimate has the same dimension as the estimate. (2. This concept cannot be used if the estimate equals zero.1) using input estimates xi for the values of the input quantities Xi. In some cases the relative standard uncertainty of measurement may be appropriate which is the standard uncertainty associated with an estimate divided by the modulus of that estimate and is therefore dimensionless.3 The standard uncertainty of measurement associated with the output estimate y.4 It should be pointed out that errors. An expanded uncertainty is obtained by multiplying the combined standard uncertainty by a coverage factor... Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.2. x N) . 2. It describes how values of the output quantity Y are obtained from values of the input quantities Xi. recognized as systematic. are also observed.3 2. Measures Measurands are particular quantities subject to measurement. If not.1) 2. XN). in essence.2.2 An estimate of the measurand Y (output estimate) denoted by y. y = f(x1. (2. No: NABL 141 Issue No: 02 Page No: 4/ 70 .1) using estimates xi of the input quantities Xi and their associated standard uncertainties u (xi). N) according to the functional relationship.…. ………….00 Amend No: 00 Amend Date: - 2.3. Some common type of these errors are due to: those reported in the calibration certificate of the reference standards /instruments used different influence conditions at the time of measurement compared with those prevalent at the time of calibration of the standard (quite common in length and d. is obtained from Eq. is the standard deviation of the unknown (true) values of the measurand Y corresponding to the output estimate y.…………. denoted by u(y). X2.3. 2.2. This.3.2) It is understood that the input values are best estimates that have been corrected for all effects significant for the model. necessary corrections have been introduced as separate input quantities. (2. 2. (2.04. It is to be determined from the model Eq.c.3.3 Various other kinds of errors. which can be recognized as systematic and can be isolated in one case. In calibration one usually deals with only one measurand or output quantity Y that depends upon a number of input quantities Xi (i = 1. Y = f(X1.

00 Amend No: 00 Amend Date: Page No: 5/ 70 .04. accuracy of measurement the closeness of agreement between a test result and the accepted reference value arithmetic mean The sum of values divided by the number of values combined standard uncertainty (uc) standard uncertainty of the result of a measurement when that result is obtained from the values of a number of other quantities. various terms and phrases are arranged in alphabetical order accepted reference value a value that serves as an agreed upon reference for comparison. National Accreditation Board for Testing and Calibration Laboratories Doc. A few terms of general interest have been taken from the “International Vocabulary of Basic and General terms in Metrology” and EAL document [3-4]. To facilitate the reader.3. equal to the positive square root of a sum of terms. may be substituted for the true value correction value added algebraically to the uncorrected result of a measurement to compensate for systematic error correction factor numerical factor by which the uncorrected result of a measurement is multiplied to compensate for a systematic error correlation the relationship between two or several random variables within a distribution of two or more random variables correlation coefficient the ratio of the covariance of two random variables to the product of their standard deviations. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02. Definitions of related terms and phrases The guide explains explicitly a large number of metrological terms which are used in practice. the terms being the variances or covariances of these other quantities weighted according to how the measurement result varies with changes in these quantities conventional true value (of a quantity) a value of a quantity which for a given purpose.

from the observations in a sample. a random component and systematic component. An error is viewed as having two components.covariance The sum of the products of the deviations of xi and yi from their respective averages divided by one less than the number of observed pairs: N 1 s xy = ∑ xi − x yi − y n − 1 i =1 ( )( ) (3. a measurement has imperfections that give rise to an error in the measurement result.00 Amend No: 00 Amend Date: Page No: 6/ 70 . error in measurement result of a measurement minus accepted reference value (of the characteristic) estimation the operation of assigning. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.04. namely. numerical values to the parameters of a distribution chosen as the statistical model of the population from which this sample is taken estimate the value of a statistic used to estimate a population parameter expanded uncertainty (U) quantity defining an interval about the result of a measurement that may be expected to encompass a large fraction of the distribution of values that could reasonably be attributed to the measurand National Accreditation Board for Testing and Calibration Laboratories Doc.1) Where n is number of observed pairs coverage factor (k) numerical factor used as a multiplier of the combined standard uncertainty in order to obtain an expanded uncertainty coverage probability or confidence level the value of the probability associated with a confidence interval or a statistical coverage interval degrees of freedom (ν) the number of terms in a sum minus the number of constraints on the terms of the sum errors In general.

5) National Accreditation Board for Testing and Calibration Laboratories Doc. the arithmetic mean or average of n independent observations zi of the random variable z. denoted by µz and which is also termed as the expected value or the mean of z. the probability density function of which is p(z) z= 1 n ∑ zi n i =1 (3. the probability that the random variable X takes value x : F(x) = Pr(X = x) (3.04.expectation the expectation of a function g(z) over a probability density function p(z) of the random variables z is defined by E[g(z)] = ∫ g(z)p(z)dz (3. It is estimated statistically by z .3) experimental standard deviation [s(qj)] for a series of n measurements of the same measurand.4) qj being the result of the jth measurement and ⎯q being the arithmetic mean of the n results considered.7) (3.6) (3. the quantity s(qj) characterizing the dispersion of the results and given by the formula : s (q j ) = ∑ (q n j =1 j −q ) 2 n −1 (3.00 Amend No: 00 Amend Date: Page No: 7/ 70 .2) the expectation of the random variable z. measurand a quantity subject to measurement probability distribution a function giving the probability that a random variable takes any given value or belongs to a given set of values probability density function the derivative (when it exits) of the distribution function : f(x) = dF(x) /dx f(x)dx is the probability element f(x) dx = Pr(x < X < x + dx) probability function a function giving for every value x. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.

random error is equal to error minus systematic error because only a finite number of measurements can be made. it is possible to determine only one estimate of random error random variable a variable that may take any of the values of a specified set of values and with which is associated a probability distribution repeatability (of results of measurements) closeness of the agreement between the results of successive measurements of the same measurand carried out under repeatability conditions repeatability conditions conditions where independent test results are obtained with the same method on identical test items in the same laboratory by the same operator using the same equipment within short interval of time. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.04. reproducibility conditions conditions where test results are obtained with the same method on identical test items in different laboratories with different operators using different equipment results of a measurement value attributed to a measurand.random error result of a measurement minus the mean that would result from an infinite number of measurements of the same measurand carried out under repeatability conditions Notes : 1.00 Amend No: 00 Amend Date: Page No: 8/ 70 . 2. reproducibility (of results of measurements) closeness of the agreement between the results of the measurements of the same measurand carried out under reproducibility conditions . obtained by measurement Note: Complete statement of the result of a measurement includes information about uncertainty in measurement sensitivity coefficient associated with an input estimate (ci) the differential change in the output estimate generated by the differential change in that input estimate standard deviation (σ) the positive square root of the variance National Accreditation Board for Testing and Calibration Laboratories Doc.

No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02. uncertainty (in measurement) parameter. variance A measure of dispersion. in general.00 Amend No: 00 Amend Date: Page No: 9/ 70 .standard uncertainty uncertainty of the result of a measurement expressed as a standard deviation systematic error mean that would result from an infinite number of measurements of the same measurand carried out under repeatability conditions minus acceptance reference value of the measurand. National Accreditation Board for Testing and Calibration Laboratories Doc. Note: Systematic error is equal to error minus random error true value (of a quantity) the value which characterized a quantity perfectly defined in the conditions which exist when that quantity is considered Note: The true value is a theoretical concept. and. can not be known exactly Type A evaluation (of uncertainty) Method of evaluation of uncertainty by the statistical analysis of series of observations Type B evaluation (of uncertainty) Method of evaluation of uncertainty by means other than the statistical analysis of series of observations. associated with the result of a measurement that characterizes the dispersion of the values that could reasonably be attributed to the measurand. which is the sum of the squared deviations of observations from their average divided by one less than the number of observations.04.

4. The readings of the test DMM may remain unchanged or undergo flicker ±1 count due to its digitizing process.4. Usually. In this case the evaluation of the standard uncertainty is based on some other scientific knowledge. there will be an observable scatter or spread in the values obtained. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.1 Evaluation of standard uncertainty in Input estimates General considerations The uncertainty of measurement associated with the input estimates is evaluated according to either a “Type A” or a “Type B” method of evaluation. 4. a 6 ½ digit stable meter calibrator is used to calibrate a device of much lower accuracy like 4 ½ digit DMM . the Type A evaluation of the uncertainty may be taken to be negligible. In such a case. α for the test and standard is taken from handbook or as per manufacturers specification.00 Amend No: 00 Amend Date: Page No: 10/ 70 . and the uncertainty on account of repeatable observations can be treated as Type B on the basis of the resolution error of the test DMM.2 4. an experiment in which a high accuracy reference standard e. In this case. The Type A evaluation of standard uncertainty is the method of evaluating the uncertainty by the statistical analysis of a series of observations.1 Type A evaluation of standard uncertainty Type A evaluation of standard uncertainty applies to situation when several independent observations have been made for any of the input quantities under the same conditions of measurement.1 4. in this case. The Type B evaluation of standard uncertainty is the method of evaluating the uncertainty by means other than the statistical analysis of a series of observations. National Accreditation Board for Testing and Calibration Laboratories Doc. In this case the standard uncertainty is the experimental standard deviation of the mean that follows from an averaging procedure or an appropriate regression analysis. one has to include the component of uncertainty associated with the thermal expansion coefficient [α = δl l] in the uncertainty budget. However.1. in a special case where high precision is needed. although the estimation of uncertainty in temperature measurement is Type A but the estimation of uncertainty in α is Type B. Case – II : Length Bar While calibrating a length bar by comparison method. in situ measurement of thermal expansion is carried out.g. Examples: Case – I : Digital multimeter (DMM) Let us consider. the evaluation of uncertainty in both temperature and α are of Type A. If there is sufficient resolution in the measurement process.2.04.

the arithmetic mean of the individual observed values qj (j = 1.1) The uncertainty of measurement associated with the estimate ⎯q is evaluated according to one of the following methods 4.68 90. 2. the estimate of Q is ⎯q.2) The positive square root of s2 (q) is termed experimental standard deviation. s 2 (q ) = 1 n ∑ qj − q n − 1 j =1 ( ) 2 (4.68 90.n).4.76 90.60 90. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02. The best estimate of the variance of the arithmetic mean ⎯q is given by s 2 (q ) s q = n 2 () (4.79 90.4) National Accreditation Board for Testing and Calibration Laboratories Doc. With n statistically independent observations (n > 1). q= 1 n ∑qj n j =1 (4.2 Let us denote by Q the repeatedly measured input quantity Xi.00 Amend No: 00 Amend Date: Page No: 11/ 70 .83 90.3) Table 4.3 An estimate of the variance of the underlying probability distribution of q is the experimental variance s2 (q) of values qj given by.63 90.2 16 121 49 64 81 484 144 16 16 49 1040 The positive square root of s2 (⎯q) is termed as estimated standard error of the mean. uq =sq () () (4.64 90.65 907.2.…….04.94 90. The standard uncertainty u (⎯q) associated with the input estimate ⎯q is the standard error.1: Data for calculation of mean and standard deviation of temperature: Observation numbers 1 2 3 4 5 6 7 8 9 10 Total Temperature 0C (t j − t × 10 −2 0 ) (t j − t × 10 −4 C -4 11 7 -8 -9 22 -12 -4 4 -7 0 ( C) 0 ) 2 2 90.2.

6) The best estimate of temperature is therefore: t = 90. 72 o C (4. We now estimate different parameters as follows: Mean Temperature: ⎛ n ⎜ t ⎜ ∑1 j j= t = ⎝ n ⎞ ⎟ ⎟ ⎠ = 90 .75 × 10 − 2 9 ( ) o C (4.00 Amend No: 00 Amend Date: Page No: 12/ 70 .8) Standard error of the mean: st = Standard uncertainty: () s (t ) 10.9) u t = 3.40 × 10−2 oC Degrees of freedom (ν) () (4. If in such a case the value of the input quantity Q is determined as the arithmetic mean ⎯q of n independent observations.5) The standard uncertainty is deduced from the value given by Eq.10) ν = n − 1 = 10 − 1 = 9 (4. the variance of the mean may be estimated by s2 q = () s2 p n (4.4) Example: Table (4.4. (4.40 × 10 −2 oC (4.72 Standard Deviation: o C (4.7) s (t ) = 1 n ∑ (t j − t n − 1 j =1 ) 2 = 1 1040 × 10 − 4 = 10.2.04.11) National Accreditation Board for Testing and Calibration Laboratories Doc.1) is shown the data from a temperature measurement.75 × 10 = n 10 2 −2 = 3. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.4 For a measurement that is well-characterized and under statistical control a combined or pooled estimate of variance s2p may be available from several sets of repeat measurements that characterizes the dispersion better than the estimated variance obtained from a single set of observations.

3. e. a single measured value. this value will be used for xi. respectively.00 Amend No: 00 Amend Date: Page No: 13/ 70 . especially in a measurement situation where a Type A evaluation is based only on a comparatively small number of statistically independent observations. It is a skill that can be learned with practice. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02. Values belonging to this category may be derived from previous measurement data . A well-based Type B evaluation of standard uncertainty can be as reliable as a Type A evaluation of standard uncertainty.3 4.04.1 Type B evaluation of standard uncertainty The Type B evaluation of standard uncertainty is the evaluation of the uncertainty associated with an estimate xi of an input quantity Xi by means other than the statistical analysis of a series of observations. experience with or general knowledge of the behaviour and properties of relevant materials and instruments .3. or a correction value.g. data provided in calibration and other certificates. (b) National Accreditation Board for Testing and Calibration Laboratories Doc.4. The standard uncertainty u(xi ) is evaluated by scientific judgment based on all available information on the possible variability of Xi. manufacturer’s specifications . 4. Otherwise it has to be calculated from unequivocal uncertainty data. The following cases must be discerned: (a) When only a single value is known for the quantity Xi. If data of this kind are not available. uncertainties assigned to reference data taken from handbooks. When a probability distribution [see Appendix – A] can be assumed for the quantity Xi. a reference value from the literature.2 The proper use of the available information for a Type B evaluation of standard uncertainty of measurement calls for insight based on experience and general knowledge. The standard uncertainty u (xi) associated with xi is to be adopted where it is given. then the appropriate expectation or expected value and the standard deviation (σ) of this distribution have to be taken as the estimate xi and the associated standard uncertainty u (xi). based on theory or experience. a resultant value of a previous measurement. the uncertainty has to be evaluated on the basis of experience taken as it may have been stated (often in terms of an interval corresponding to expanded uncertainty).

in this case. at confidence level of 99. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02. the multiple becomes the specific factor (see Appendix – A). The standard uncertainty of the standard slip gauge is then given by u(SG) = 72 nm /3 = 24 nm (4.58. The corresponding relative standard uncertainty u(Rs )/ Rs = 5 × 10-6 The estimated variance is u2 = (50 µΩ)2 = 2. The uncertainty of this value is 72 nm. The standard uncertainty is then u(m) = 300 / 1. the quoted uncertainty defines an interval having a 90% level of confidence. Case I: A calibration certificate states that the mass of a given body of 10 kg is 10.14) Where we have taken 1. The standard uncertainty of the resistor may be taken as u(Rs ) = 129 µΩ / 2.18) (4. u(m) = 300 / 2 = 150 mg (4.15) Therefore. In such a case.12) and estimated variance is u2(m) = 0.64 = 182.64 as the factor corresponding to the above level of confidence.17) (4. assuming the normal distribution unless otherwise stated. the standard uncertainty is simply.000002 mm.45 %) is given by 300 mg.000650 kg.7 % (corresponding to 3 times of standard deviation).5 × 10-9 Ω2 Case IV: A calibration certificate states that the length of a standard slip gauge (SG) of nominal value 50 mm is 50.Examples: In cases.9 mg (4.0225 g2 (4. where the uncertainty is quoted to be particular multiple of standard deviation (σ).13) Case II: Suppose in the above example. specific factor is 2. Case III: A calibration certificate states that the resistance of a standard resistor.00 Amend No: 00 Amend Date: Page No: 14/ 70 .000742 Ω ± 129 µΩ at 23 0 C and that the quoted uncertainty of 129 µΩ defines an interval having a level of confidence of 99%.04. The uncertainty at 2 σ (at confidence level of 95 .16) National Accreditation Board for Testing and Calibration Laboratories Doc.58 = 50 µΩ (4. Rs of nominal value 10 Ω is 10.

19) u 2 (x i ) = 1 (a + − a .025 3 = 0.25 % Full Scale Deflection Assuming that with the above specifications.20) yields u 2 (x i ) = Examples: 1 (a )2 3 (4.g. (a + − a− ) 2 (4. a= Here . a temperature range. a rounding or truncation error resulting from automated data reduction). Therefore. Eq. there is an equal probability of the true value lying anywhere between the upper (a + ) and lower (a − ) limits.25) u= a 3 = 0.025 bar and a − = −(0. (4. manufacturer’s specifications of a measuring instrument.26) National Accreditation Board for Testing and Calibration Laboratories Doc.)2 12 (4.05 / 2 = 0.025 bar a = 0. a probability distribution with constant probability density between these limits (rectangular probability distribution) has to be assumed for the possible variability of the input quantity Xi . Scale : 1 division = 0.) 2 (4. (4.(c) If only upper and lower limits a + and a .25% × 10) bar = −0. for rectangular distribution.21) The specifications of a dial type pressure gauge are as follows : Range : 0 to 10 bar.23) a + = (0.24) (4. Hence the standard uncertainty is given by .05 bar.0144 bar (4.00 Amend No: 00 Amend Date: Page No: 15/ 70 .025 bar .20) for the square of the standard uncertainty .22) (4.04.25% × 10) bar = 0. Resolution : ½ division = 0. According to case (b) above this leads to xi = for the estimated value and 1 (a + + a .025 bar Accuracy : ± 0. If the difference between the limiting values is denoted by 2a . No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.can be estimated for the value of the quantity Xi (e.

2) is given by the difference of the input estimate N corresponding sum or y = ∑ pi x i i =1 (5. by calculating the change in the output estimate y due to a change in the input estimate xi of + u(xi) and -u(xi) and taking as the value of ci the resulting difference in y divided by 2u (xi) . (5.3) or by using numerical methods. (2.5) whereas the sensitivity coefficients equal to pi and Eq.e. n) is the contribution to the standard uncertainty associated with the output estimate y resulting from the standard uncertainty associated with the input estimate xi. xi ± u (xi).6) National Accreditation Board for Testing and Calibration Laboratories Doc. (5. c i = (∂f /∂xi ) = (∂f /∂Xi ) at Xi = xi 5. i.…. ui (y) = ci u(xi) (5. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.3 f (X1 .2) where ci is defined as sensitivity coefficients associated with the input estimate xi i. 2. evaluated at the input estimates xi .2 (5. the partial derivative of the model function f with respect to Xi .5.g.e.1) converts to u 2 (y ) = ∑ p i2u 2 (x i ) i =1 N (5.3) The sensitivity coefficient ci. describes the extent to which the output estimate y is influenced by variations of the input estimate xi. Sometimes it may be more appropriate to find the change in the output estimate y from an experiment by repeating the measurement at e.00 Amend No: 00 Amend Date: Page No: 16/ 70 .X2 .1) The quantity ui(y) (i = 1.4) the output estimate according to Eq. If the model functions is a sum or difference of the input quantities Xi.1 Evaluation of standard uncertainty in output estimate For uncorrelated input quantities the square of the standard uncertainty associated with the output estimate y is given by. 5. ………XN) = ∑p X i =1 i i (5. u 2 ( y ) = ∑ ui2 ( y ) i =1 n (5.04. N 5. It can be evaluated from the function f by Eq.

w 2 (y ) = ∑ p w (x ) i =1 2 i 2 i n (5.8) The sensitivity coefficients equal piy/xi in this case and an expression analogous to Eq...9) National Accreditation Board for Testing and Calibration Laboratories Doc.1)... if relative standard uncertainties w(y) = u(y)/y and w (xi) = u (xi) / xi are used.00 Amend No: 00 Amend Date: Page No: 17/ 70 . X N ) = c ∏ X ipi i =1 N (5..7) the output estimates estimate again is the corresponding product or quotient of the input y = c ∏ X ipi i =1 N (5.6) is obtained from Eq.04.. (5... No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02. X 2 . (5.4 If the model function f is a product or quotient of the input quantities XI f ( X 1.5.

5 6.e. it may still occur that the standard uncertainty associated with the output estimate is of insufficient reliability. The assigned expanded uncertainty corresponds to a coverage probability of approximately 95 %. the reliability criterion is always met if none of the uncertainty contributions is obtained from a Type A evaluation based on less than ten repeated observations. obtained by multiplying the standard uncertainty u(y) of the output estimate y by a coverage factor k. However. If one of these conditions (normality or sufficient reliability) is not fulfilled. The reliability of the standard uncertainty assigned to the output estimate is determined by its effective degrees of freedom (see Appendix B). in this case.4 6. However. Even if a normal distribution can be assumed. the standard coverage factor k = 2 shall be used. 6. it is not expedient to increase the number n of repeated measurements or to use a Type B evaluation instead of the Type A evaluation of poor reliability. the conditions of the central limit theorem are met and it can be assumed to a high degree of approximation that the distribution of the output quantity is normal. information on the actual probability distribution of the output estimate must be used to obtain a value of the coverage factor k that corresponds to a coverage probability of approximately 95 %. the standard coverage factor k = 2 can yield an expanded uncertainty corresponding to a coverage probability of less than 95 %. In these cases. when evaluating the results of an interlaboratory comparison or assessing compliance with a specification. 6. If.2 The assumption of a normal distribution cannot always be easily confirmed experimentally. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02. U = ku (y) (6. in the cases where several (i.e. The use of approximately the same coverage probability is essential whenever two results of measurement of the same quantity have to be compared. the method given in Appendix – B should be used. other procedures have to be followed. For the remaining cases.00 Amend No: 00 Amend Date: Page No: 18/ 70 .g.3 6. e.6 National Accreditation Board for Testing and Calibration Laboratories Doc.04.1) In cases where a normal (Gaussian) distribution can be attributed to the measurand and the standard uncertainty associated with the output estimate has sufficient reliability. e.g. i. 6. in order to ensure that a value of the expanded uncertainty is quoted corresponding to the same coverage probability as in the normal case.6. normal distributions or rectangular distributions. N ≥ 3) uncertainty components derived from well–behaved probability distributions of independent quantities. all cases where the assumption of a normal distribution cannot be justified.1 Expanded uncertainty in measurement Calibration laboratories shall state an expanded uncertainty in measurement (U). contribute to the standard uncertainty associated with the output estimate by comparable amounts.

To this an explanatory note must be added which in the general case should have the following content: The reported expanded uncertainty in measurement is stated as the standard uncertainty in measurement multiplied by the coverage factor k = 2. For the process of rounding. the rounded up value should be used.00 Amend No: 00 Amend Date: Page No: 19/ 70 . in cases where the procedure of Appendix A has been followed.7. 7.3 National Accreditation Board for Testing and Calibration Laboratories Doc.1 Statement of uncertainty in measurement In calibration certificates the complete result of the measurement consisting of the estimate y of the measurand and the associated expanded uncertainty U shall be given in the form (y ±U). No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02. (See Appendix – B). However. 7.2 7. the usual rules for rounding of numbers have to be used. which for a normal distribution corresponds to a coverage probability of approximately 95 %. The numerical value of the uncertainty in measurement should be given to at most two significant figures.04. The numerical value of the measurement result should in the final statement normally be rounded to the least significant figure in the value of the expanded uncertainty assigned to the measurement result. the additional note should read as follows: The reported expanded uncertainty in measurement is stated as the standard uncertainty in measurement multiplied by the coverage factor k which for a t-distribution with νeff effective degrees of freedom corresponds to a coverage probability of approximately 95 %. However. if the rounding brings the numerical value of the uncertainty in measurement down by more than 5 %.

A formal example of such an arrangement is given as Table (8. In this table all quantities should be referenced by a physical symbol Xi.1 Apportionment of standard uncertainty The uncertainty analysis for a measurement-sometimes called the Uncertainty Budget of the measurement-should include a list of all sources of uncertainty together with the associated standard uncertainties of measurement and the methods of evaluating them. or a short identifier.00 Amend No: 00 Amend Date: Page No: 20/ 70 . The degrees of freedom have to be mentioned. 8. For each of them at least the estimate xi. the sensitivity coefficient ci and the different uncertainty contributions ui(y) should be specified. the associated standard uncertainty in measurement u (xi).8. it is recommended to present the data relevant to this analysis in the form of a table. For the sake of clarity.1: Schematic view of an Uncertainty Budget Source of Uncertainty Xi X1 X2 X3 Estimates Limits xi ± ∆ xi Probability Distribution .2 Table 8. Similarly.04. For repeated measurements the number n of observations also has to be stated. The dimension of each of the quantities should also be stated with the numerical values in the table.Type A or B -Type A or B -Type A or B -Type A or B Standard Sensitivity Uncertainty coefficient u(xi) ci u(x1) u(x2) u(x3) c1 c2 c3 Uncertainty contribution ui(y) u1(y) u2(y) u3(y) Degree of freedom νi ν1 ν2 ν3 νN νeff x1 x2 x3 ∆ x1 ∆ x2 ∆ x3 ∆ xN XN Y xN y -Type A or B u(xN) cN uN(y) uc(y) National Accreditation Board for Testing and Calibration Laboratories Doc.1) applicable for the case of uncorrelated input quantities. νeff has to be evaluated as mentioned in Appendix –B. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02. The standard uncertainty associated with the measurement result u(y) given in the bottom right corner of the table is the root sum square of all the uncertainty contributions in the outer right column. 8.

3. For single values.g. In the case of a direct comparison of two standards the equation may be very simple. adopt the standard uncertainty where it is given or can be calculated according to paragraph 4. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.2(c). If no data are available from which the standard uncertainty can be derived. (9. (5. Calculate for each input quantity Xi the contribution ui (y) to the uncertainty associated with the output estimate resulting from the input estimate xi according to Eqs.1) Step 4 () Step 5 Step 6 Step 7 Step 8 Step 9 National Accreditation Board for Testing and Calibration Laboratories Doc.1).1) to obtain the square of the standard uncertainty u(y) of the measurand.3. state a value of u (xi) on the basis of scientific experience.3) and sum their squares as described in Eq. Y = X1 + X2 Step 2 Step 3 Identify and apply all significant corrections to the input quantities.2. (2.04. If only upper and lower limits are given or can be estimated.2) and (5. Calculate the expanded uncertainty U by multiplying the standard uncertainty u(y) associated with output estimate by a coverage factor k chosen in accordance with Section 6. resultant values of previous measurements. e.00 Amend No: 00 Amend Date: Page No: 21/ 70 .9.g. Step-by-step procedure for calculating the uncertainty in measurement The following is a guide to the use of this document in practice: Step 1 Express in mathematical terms the dependence of the measured (output quantity) Y on the input quantities Xi according to Eq. calculate the standard uncertainty u (xi) in accordance with paragraph 4.2(a). Pay attention to the uncertainty representation used.3. the associated expanded uncertainty U and the coverage factor k in the calibration certificate in accordance with Section 7.2 (b). calculate the expectation and the standard uncertainty u (xi) according to paragraph 4. e. Report the result of the measurement comprising the estimate y of the measurand. List all sources of uncertainty in the form of an uncertainty analysis in accordance with Section 8. For input quantities for which the probability distribution is known or can be assumed. correction values or values from the literature. (5. Calculate the standard uncertainty u q for repeatedly measured quantities in accordance with sub-section 4.

Appendix A .

00 Amend No: 00 Amend Date: Page No: 22/ 70 .σ µ µ+σ Figure A. Values of the coverage factor for various level of confidence for a normal distribution are as follows: National Accreditation Board for Testing and Calibration Laboratories Doc.04.1. In such cases. σ 2π [ ] -∞<x <+∞ (A. In the absence of any specific knowledge about this distribution. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02. the quoted uncertainty in an input or output quantity is stated along with level of confidence.1 Normal distribution The probability density function p(x) of the normal distribution is as follows: P (x ) = 1 2 exp − (x − µ ) / 2σ 2 .Probability distribution A.1) represents such a distribution. 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. µ. The value of the coverage factor depends upon the distribution of the (input or output) quantity. Figure (A.1: Schematic view of the normal (Gaussian) distribution A.1 When to use normal distribution In some situation.1) Where µ is the mean and σ is the standard deviation. one may assume it to be normal.

one can only assume that it is equally probable for Xi to lie anywhere within this interval .48 a (A.2) If based on available information.27 % level Coverage 1.960 95.5) Var (X i ) = a 2 / 3 .2) represents such a distribution.2. In such a situation rectangular distributions is used. a − < x < a + .1 When to use rectangular distribution (A.645 95 % 1. where a = (a + − a − ) / 2 A. with u(x i ) = a (A. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.576 99. with u(x i ) = 1.04. it can be stated that there is 50 % chance that the value of input quantity Xi lies in the interval between a − and a + and also it is assumed that the distribution of Xi is normal.73 % 3. National Accreditation Board for Testing and Calibration Laboratories Doc.000 99 % 2. and also it is assumed that the distribution of Xi is normal.2 Rectangular distribution The probability density function p(x) of rectangular distribution is as follows: P (x ) = 1 . where it is possible to estimate only the upper and lower limits of an input quantity Xi and there is no specific knowledge about the concentration of values of Xi within the interval .45 % 2.6) In cases.3) A.000 factor (k) 90 % 1.00 Amend No: 00 Amend Date: Page No: 23/ 70 . then the best estimate of Xi is : x i = a.1: Confidence Level and the corresponding Coverage factor (k) Confidence 68.Table A. where a = (a + − a − ) / 2 2a (A.000 If based on available information.4) Figure (A. it can be stated that there is 68% chance that the value of input quantity Xi lies in the interval of a − and a + . The expectation of Xi is given as xi E (X i ) = x i = (a + + a − ) / 2 and its variance is (A. then the best estimate of Xi is: x i = a = (a − + a + ) / 2 .

8) When β → 1.7) Var (X i ) = a 2 (1 + β)2 /6 (A. For such cases. it is more reasonable to assume that Xi can lie anywhere within a narrower interval around the midpoint with the same probability while values nearer the bounds are less and less likely to occur. and its variance is (A. and a top of width 2β a . a base of width a + − a − = 2a.µ .2: Schematic view of the rectangular distribution A. where 0 ≤ β ≤ 1 is used . No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02. The expectation of Xi is given as: E (X i ) = (a + + a − ) / 2 .3 Symmetrical Trapezoidal Distribution The above rectangular distribution assumes that Xi can assume any value within the interval with the same probability.00 Amend No: 00 Amend Date: Page No: 24/ 70 . the probability distribution is represented by a symmetric trapezoidal distribution function having equal sloping sides (an isosceles trapezoid).a /√3 µ µ + a /√3 Figure A. However. in many realistic cases. the symmetric trapezoidal distribution is reduced to a rectangular distribution. National Accreditation Board for Testing and Calibration Laboratories Doc.04.

No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02. The expectation of Xi is given as.10) Figure A.A. E (X i ) = (a + + a − ) / 2 .3. the symmetric trapezoidal distribution is reduced to a triangular distribution. When the greatest concentration of the values is at the center of the distribution.9) Var (X i ) = a 2 / 6 (A. then one must use the triangular distribution.3) shows such a distribution. and its variance is (A.04.00 Amend No: 00 Amend Date: Page No: 25/ 70 .3: Schematic view of the triangular distribution National Accreditation Board for Testing and Calibration Laboratories Doc.1 Triangular Distribution When β = 0. Figure (A.

and reflection occurs when the impedances do not match.00 Amend No: 00 Amend Date: Page No: 26/ 70 . The mismatch uncertainty is given by 2Γs ΓL where Γs and ΓL are the reflection coefficients of the source and the load respectively.04.A. At high frequency the power is delivered from a source to a load. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.4: Schematic view of the U-shaped distribution National Accreditation Board for Testing and Calibration Laboratories Doc.4).11) Figure A. The standard uncertainty is computed as: u2 (xi) = (2 ΓS ΓL)2 / 2 (A.4 U-Shaped Distribution This U-shaped distribution is used in the case of mismatch uncertainty in radio and microwave frequency power measurements (shown in figure (A.

Appendix B .

is a measure of the reliability.N) defined in Eqs.Coverage factor derived from effective degrees of freedom B. are the contributions to the standard uncertainty associated with the output estimate y resulting from the standard uncertainty associated with the input estimate x i which are assumed to be mutually statistically independent .1) νi where u i (y) (i = 1. Similarly.. Note : The calculation of the degrees of the freedom ν for Type A and Type B of the evaluation may be as follows: National Accreditation Board for Testing and Calibration Laboratories Doc. (5.00 Amend No: 00 Amend Date: Page No: 27/ 70 . which is approximated by an appropriate combination of the νi for its different uncertainty contributions u i (y). Estimate the effective degree of freedom νeff of the standard uncertainty u(y) associated with the output estimate y from the Welch-Satterthwaite formula. it is truncated to the next lower integer and the corresponding coverage factor k is obtained from the table.……. B. The procedure for calculating an appropriate coverage factor k : Step 1 Step 2 Obtain the standard uncertainty associated with the output estimate. That means taking into account how well u(y) estimates the standard deviation associated with the result of the measurement.1 To estimate the value of a coverage factor k corresponding to a specified coverage probability requires that the reliability of the standard uncertainty u(y) of the output estimate y is taken into account.1) and (5. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02. and the νi is the effective degrees of freedom of the standard uncertainty contributions u i (y). the degrees of freedom of the estimate.2) .3 Welch-Satterthwaite formula is as follows: ν eff u4 (y) = N 4 ui ( y ) ∑ i =1 (B.2. Obtain the coverage factor k from the table of values of student “t” distribution.3. which depends on the size of the sample on which it is based. For an estimate of the standard deviation of a normal distribution.04.2 Step 3 B. a suitable measure of the reliability of the standard uncertainty associated with an output estimate is its effective degrees of freedom νeff. If the value of νeff is not an integer.

Further interpretation on the above is given on page 29 & 30 National Accreditation Board for Testing and Calibration Laboratories Doc. when lower and upper limits are known νi → ∞ (B. since it is a common practice to chose a. Concerned laboratories should refer to Annexure – G (with special emphasis on table G-2 ) and Annexure –H for related examples.3) (B.00 Page No: 28/ 70 .Type A Evaluation For the results of direct measurement (Type A evaluation). νi = n .to a+ is extremely small. However.00 Amend No: 01 Amend Date: 18.2) It is suggested that νi should always be given when Type A and Type B evaluations of uncertainty components are documented. the accredited calibration laboratories shall be required to follow ISO Guide to the expression of uncertainty in Measurement (1995). the degree of freedom is related to the number of observations (n) as. Where high precision measurements are undertaken.04.08. assuming that νi → ∞ is not necessarily unrealistic. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.1 Type B Evaluation For this evaluation.and a+ in such a way that the probability of the quantity lying outside the interval a.

This shows that even when many measurements are taken.00 Amend No: 02. ν = is only 2. ∆u(xi)/ u(xi) = 0. In such instances. Every component of uncertainty can have an appropriate number of degrees of freedom. It is of the interest to note that equation (1) tells us that when we have made 51 measurements and taken the mean. leads to this instance. for example ν = n . A high number of degrees of freedom is associated with a large number of measurements or a value with a low variance or a low dispersion associated with it. The question is how to assign components evaluated by Type B processes. the reliability of the uncertainty is not necessarily any better than when a type B assessment is make. most spreadsheets provide the standard deviation of the fit when data is fitted to a curve. the relative uncertainty in the uncertainty of the mean is 10%. ⎯x. if relative uncertainty is 10%.Interpretation on Effective Degrees of Freedom “Whilst the reason for determining the number of degrees of freedom associated with an uncertainty component is to allow the correct selection of value of student’s t. the limits may be determined so that we have complete confidence in their value. For a relative uncertainty of 25 % then ν = 8 and for relative uncertainty of 50 %.00 Page No: 29/ 70 ……… ………… 1 . This standard deviation may be used as the uncertainty in the fitted value due to the scatter of the measurand values.1. The assigning of limits. it is better to try to determine the limits more definitely. then a lesser number of degrees of freedom must be assigned. but may for convenience be thought of as a percentage or a fraction. For some distributions. It also shows why we restrict uncertainty to two digits. 03 Amend Date: 18. National Accreditation Board for Testing and Calibration Laboratories Doc. The smaller the number. It is equation G. For the mean. If the limits themselves have some uncertainty.1 Then it can be shown that the number of degrees of freedom is 50.04. Indeed. it also gives an indication of how well a component may be relied upon. For example.3 that is: ν ≈ ½ [ ∆u(xi)/ u(xi) ]-2 Where : ∆u(xi)/ u(xi) is the relative uncertainty in the uncertainty This is a number less than 1. For example. it is usually better to rely on prior knowledge rather than using an uncertainty based on two or three measurements.ν. where n is a number of repeated measurements.e. namely infinite degrees of freedom. the process is also quite straightforward. For other Type A assessments. the number of degrees of freedom is effectively infinite. i. assigned to it. The ISO Guide to the expression of uncertainty in measurement (GUM) gives a formula that is applicable to all distributions. Rather than become seduced by the elegance of mathematics. which are worst case. A low number of degrees of freedom corresponds to a large dispersion or poorer confidence in the value. particularly if the uncertainty is a major one.08. the better defined is the magnitude of the uncertainty. and simplifies the calculation of effective degrees of freedom of the combined uncertainty. The value is usually not reliable enough to quote to better than 1 % resolution. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.

No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02. National Accreditation Board for Testing and Calibration Laboratories Doc.04. which is: n νeff = [uc4 (y)/ ∑{ ui4 (y)/ νi}] 1 Where: νeff νi is the effective number of degrees of freedom for uc the combined uncertainty is the number of degrees of freedom for u i . Adopted from NATA document on “Assessment of uncertainties in Measurement”.00 Amend No: 03 Amend Date: 18. This is calculated using the Welch-Satterthwaite equation. 1999. it remains to find the number of degrees of freedom in the combined uncertainty.00 Page No: 30/ 70 . 2 u i (y) is the product of c i u i “The other terms have their usual meaning”.08. The degrees of freedom for each component must also be combined to find the effective number of degrees of freedom to be associated with the combined uncertainty.Once the uncertainty components have been combined. the ith uncertainty term ….

Degrees Freedom (ν) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 25 30 ∞ 68.03 1.000 99 63.84 1.13 2. k is 1.25 3.21 9.20 1. and 3.05 1.27 3.04 1.11 2.92 2.75 1.09 2.72 1.45 13. 95.01 2.11 1.03 3.16 2.71 1.31 2.62 5.14 2.000 90 6.81 1.71 3.31 2.26 2.43 2.60 4.31 2.1: Student t-distribution for degrees of freedom ν.18 2.74 1.14 1.71 4.03 1.960 95.84 4.27 %.94 1.21 2. 2.79 2.576 99.86 2.23 2.28 2.17 3.23 2.86 1.04 1.45 3.80 1.04 1.14 2.09 2.18 2.97 4.48 3.16 2.11 3.33 3.76 3.25 2.20 2. The t-distribution for ν defines an interval -t p (ν) to + t p (ν) that encompasses the fraction p of the distribution.73 %.22 6.09 2.54 3.57 2.50 3.645 Fraction p in percent 95 12.85 3.Table B.87 2.13 2.73 1.77 1.05 3.02 1.59 3.51 4.10 2.92 2.03 1.13 2.90 4.95 2.000 National Accreditation Board for Testing and Calibration Laboratories Doc.96 3.08 1.11 2.80 19.53 3.06 1.06 2.07 1.32 1.12 2.64 3.37 2.90 2. respectively.45 %.92 5.88 2. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02. For p = 68.02 1.05 1.15 2.83 1.70 1.27 1.51 3.75 2.42 3.35 2.18 2.66 9. and 99.03 1.89 1.36 3.69 3.02 1.17 2.04.32 2.03 1.45 2.98 2.75 1.09 1.20 2.73 235.52 2.78 1.28 4.73 1.09 3.78 2.00 Amend No: 00 Amend Date: Page No: 31/ 70 .53 4.85 2.03 1.36 2.30 3.04 1.65 2.76 1.

Appendix C Solved examples showing the applications of the method outlined here to eight specific problems in different fields .

00 Amend No: 00 Amend Date: Page No: 32/ 70 . Standard reference temperature (Tr e f) = 20 0 C. Mathematical model YGUT = XSTD + ∆X (C.C. XSTD is the gauge block size and ∆X is the error or the difference between the micrometer reading and gauge block size. Uncertainty equation The combined standard uncertainty equation is given by. and Least count of thermometer used = 1 0 C.00008 mm. Actual calibrating temperature (Tc) = 23 0 C.00010 ± 0. The detailed specifications of the slip gauge are as follows: Range = 25 mm.1) Where YGUT is the micrometer reading [Gauge under test (GUT). ⎡ ⎧ δΥ ⎫ ⎤ ⎡ ⎧ δΥ ⎫ ⎤ uc (YGUT ) = ⎢ ⎨ GUT ⎬{u (Χ STD )}⎥ + ⎢ ⎨ GUT ⎬{u (∆X )}⎥ ⎦ ⎣ ⎩ δΧ STD ⎭ ⎦ ⎣ ⎩ δ∆Χ ⎭ 2 2 (C.1 Micrometer calibration using “0” grade slip gauge at 25mm Introduction The instrument under calibration is a micrometer of 0 – 25 mm range with a slip gauge of 25 mm of nominal size. Calibrated value = 25.04. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.2) National Accreditation Board for Testing and Calibration Laboratories Doc.

000 0. 4.4) Table C. 0.0006mm j =1 n (C.47 × 10 −4 = = 2.6) Degrees of freedom (νi) ν= n–1 = 5 –1 = 4 (C.001 0.446 × 10 −4 mm = 0.47 × 10−4 mm 4 (C.6 1.Measured results Type A evaluation Five readings are taken and the deviation from the nominal value is as follows.7) National Accreditation Board for Testing and Calibration Laboratories Doc. 3.001 0.000 0. 2446 µ m () (C. 5.04.001 0.6 Observation numbers Deviation from nominal value (xj) (mm) 1.00 Amend No: 00 Amend Date: Page No: 33/ 70 .1 : Data for calculation of mean and standard deviation ⎯x (mm) (xj .6 1. Mean Deviation: x = (∑ x j ) /n = 0.5) u x = 0 .6 3.0006 Standard deviation of the mean: sx = Standard uncertainty: () s 2 ( x ) 5.3) Standard deviation : s( x ) = 1 n ∑ xj − x n − 1 j =1 ( ) 2 = 1 (12 × 10−7 ) = 3 × 10−7 mm = 5. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.6 3.2446µm n 5 (C.⎯x)2 × 10-7 (mm) 1. 2.

9) Standard uncertainty (u3) due to difference in thermal expansion coefficient of the slip gauge and micrometer It is assumed that the difference in thermal expansion coefficient of standard slip gauge and the micrometer screw is amounting to 20 %.08 µm.2875 µm (C. the standard uncertainty is u6 = 0.08 µm = 0.5µm ⎝ 2 ⎠ Standard uncertainty (u5) due to the parallelness of micrometer faces’ (C. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02. Standard uncertainty (u1) due to the temperature measurement ±1 0 C.8 ) Standard uncertainty (u2) due to difference in temperature of micrometer and slip gauge Assuming the temperature of the slip gauges and micrometer are the same but still it can have a difference ±1 0 C.10) ⎛ 1 ⎞ say ⎜1 fringe ⎟ ≈ u 4 = 0.2875 µm (C.046µm 3 (C.5 × 10-6 mm = 287.00 Amend No: 00 Amend Date: Page No: 34/ 70 . Standard thermal expansion coefficient of the gauge block is 11.04.11) ⎛ 1 ⎞ say⎜1 fringe ⎟ ≈ u5 = 0.5 × 10-6 mm = 0. Hence.5×10-6/ 0C u 1 = 25 × 1×11.5 × 10-6 × (20 /100) mm = 0.Type B evaluation The uncertainty quoted in the gauge block calibration certificate is considered to be Type B uncertainty of normal distribution. again uncertainty component u2 = 0. hence the uncertainty component [∆T = Tc – Tref = 30 C]. u3 = 25 × 3 × 11.12) The uncertainty in the value of the standard is taken from the calibration certificate say 0. Assuming rectangular distribution.5 µm ⎝ 2 ⎠ Standard uncertainty (u6) due to the Standard used for calibration (C.1725 µm Standard uncertainty (u4) due to the flatness of micrometer faces’ (C.13) National Accreditation Board for Testing and Calibration Laboratories Doc.

099)4 4 ∞ ∞ ∞ + (0.531)4 (0.046)4 ∞ (C.099 )2 + (0.165 )4 + (0.288)2 + (0.288)2 + (0. Degrees of freedom (νeff ) ν eff = ∑ uc (y ) 4 n (u c (y )) 4 i =1 νi Veff = (0.14) = 0.15) (C.165 )4 + (0.2446)2 + (0.2446)4 + (0.00358 4 ≅ 89 ≅ ∞ Combined uncertainty Combined uncertainty [uc (YGUT)] is . No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.04. = (0.288 )4 ∞ + (0.288 )4 ∞ + (0.165)2 + (0.046)2 µm uc (YGUT ) = 0.07950 0.00 Amend No: 00 Amend Date: Page No: 35/ 70 .531 µm (C.165)2 + (0.16) National Accreditation Board for Testing and Calibration Laboratories Doc.The sensitivity coefficients (ci) are 1 and degree of freedom is νi = ∞ [Type B components ] in all six cases .

287 0.288 1.25 0.165 1.04.531 ∞ k=2 1.Type B .√3 Rectangular .172 0.0 0. National Accreditation Board for Testing and Calibration Laboratories Doc.5 0.165 ∞ ∞ ∞ ∞ ∞ ∞ u1 u2 0.4 % and for ν eff = ∞.143 Probability Distribution – Type A or B .086 0.04 0.4 % and for ν eff = ∞ .165 u3 0.062 µm Reporting of results The value at 25 mm is 25.531 µm = 1. for the confidence level of 95.046 1.288 u6 0.Table C.√3 Rectangular .143 0.0 0.√5 Sensitivity Uncertainty Degree Standard Uncertainty coefficient contribution of ci u i (y) u(x i) freedom (µm) (µm) νi 0.046 Repeatability u(⎯x) 0.Type B .√3 Normal .08 0. the coverage factor k = 2.√3 Rectangular .0 0.099 u4 0.165 1.factor Rectangular . U = k uc (YGUT ) = 2 × 0.Type A .062 ∞ Expanded Uncertainty (U) From the student’s distribution table.00 Amend No: 00 Amend Date: Page No: 36/ 70 .2: Uncertainty Budget: Source of Uncertainty Xi Estimate s xi (µm) 0.0 0.25 0.Type B .Type B .Type B .Type B .547 1.√3 Rectangular .099 1.287 Limits ± ∆x i (µm) 0.2446 4 uc (YGUT ) Expanded uncertainty 0.288 u5 0.0 0. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.0 0.062 µm with coverage factor k = 2 for confidence level of 95.0 0.√3 Rectangular .5 0.288 1.00010 mm ± 1.

Reference standard (ms): The calibration certificate for the reference standard gives a value of 10. observed difference in mass between the unknown mass and the correction for eccentricity and magnetic effects. is obtained as mx = ms+ δmD +δ m + δmc + δA where ms δmD δm standard δmc δA (C. Comparator (δm): A previous evaluation of the repeatability of the mass difference between two weights of the same nominal value gives a pooled estimate of standard deviation of 25 mg. Eccentricity and magnetic effect (δmc): The variation of mass due to eccentric load and magnetic effect is found to be ± 10 mg.C. 2. Mathematical model The unknown conventional mass mx.04. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02. drift of the value of the standard since its last calibration.00 Amend No: 00 Amend Date: Page No: 37/ 70 .2 Calibration of weight of nominal value 10 kg Introduction The calibration of a weight of nominal value 10 kg of OIML class M1 is carried out by comparison to a reference standard (OIML class F2 ) of the same nominal value using a mass Comparator whose performance characteristics have previously been determined. correction for air buoyancy. National Accreditation Board for Testing and Calibration Laboratories Doc. 4. the combined standard uncertainty is given by n ⎡ δf ⎤ 2 u c ( y ) = ∑ ⎢ ⎥ u 2 ( xi ) i =1 ⎣ δx ⎦ 2 (C.000.005 g with an associated expanded uncertainty of 45 mg (coverage factor k = 2) Drift of the value of the standard (δmp): The drift of the value of the reference standard is estimated from the previous calibrations to be zero within ± 15 mg. 3. 5.18) Details of the specifications 1. Air buoyancy (δA): The limit of air buoyancy correction is found to be 10 mg. For uncorrelated input quantities.17) conventional mass of the standard.

Table C. u (δm c ) = 10 mg = 5.040 0. 5.010 B reading (g) 0.040 0. 45 mg = 22. From the calibration certificate of the standard (A). Assuming a rectangular distribution. the standard uncertainty = 25 mg = 11. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.025 0.015 0.025 g (obtained from prior evaluation) Standard uncertainty uA = u(δm) = s(⎯δm ) Degrees of freedom = 5 – 1 = 4 Type B evaluation 1.00 Amend No: 00 Amend Date: Page No: 38/ 70 . distribution.010 0.015 0. the standard uncertainty u (δm D ) = 15 mg = 8.045 0.020 0. 5 u (ms ) = 2. 1 Type A evaluation Five observations of the difference in mass between the unknown mass (B) and the standard (A) are obtained using the substitution method and the ABBA weighing sequences: Arithmetic mean ⎯δ m = 0.010 0.020 0.025 0.010 0.010 Observed difference (g) 0.5mg 2 Assuming a rectangular Drift in the value of the standard is quoted as ±15 mg.18 mg. pooled estimate of standard deviation sp (δ m) = 0.030 B reading (g) 0.030 0. 2.020 0. A reading (g) 0. the expanded uncertainty (U) is certified as 45 mg with a coverage factor k = 2. the variation of mass is found to be ± 10 mg.010 0.015 1.010 0.66mg 3 3.020 0.3 : Observations No. the standard uncertainty.020 A Reading (g) 0.030 0.020 0. Due to eccentricity and magnetic effects.017 g.015 0. 4. 3.77mg 3 National Accreditation Board for Testing and Calibration Laboratories Doc.04.

4.20) Expanded uncertainty U = ku c (m) = 2 × 27.8)4 + (22.66)2 + (5.18)2 + (25.5)4 + (8.8)4 (11.77)4 4 ∞ ∞ ∞ (5.66)4 + (5.45mg Degrees of freedom (νi): In all these four cases.00 Amend No: 00 Amend Date: Page No: 39/ 70 . No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.1 4 ~ ∞ ~ 153.77)4 + ∞ = 598029.04.77mg 3 (C.44 15623.6 mg National Accreditation Board for Testing and Calibration Laboratories Doc.8mg (C.77 )2 + (5. distribution. Error in air buoyancy correction is reported to be ±10 mg.45)2 = 27.77 )2 mg = 25. the degree of freedom is νi = ∞ Degrees of freedom νeff v eff = (27.1 Combined standard uncertainty Combined standard uncertainty is 2 2 uc = u A + u B = (11.19) (22.8 mg = 55.5)2 + (8. the standard uncertainty Assuming rectangular u (δΑ ) = uB = 10 mg = 5.

Type B .0 8.000005 kg 15 mg ± ∆x i Probability Distribution -Type A or B .5 %.5 ci 1.77 11.4: Uncertainty Budget: Source of Uncertainty Estimates Limits xi Xi ms δm d δm c δA 10.Type B . Table C.77 5.18 1 11.Type B . No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.00 Amend No: 00 Amend Date: Page No: 40/ 70 .√3 Normal .0 5.0 5.Factor Standard Sensitivity Uncertainty Uncertainty coefficient contribution u(x i) (mg) 22.77 1.77 1.5 mg 10 mg 5 mg 10 mg 5 mg Repeatabilit y Normal .2 Rectangular . The reported expanded uncertainty of measurement is stated as the standard uncertainty in measurement multiplied by the coverage factor k = 2.√5 8.Type A .6 ∞ National Accreditation Board for Testing and Calibration Laboratories Doc. which for a normal distribution corresponds to a coverage probability 99.5 Degree of freedom vi ∞ ∞ ∞ ∞ 45 mg 7.000025 kg ± 56 mg.04.66 5.√3 Rectangular .8 ∞ Expanded uncertainty k=2 55.√3 Rectangular .0 u i (y) (mg) 22.Type B .18 4 uc (m ) 27.66 1.Reported result The measured mass of the nominal 10 kg weight is 10.

For example.04. Repetition: . the piston of the secondary standard gauge (SPC) as well as the industrial DWT have been rotated at a constant rpm with a synchronous motor to relieve friction.2 – 100 MPa Class ⎯ S The data obtained from the characterization of the piston gauge is as follows: Resolution .the instrument has been leveled so that the axis of rotation of the piston is vertical. at 23 0C.1 % of full scale pressure SECONDARY STANDARD – STANDARD PISTON CYLINDER ASSEMBLY Range ⎯ 0. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.value. gNPL is 9. which is used as the secondary standard.1 – 60 MPa Serial No: XXXXXXX Make: XXXXXXXX Manufacturer’s data : Resolution . Acceleration due to gravity (g) correction: . The specifications of the industrial DWT and SPC are as follows: Dead Weight Tester Range: 0.± 0.00 Amend No: 00 Amend Date: Page No: 41/ 70 .01 MPa in the whole studied range of pressure Calibration Procedure The calibration is carried out by crossfloat method.3 Calibration of a Industrial Dead Weight Tester An industrial Dead Weight Tester (DWT) up to 60 MPa is calibrated against a standard piston cylinder (SPC) assembly.± 0.1 MPa Accuracy .791241/gManufac).C.during the calibration. This method starts with the following steps: Leveling: .The correction has been incorporated whenever there is a difference in g. Rotation: . National Accreditation Board for Testing and Calibration Laboratories Doc.the temperature of SPC and DWT has been maintained near 230C and suitable temperature correction has been incorporated to make all the results at the same temperature. Temperature: . that is.the calibrations have been carried out at least five times under identical condition both at the increasing pressure as well as decreasing pressure.791241 m/s2 therefore the correction factor is (9.± = 0.01 MPa Standard uncertainty – ± 0.

it is normally observed that the ∆P is a linear function of PSPC . Mathematical Model The mathematical relationship can be modeled as: PDWT = PSPC + ∆ P . ∆P = ∆Po + where ∆P0 and S 1 S1 × PSPC .00 Amend No: 00 Amend Date: Page No: 42/ 70 . therefore. However partial derivative of PDWT with S1 is PSPC . is the average Where PDWT is the pressure as measured by the industrial DWT . we have to take into account the value of PSPC standard uncertainty. In the simplest case and also in this limited pressure range up to 100 MPa. The average sign indicate the arithmetic mean of several repetitive measurement under identical condition. Therefore. (C.23) where the bracketed quantities are the standard uncertainties due to the repeatability of the readings and the standard gauge.22) are assumed to be constants . which is.The pressure generated at the reference level both at the secondary standard (SPC) and test gauge (DWT) has been maintained constant. The combined standard uncertainty equation is then given by ⎡⎧δΡ ⎫ ⎤ ⎡⎧δΡ ⎫ ⎤ ⎡⎧δΡ ⎫ ⎤ uc (PDWT ) = ⎢⎨ DWT ⎬{u(PSPC )}⎥ + ⎢⎨ DWT ⎬{u(∆Ρ )}⎥ + ⎢⎨ DWT ⎬{u(S1 )}⎥ O O ⎦ ⎣⎩ δΡSPC ⎭ ⎦ ⎣⎩ δ∆Ρ ⎭ ⎦ ⎣⎩ δS1 ⎭ 2 2 2 (C. It may be mentioned here that the partial derivatives of PDWT with PSPC and PDWT with ∆Po is 1 . Table (C. the sensitivity coefficients also for both these cases equal to one. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.Reference level: . for the estimation of combined National Accreditation Board for Testing and Calibration Laboratories Doc.4) represents the data where σ represents the standard uncertainty in each set of readings at a given pressure. ∆P is the difference between the two readings of the PDWT and the PSPC . respectively.04. PSPC standard pressure as obtained from the standard gauge PSPC .

001199MPa (C.9782 39.9965 9.9549 54.9706 29. 1-9 PW .0 10.728 0.9603 54. s(qk) = 0.9991 0.9702 29.9988 (kPa) 0.364 1. 1-3 PW.9991 4.9965 9. 7-8 PW 0.9513 59.44 29.162 0.80 13.9568 49.9935 19.0 20. As mentioned.9561 49.50 21.9561 59.0 60.603 0.9861 29.9659 39.9545 49.5: Increasing and Decreasing Pressure Weight used Nominal Pressure PDWT (MPa) 1. 1-9 PW .9975 0.0 50.9861 9. 1-3 PW .9799 19.0 45.9611 54.0 60. The PSPC is the average of these five readings.0 1.896 0.682 0.1 PW 1-2 PW .467 0.9671 39.0 5.9935 4.9606 54.9961 0.9939 4.0 40.9576 49.619 0.9782 19.9971 0.0026827 MPa (C.08 National Accreditation Board for Testing and Calibration Laboratories Doc.000 0.9659 49.9494 59.9663 49.96 54.9497 54.993 19.9665 49.9654 39.25) PSPC (MPa) 0.9933 19.9611 44.9931 19.9595 54.9498 59.9978 0.9787 39.9961 9.0 55. 6 PW .9865 29.422 0. PW represents Piston and Weight hanger.9704 29.0 30. 1-4 .88 13.9561 49.9869 9.9701 29.9532 54.9641 44.9871 29.9931 19.9669 49.9556 59.9528 54.9978 4.0 30.9867 29.24 1.9992 4.392 0.9545 59.477 0.9967 9.9532 59.9557 59.971 44.9595 44.9871 9.9532 59.116 (kPa) 1.00 Amend No: 00 Amend Date: Page No: 43/ 70 .88 38.9989 4. 1-6 PW.9665 39.9691 44.9971 9. 7-8 PW.9867 9.9654 49.0 20.4) shows the data as obtained from the experiment.1 = 4 Table C.675 0.9700 44.10 6.9875 9. 1 PW .911 1.96 33.0026827 Degree of freedom νi = 5 .819 2.0 55.9791 39.626 1.9512 54.9801 19.9791 19.0 45.780 0. 6 PW .9669 39.9706 44. we have taken five readings (n = 5) of the same pressure point while increasing / decreasing the pressure cycle.9799 39. 1-4 PW.9787 19.76 20. 1-2 PW .9548 59.0 40.04.28 48.9611 44.9604 44.9512 59. the standard uncertainty is given by.993 4.Experimental Results Table (C.24 3.16 43.9691 29. u1(s ) = 0.08 29.425 0. 1-5 PW.64 47.971 29.44 6.9967 0.9787 19. ∆P is the error or the difference between PDWT and PSPC .0 10.9933 4.64 39.9576 59.9967 0.9932 19.9931 4.9694 44. 1-6 PW.9865 29.0 50.9675 39.9606 44. 1-5 PW .9861 29.9988 4.9987 4. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.9964 0.9671 49.1-4 PW.9556 49.9789 39.9964 9.9992 (MPa) 0.943 0.9552 54. Type A evaluation of standard uncertainty (1) Repeatability We have taken 5 repeatable readings at each and every pressure or n= 5.5) is. 1-4 .0 Reading of SPC 5 = 0.0 5. The maximum standard uncertainty (σ) from the table (C.86 33.9604 54.64 3.68 44.9787 39.24) Hence.9701 44.9997 0.9933 4.9871 9.9989 σ ∆P PW PW .

0012762 (MPa) 0.000014 )2 = 0. as mentioned earlier.001199 )2 + (0. equals to PSPC . No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.001527 )2 = 0.28) Therefore. (C.001527 MPa (C.23) with the sensitivity coefficient.04.000802 0. we have 20 data points as are shown in Table (C.(C. this ∆P can be fitted with PSPC in a linear fitting program. In the present case. there is a difference in pressure (∆P) at each pressure of PSPC [(Eq. ∆P is equal to 0.6).001304 (MPa) 0.27) At the maximum pressure [60 MPa].0012762 )2 + (60 × 0.22)] . Thus.000802 × PSPC (MPa ) − 0.04682 MPa at 60 MPa. Table C.0013(MPa ) (C. From the above Eq. It is therefore ∆P is maximum at 60 MPa but reduces as we decrease the pressure.26).000014 19 The standard uncertainty in ∆P is evaluated from Eq. ∆Ρ = 0. u2 (∆Ρ ) = (0. estimated Type A standard uncertainty is u A = u1 (s ) + u (∆Ρ ) 2 2 = (0.00 Amend No: 00 Amend Date: Page No: 44/ 70 .26) The different fitting parameters with standard uncertainty are shown in Table (C.(2) Data Analysis As mentioned in the mathematical model. (C. the fitted equation reduces to.29) National Accreditation Board for Testing and Calibration Laboratories Doc. u2 (∆Ρ ) = u (∆Ρo ) + PSPC × u (S1 ) 2 ( ) 2 (C.6: Regression Output: Source Parameters ∆Po U(∆Po) (1 σ) S1 u(S1 ) (1 σ) Degrees of Freedom Fitted Value -0.00194 MPa (C. the standard uncertainty reduces to.5).

00570 1.Type B √3 Normal -Type A k = 2.01 0.factor Standard uncertainty u(x i ) (MPa) Sensitivity coefficient Uncertainty contribution u i (y) (MPa) Deg.01MPa 2 (C.31) Degree of freedom νi = ∞ Table C.7: Summary of standard uncertainty components Source of uncertainty (X i) Estimates ( xi ) (MPa) Limits ±∆ x i (MPa) Probability Distribution –Type A or B .32) (C. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02. the Type B component is given by.0057 ∞ 0.00194 )2 + (0.04.00 Amend No: 00 Amend Date: Page No: 45/ 70 . the boundaries of this rectangular distribution is given by a = a+ − a− = 0.00194 1.01 uA uc (PDWT ) Expanded uncertainty Rectangular .00602 0.01 = = 0.00 0.0057MPa 3 3 (C. of freedom (νi ) uB 0.0 0. there is an equal probability of the value lying anywhere between the lower (a-) and upper (a+) limits.00602 MPa National Accreditation Board for Testing and Calibration Laboratories Doc.0057 )2 MPa = 0.0 0.012 19 ∞ ∞ Combined standard uncertainty The combined standard uncertainty is then given by 2 2 uc (PDWT ) = u A + u B (C.00194 0.30) Hence.33) = (0.Type B Evaluation of Standard uncertainty From the specifications of the standard piston cylinder assembly. uB = a 0.

00602 MPa ≈ Reporting of results For the range 0.00 × 0.0 for a confidence level of approximately 95.012 MPa which is approximately 0.001527)4 + (0.0057)4 4 19 ∞ ≈ Expanded uncertainty ∞ (C.45 %. Therefore. the expanded uncertainty is given by U= k × uc (PDWT) = 2. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.02 % of the full scale pressure.Effective Degree of freedom (νeff) The effective degree of freedom of the combined standard uncertainty is given by ν eff = (uc )4 4 4 u14 u2 u B + + v1 v1 ∞ = (0.012 MPa National Accreditation Board for Testing and Calibration Laboratories Doc.34) Using the student’s t-distribution table. 0. This is determined from a combined standard uncertainty uc = 0.00 based on students distribution for ν = ∞ degrees of freedom and estimated to have a level of confidence of 95.00602)4 (0. k = 2.60 MPa.00 Amend No: 00 Amend Date: Page No: 46/ 70 .04. the uncertainty U is ± 0.001199)4 + (0.00602 MPa and a coverage factor k = 2.45%.

= Φ s × E s ⎡ E AS ⎤ × ET ⎢ E AT ⎥ ⎣ ⎦ (C. the indirect illuminance EAT is measured. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02. 7. It should remain switched on always to avoid warm up period. After burning –in period. The indirect illuminance EAS is measured. The switched on auxiliary lamp is moved into the sphere. the indirect illuminance from the auxiliary lamp EAS and EAT with standard lamp inside the integrating sphere and test lamp inside the sphere respectively and the indirect illuminance ES and ET produced by the standard lamp and the test lamp respectively. E AS. E S. 5.36) The factor E AS E AT considers the effect of different sizes and types of test lamps and the standard lamps. 2. of the lamp can be calculated from the luminous flux ΦS of the standard lamp according to the following relation ΦT .C. Mount the standard lamp into the sphere center. can be written as ΦT. 35) The luminous flux ΦT. From the above it is clear that the luminous flux of the test lamp is a function of the luminous flux of the standard lamp ΦS. the indirect illumination by standard lamp Es is measured. a substitution method is applied in which a test lamp substitutes a luminous flux standard and the luminous flux of a test source is evaluated by comparing the indirect illuminance in the two cases. E T.00 Amend No: 00 Amend Date: Page No: 47/ 70 . The functional dependence of ΦT. The standard lamp is moved out of the sphere and the test lamp to be measured is mounted into the sphere center with the auxiliary lamp still burning.04. Switch on the measuring equipment and let the auxiliary lamp warm up for 15 minutes. 3. Turn on the test lamp to be measured. National Accreditation Board for Testing and Calibration Laboratories Doc. After burning in period the indirect illuminance ET is measured. 6. 4. = f (ΦS. Turn the supply voltage down and switch off the standard lamp. The luminous flux measurement has to be conducted as follows: 1. E AT) (C.4 Estimation of measurement uncertainty in luminous flux measurement For Luminous flux measurement of light sources with integrating sphere.

254 10.12 l m.E AS)2 400 × 10-6 9 × 10-6 484 × 10-6 196 × 10-6 25 × 10-6 16 × 10-6 100 × 10-6 81 × 10-6 484 × 10-6 441 × 10-6 S. (C. respectively . the expectation value of ES / ET and EAS / EAT will be close to 1.254 10. E AS 10.76 × = 1086.14 lux and 83.36) comes out to be Φ T = 1045 × 10.04.00 Amend No: 00 Amend Date: Page No: 48/ 70 .272 10. the value ΦT can be calculated from Eq. we will calculate value of ΦT and standard uncertainty in u (ΦT) by the following example in which the uncertainties in the measurements of EAS and EAT and ES and ET are calculated by statistical method and is an example of Type A evaluation of standard uncertainty. 7.Since the quantities on the RHS of the Eq. 10.271 10.36).285 10. 6.279 10.76 lux . the equation for the combined standard uncertainty can be expressed as an estimated relative combined variances u 2 (Φ T ) u 2 (Φ S ) u 2 (E S ) u 2 (ET ) u 2 (E AS ) u 2 (E AT ) = + + + + 2 2 2 2 2 ΦT Φ2 ES ET E AS E AT S (C.277 E AS 10. same electrical parameters and same colour temperature.38) National Accreditation Board for Testing and Calibration Laboratories Doc. from the measurement of expectation value of ES / E T and E AS / E AT. ES and ET are 10. (C. 10. if the expectation or the average values of the EAS .14 (C. No. EAT .20 lux.20 81. Hence the value of ΦT will be almost equal to ΦS and the uncertainty u (ΦT) can be calculated from Eq. for identical standard lamp and test lamp of identical shape and size . (C.36) are in product form.276 For identical standard and test lamp of identical size. Table C. 2. However.286 10. 81. (C. 1. 8. the value of ΦT from Eq.6 lm 10.290 10. shape. 9.37) If ΦS for standard source is given to be 1045 l m and the standard uncertainty is ± 9. 5.296 10.37). In the case study . 3.276 83.276 lux .8: Observations (E AS . No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02. 4.

Type A Evaluation We will illustrate by an example the calculation of the standard uncertainty u(E AS ) and relative standard uncertainty u(E AS) / E AS for one of the parameters. E S and E T. E AS by Type A evaluation.41) u E AS = ( ) 16.08 × 10 −3 10 (C.1 = 10 – 1 = 9 ES u (ET ) = 1. ∑ (E 10 i =1 AS − E AS ) 2 = 2326 × 10 −6 (C.1 = 10 – 1 = 9 ET National Accreditation Board for Testing and Calibration Laboratories Doc.00 Amend No: 00 Amend Date: Page No: 49/ 70 .6 × 10 –2 and degree of freedom in this case is = νi = n .39) Variance s (E AS 2 ( )= ∑ E 10 i =1 AS − E AS 9 ) 2 = 2326 × 10 −6 9 (C.2 × 10 –2 and degree of freedom in this case is = νi = n . The values of the relative uncertainties are u E AT E AT ( ) ( ) = 6 × 10 –4 and degree of freedom in this case is = νi = n .1 = 10 – 1 = 9 u ES = 1.04.g.08 × 10 –3 Standard deviation of the mean which is known as standard uncertainty is (C. can be evaluated by Type A evaluation.44) Similarly the relative uncertainties for E AT. The degree of freedom in each case is also 9 as the total number of observations made in each case are 10. e.40) Standard Deviation s (E AS ) = 16. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.42) The relative standard uncertainty is u E AS = 5 × 10 −4 E AS The degree of freedom in this case is νi = n – 1 = 10 – 1 = 9 ( ) (C.43) (C.08 × 10 −3 = 5.

Type B Evaluation The uncertainty in the value of ΦS is ± 9.00 Amend No: 00 Amend Date: Page No: 50/ 70 .37). (C.06 × 10-2 ΦT Effective degrees of freedom ν eff ⎡ u c (Φ T ) ⎤ ⎢ Φ ⎥ T ⎣ ⎦ = 4 4 4 ⎡ u (Φ S )⎤ ⎡ u (E S ) ⎤ ⎡ u (ET ) ⎤ ⎢ Φ ⎥ ⎢ E ⎥ ⎢ ⎥ S ⎦ ⎣ ⎣ S ⎦ + ⎣ ET ⎦ + + ∞ 9 9 4 ⎡ u (E AS )⎤ ⎢ E ⎥ ⎣ AS ⎦ + 9 4 ⎡ u (E AT ) ⎤ ⎢ E ⎥ ⎣ AT ⎦ 9 4 (C.25 × 10 −6 2 ΦT 2 (C.45) degree of freedom in this case is νi = ∞ The relative standard uncertainty is u (Φ s ) 5.12 = 5. the standard uncertainty in the value of ΦS is : u (Φ s ) = 9. Assuming rectangular distribution.3lm 3 (C.04.46) uc (Φ T ) for the value of ΦT is calculated using ΦT uc (Φ T ) = 25 × 10 −6 + 144 × 10 −6 + 256 × 10 −6 + 0. (C.47) Therefore.12 l m and is taken from the certificate of the calibration of the standard lamp.36 × 10 −6 + 0. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.48) National Accreditation Board for Testing and Calibration Laboratories Doc.3 = = 5 × 10 −3 Φs 1045 Combined standard uncertainty The value of the relative combined uncertainty Eq. uc (Φ T ) = 2.

Type A . the coverage factor k = 2.6 × 10 –2 5.Factor Standard Uncertainty u(xi ) Sensitivity Uncertainty coefficient contribution ci ui(y) Deg.0 k = 2.√10 Normal .2 × 10 –2 1.2 × 10 −3 4 −2 4 ∞ 9 [2.6 × 10 –2 5 × 10 –3 1.2 × 10 –2 1.ν eff = [5 × 10 ] + [1.Type A . Reporting results The value of φT = (1086.8 = 19 (C. the value of φT = (1086.6 × 10 –2 5 × 10 –3 1.0 1.0 9 1.6 ± 22.8) l m Table C.9: Details of the uncertainty budget Source of Uncertainty Xi Estimates xi Limits ±∆xi Probability Distribution .06 × 10 ] ] + [1.04.0 9 ∞ 19 5. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.4) l m and expanded uncertainty with coverage factor k = 2.6 × 10 ] + [6 × 10 ] + [5 × 10 ] −2 4 −2 4 −4 4 −5 4 (C.4 l m 9 1.Type A . and for νeff = 18.3 l m 22.6 ± 46.6 × 10 –3 1.√10 Rectangular .3 lm 1. for confidence level of 95 %.Type A .Type B 1.√10 Normal .8 = 19 .09 46.00 Amend No: 00 Amend Date: Page No: 51/ 70 .Type A or B .8 l m 19 National Accreditation Board for Testing and Calibration Laboratories Doc.50) Expanded uncertainty (U) From Student’s t distribution.49) 9 9 9 νeff = 18.09.√10 Normal . of freedom νi ES u(ES ) ET u(ET ) EAS u(EAS ) EAT ΦS Combined uncertainti es Expanded uncertainti es u(EAT ) u(ΦS ) uc(ΦT ) Normal .09.0 9 1.

50 C. Correction = Correction due to the digital thermometer and Type K thermocouple Uncertainty evaluation The combined standard uncertainty (uc ) includes uncertainties of the repeatability of the displayed readings. uc = combined standard uncertainty in the measurement. the readings were taken after a stabilization time of half an hour.04. u3 = standard uncertainty in the thermocouple National Accreditation Board for Testing and Calibration Laboratories Doc. Ten measurements were taken as recorded in Table (C. The temperature controller of the chamber was set at 5000C . u2 = standard uncertainty in the digital thermometer.C. u1 = standard uncertainty in the repeatability of measured readings. Digital thermometer specification Resolution: 0.51) (C.10) Mathematical Model The mathematical model is represented as follows: T = D + Correction Where T= Temperature measured. the digital thermometer and the thermocouple. The correction for the thermocouple at 500 0 C is 0. Measurement record When the temperature chamber indicator reached 500 0 C.0 0 C at confidence level of 99 %.5 Temperature measurement using thermocouple Introduction A digital thermometer with a Type K thermocouple was used to measure the temperature inside a temperature chamber.10 C Type K accuracy (one year): ± 0. The last calibration report provided an uncertainty of ± 2.52) where. D = Displayed temperature of the digital thermometer.6 0 C Thermocouple The Type K thermocouple is calibrated every year. This can be represented in the equation below: 2 2 uC = u12 + u2 + u3 (C. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.00 Amend No: 00 Amend Date: Page No: 52/ 70 .

No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.9 500.Table C.56) Thus the standard uncertainty (u1) is equal to 0.0106 0 C 2 9 (C.10) shows the data obtained from the experiment.03 0 C.54) Standard deviation s (Ti) = 0.10).9 499. Degrees of freedom (νi ) = n .1 = 10 – 1 = 9.030 C 10 n (C.53) where Ti are the 10 measurements taken as listed in Table (C.1 500. Mean value.1 500.0 500.103 0 C Standard deviation of the mean is as follows: (C.0 Analysis of measurement uncertainty components Type A evaluation (A) Standard uncertainty in the readings (u1) Table (C.096) = 0.2 499.55) u1 = s T = () s (Ti ) 0.10: Measurements record Measurement (i) 1 2 3 4 5 6 7 8 9 10 T in 0 C 500. The temperature of the chamber after taking into consideration the correction of the thermocouple is 500. T= 1 10 ∑ Ti = 500.1 499.103 0C = = 0. National Accreditation Board for Testing and Calibration Laboratories Doc. The variance is calculated as follows: s 2 (Ti ) = 1 n ∑ Ti − T n − 1 i =1 ( ) 2 = 1 (0.0 501.00 Amend No: 00 Amend Date: Page No: 53/ 70 .04.50C.02 10 i =1 (C.9 500.

Assuming rectangular distribution. the uncertainty in the thermocouple is ± 2. 6 = 0.00 Amend No: 00 Amend Date: Page No: 54/ 70 . u3 = 2.60 C.Type B evaluation Standard uncertainty (u2) From specifications. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.78)2 = 0.58) degrees of freedom (νi ) = ∞ Combined standard uncertainty The value of the combined standard uncertainty is calculated using Eq.03)4 9 =∞ (C.85 0 C (C.59) Effective degrees of freedom ν eff = (0.85)4 (0. the standard uncertainty in the digital thermometer (u2) is.0 0 C. 0 = 0.61) National Accreditation Board for Testing and Calibration Laboratories Doc.57) degree of freedom (νi ) = ∞ Standard uncertainty (u3) From calibration report.780 C 2.03)2 + (0.52) uc = (0.7 0 C (C.60) Expanded uncertainty U = k × uc = 2 × 0. with a confidence level of 99 % (k = 2.85 = 1. the uncertainty in the digital thermometer is ± 0.58 (C. u2 = 0.35 0C 3 (C.58).35)2 + (0. (C.04.

0 0. The reported measurement uncertainty is estimated at a level of confidence of approximately 95 % with a coverage factor k of 2.7 0 C. National Accreditation Board for Testing and Calibration Laboratories Doc. of freedom νi ∞ ∞ 9 ∞ ∞ 0.Type B .78 0.85 1.Table C.7 Reporting of results The temperature of the chamber was measured to be 500.Type B . No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.6 0 0 C 0. The measurement uncertainty is ± 1.2.0 0.3 2.35 Deg.Type A √10 Standard Uncertainty u(xi ) 0 C 0.Factor Rectangular .0 Probability Distribution .0 0.04.00 Amend No: 00 Amend Date: Page No: 55/ 70 .58 Normal .Type A or B .03 Combined uncertainty Expanded uncertainties uc U k=2 0.03 1.78 1.√3 Normal .35 Sensitivity Uncertainty coefficient contribution ui(y) ci 0 C 1.5 0 C.11 : Statement of the uncertainty budget Source of Uncertainty Xi Digital Thermometer Thermocoupl e Repeatability Estimates Limits xI ±∆xI C 0.

Three months’ stability data from the manufacturer’s specifications is 5.500000 V on the DMM and the emf ex indicated by the nanovoltmeter is noted.008 %. with above precautions taken. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.5 V being nearly equal to 1 µV. So.8 × 10-6 at 95 % confidence level. The polarity of the DC voltage is reversed and above process is repeated and DC calibrator output voltage V. ∆VDC is error of DC calibrator due to its three months stability from the manufacturer’s data. V DC = ν + + ν − 2 is average of two polarity DC voltage output of calibrator. Mathematical model The mathematical model used is VAC = ( VDC + ∆VDC + ∆Vth ) ( 1 + δ ) (C.62) VAC is the voltage estimated for an indicated value of 0. 2. the error contribution by above factors are very low (≤ 2to 3 × 10-6) and can be neglected. The DC calibrator is regularly calibrated at intervals of six months. The precaution is that the interconnecting leads are coaxial shielded and are kept very small. This is very small as compared to 0.0 ×10-6.is recorded.04.6 Calibration of a 6 ½ digit DMM on its 1 Volt AC range Introduction We discuss the method of calibration of a 6 ½ digit DMM on its 1 Volt AC range at a nominal 0. The AC/DC transfer uncertainty is ± 0.5 V level at 1 kHz using 0. as the calibrator was calibrated three months before and ∆Vth is error due to thermal emf which comes from the fact that the polarity of DC voltage is reversed. For the range of 1V the uncertainty in the calibrator from its calibration certificate is ± 5.00 Amend No: 00 Amend Date: Page No: 56/ 70 . δ is AC/DC transfer correction factor of the TVC at the frequency of calibration. The whole measurement process is repeated several times.500000 V on DUC. The inputs are : 1. National Accreditation Board for Testing and Calibration Laboratories Doc.C. At frequencies up to 10 kHz.01 % at 95 % confidence level.5 V calibrated thermal voltage converter (TVC) . These precautions minimize the loading as well as transmission errors. The AC /DC transfer correction factor for the thermal converter is + 0. A DC voltage of positive polarity is applied to TVC and is adjusted so as to repeat a reading of ex on the nanovoltmeter. The assumption is that the drift in the values of δ is small and is also neglected and error of DMM in 1 volt range due to ± 1 count is ± 1µV and is also neglected. finally the equation becomes VAC = VDC + ∆VDC + δVDC + δ∆VDC (C. The output of the DC calibration is noted as V+. The AC Calibrator is replaced by a calibrated DC calibrator and the DUC is disconnected. AC voltage from highly stable AC Calibrator is applied to both DMM (DUC) and the standard (TVC) connected in parallel via a coaxial switch and a Tee adaptor for an indication of 0.63) The product δ × ∆VDC is extremely small and is neglected. The reference plane of measurement (mid point of Tee) is brought close to input plane of DMM (DUC).

The observations are average of two polarity DC voltages. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.499991 0. and c1 = =1 δV DC δ∆V DC δδ (C.499994 0.12: Experimental observations Serial Number 1 2 3 4 5 Readings (V) 0.499982 0. c 2 = = 1.65) The components of total measurement uncertainty comprise of u1 (V) u2 (V) u3 (V) u4 (V) = = = = DC calibrator’s applied voltage uncertainty as mentioned in its calibration certificate DC calibrator’s uncertainty due to its stability Uncertainty in the AC/DC transfer and Uncertainty due to repeatability and the corresponding sensitivity coefficients are c1 = δV AC δV AC δV AC = 1.00 Amend No: 00 Amend Date: Page No: 57/ 70 . Table C.04.499993 Uncertainty evaluation VAC = VDC + ∆VDC + δVDC For uncorrelated input quantities.66) National Accreditation Board for Testing and Calibration Laboratories Doc.499986 0.64) ⎡ δf ⎤ u = ∑ ⎢ ⎥ u 2 ( xi ). i =1 ⎣ δxi ⎦ n 2 c 2 (C. the combined standard uncertainty is (C.

Standard deviation of mean or standard uncertainty sq = Degrees of freedom () 0. the standard uncertainty u 2 (V ) = Degrees of freedom = ∞ 3.96 1. uncertainty due to 3 months stability data a2 = ±5. For rectangular distribution.0 3 = 2.69) From DC calibrator’s specifications. distribution is normal and coverage factor = 1.0 × 10-6.96 (C.67) νi = 5 – 1 = 4 Type B evaluation 1.96× 0.Type A evaluation Mean DC Voltage = 0.23 × 10 −6V 5 (C.71) National Accreditation Board for Testing and Calibration Laboratories Doc.5 µV = 25.5 µV 1. a2 3 = 5.499989 V.44 µV (C.5 µV =1. The distribution is normal and the coverage factor for 95 % confidence level is 1. u1(V ) = Degrees of freedom = ∞ 2.96 1.5 µV = 1. Standard deviation = 0.96 Standard uncertainty u 3 (V ) = Degrees of freedom = ∞ a3 1 × 100 = = 51. (C. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.8 = = 2.70) From AC/DC transfer at 95 % confidence level a3 = 100 × 10-6.0000005 V.02 × 0.96 (C.04.68) Uncertainty of DC calibrator from its calibration certificate.000005 = 2.89 × 0.00 Amend No: 00 Amend Date: Page No: 58/ 70 . a1 5.48 µV 1.96.

75) 4 Expanded uncertainty For 95.48)4 + (1.5]4 (1.04 = 28.5 = 57µV (C.00 Amend No: 00 Amend Date: Page No: 59/ 70 .44)4 + (2.5)4 ∞ ∞ ∞ = ∞ (C.76) National Accreditation Board for Testing and Calibration Laboratories Doc.5 + (1. thus U = kuc (V) = 2 x 28.5 + u12 + u2 + u4 = 25.04.5 µV (C. 2 2 uc = 25.45 % level of confidence.72) = Effective degrees of freedom (veff) 25.44 )2 + (2.5 + 3. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.74) 4 ν eff = [28. the coverage factor k = 2.73) ν eff = [uc ]4 (u1 )4 + (u2 )4 + (u3 )4 + (u4 )4 ∞ ∞ ∞ (C.Combined standard uncertainty There is a dominant factor = 25.5 µV.23)4 + (25.48)2 + (1.23)2 (C.

44 -Type B -√3 Normal 25.0 25.48 -Type B -1.5 28.5 Repeatabilit y uc(Vac) Expanded uncertainty 28.Factor µV Normal 1.5 ± ∆ xi µV 2.13: Uncertainty Budget: Source of Uncertainty Xi Estimates Limits xi V 0.000008 ) V ± 57 µV = 0.0 2.0 50. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.499993V ± 57 µV (C.Type A or B u(xi) .Table C.04.0 1.77) National Accreditation Board for Testing and Calibration Laboratories Doc. VAC = 0.0 Reporting of result The measured average AC Voltage corresponding to 0.00 Amend No: 00 Amend Date: Page No: 60/ 70 .500000 V indicated by the DMM.5 -Type B -1.96 Rectangular 1.0 1.499989 (1 + 0.0 1.5 57.0 Sensitivity Uncertainty Degree coefficient Contribution of ci ui (y) freedom vi µV 1.48 ∞ 1.9 Standard Probability Distribution Uncertainty .5 u3 0.96 Normal -Type A k=2.44 ∞ ∞ 4 ∞ ∞ u1 u2 0.

± 2ΓGΓx for the unknown. Method of measurement Five separate measurements were taken which involved disconnection and reconnection of both the unknown sensor and the standard sensor on a power transfer system. National Accreditation Board for Testing and Calibration Laboratories Doc. monitored source of known reflection coefficient.00 Amend No: 00 Amend Date: Page No: 61/ 70 . drift in standard sensor since the last calibration. and ratio of mismatch losses. (C.78) calibration factor of the standard sensor. Γs and Γx are the reflection coefficients for source.7 Calibration of a RF Power Sensor at a Frequency of 18 GHz Introduction The measurement involves the calibration of an unknown power sensor against a standard power sensor as reference standard by substitution on a stable.80) ΓG. Mismatch uncertainty: As the source is not perfectly matched and the phase relation of the reflection coefficients of the source.79) (C. ratio of reference power source (short-term stability of 50 MHz). All measurements are made in terms of voltage ratios that are proportional to calibration factor. The calibration factor Kx of the unknown power sensor is determined by Kx Where Ks Ds δDC δM δREF = = = = = = (Ks + Ds) × δDC × δM × δREF. there is an uncertainty due to mismatch for each sensor at the calibration frequency as well as reference frequency. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02. the standard and the unknown. The corresponding limits of deviation is calculated from the well known formula: Mismatch uncertainty = Mismatch uncertainty = ± 2ΓGΓS for the standard sensor. respectively. ratio of DC voltage outputs.C. (C. The measurement is made in terms of calibration factor which for a matched source is defined as the ratio of incident power at the calibration frequency to the incident power at the reference frequency of 50 MHz under the condition that both incident powers give equal power sensor response. None of uncertainty contributions is considered to be correlated.04. the unknown and the standard sensors are not known.

9496 ≅ 0. Γs at 50 MHz = 0. at 18 GHz = 0.000 with deviations ± 0. The mean value is K x = 0.004.965 ± 0. at 18 GHz = 0. and Γ x at 50 MHz = 0.06. This uncertainty has been accounted for by adding it in quadrature with the actual measured values.02 . Uncertainty evaluation Type A evaluation The measured values of calibration factor for the unknown power sensor are shown in Table (C. Standard sensor: The standard sensor was calibrated 6 months ago. at 18 GHz = 0.For this case ΓG at 50 MHz = 0.02.14).82) (C.83) The long-term stability from the results of five annual calibrations was found to have limits not greater than ± 0.10 (C.02 .4 % per year.84) National Accreditation Board for Testing and Calibration Laboratories Doc.950.09.04. (C. The values of reflection coefficients are themselves uncertain.1 % from measurements against a reference attenuation standard up to ratios of 2:1 at confidence level of 95 %.012 at confidence level of 95 %. The value of calibration factor from its calibration certificate is 0.00 Amend No: 00 Amend Date: Page No: 62/ 70 .81) C. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02. The instrumentation linearity uncertainty has been estimated to lie within ± 0. Reference source: The ratio of power outputs of the reference source has been estimated to be 1.

002. u(Ds ) = Degrees of freedom = ∞ 3.1 = 4. u(Ks) = 0. Uncertainty reported in calibration certificate of the standard sensor = ± 0.958 0.004.012 / 1. 002 = 0 .943 s(K x ) = 0.96 = 0.951 0.012 at the confidence level of 95 %.04. and 3 (C. This is a rectangular distribution and the standard uncertainty.0061.88) Uncertainty due to the stability of 50 MHz reference source is ± 0.0057 Standard uncertainty in u (s )K x is (C.0023. (C. This is a rectangular distribution and the standard uncertainty u (δREF) u(δ REF ) = 0.950 0. Calibration Factor 0. The standard uncertainty u(Ks) is Degrees of freedom = ∞ 2.004 = 0. 0 .14: Calibration factor for the unknown sensor Number 1 2 3 4 5 The experimental standard deviation [s(Kx)].Table C.0057 = 0. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.86) Degree of freedom = 5 . 0012 .946 0.87) Uncertainty in drift in calibration factor since its last calibration is ± 0. and 3 (C.85) u( K x ) = 0. Type B evaluation 1.0025 5 (C.89) Degrees of freedom = ∞ National Accreditation Board for Testing and Calibration Laboratories Doc.00 Amend No: 00 Amend Date: Page No: 63/ 70 .

(C. and 1.0012)2 + (0.0025)2 or.0023.00056 at 50MHz 2 0.93) (C.00 Amend No: 00 Amend Date: Page No: 64/ 70 .0085 at 18GHz (C.94) Combined standard uncertainty: u2c (Kx) = u2 (Ks) + u2 (Ds) + u2 (δREF) + u2 (δ) + u2 (MSX) + u2 (Kr) u2c (Kx) = (0. Uncertainty due to the instrument linearity is ± 0. This is a normal distribution and the standard Uncertainty u (δDC) u(δ DC ) = 0.001.0061)2 + (0.0023)2 + (0.92) u( M s ) = u( M x ) = (C. Uncertainty due to mismatch: a) b) c) d) Standard sensor at 50 MHz = ± 0.95) u c (K x ) ≈ 0.91) u(M x ) = (C.0076 at 18GHz 2 0.012 This is U-shaped and the corresponding associated standard uncertainty figures are: u(M s ) = 0.90) Degrees of freedom = ∞ 5.0008 = 0.4. veff is estimated and is found to be approximately 3301 or ∞.0108 = 0.0076)2 + (0.001 = 0.96) From Welch-Satterthwaite formula.0008 unknown sensor at 50 MHz = ± 0.0085)2 + (0.0008 Standard sensor at 18 GHz = ± 0.012 2 = 0.0134 Effective degrees of freedom νeff (C. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.0056)2 + (0.00056 at 50MHz 2 0.04.0005)2 + 2(0.0108 unknown sensor at 18 GHz = ± 0. National Accreditation Board for Testing and Calibration Laboratories Doc.0008 = 0.96 (C.

0061 0.96 0.00056 0.004 Normal -Type B Rectangular -Type B Normal -Type B Rectangular -Type B 0.0012 0.001 0.0085 0.0108 0.0 0.Table C.0005 0.012 0.0085 0.00056 0.0 1. (C. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.0061 0.027 Reported Result: The calibration factor of unknown power sensor at 18 GHz is 0.0268 ≈ 0.0076 0.450 ± 0.002 0.965 0.0 0.002 1.0023 ∞ ∞ ∞ ∞ 0.15: Uncertainty Budget: Source of Uncertainty Xi Estimates Limits xi ± ∆ xi Probability Distribution .0263 3301 ∞ Expanded uncertainty U (Kx) U (Kx) = 0.0 0.0 1.0008 -U shaped -do- 0.0005 0.0 1.96.0 0.0 1.0012 0.0 1.0025 0. which for a normal distribution corresponds to a coverage probability or confidence level of 95 %.00 Amend No: 00 Amend Date: Page No: 65/ 70 .00056 1.0076 0.0025 1.0023 1.0 0.012 -do-do-Type B Normal -Type A √5 k = 1.027 The reported expanded uncertainty of measurement is stated as the combined standard uncertainty multiplied by the coverage factor 1.0134 × 2 = 0.Type A or B Standard Sensitivity Uncertainty Coefficient u(xi) ci Uncertainty Degree of Contribution Freedom ui (y) vi Ks Ds δDC δREF Mismatch at 50 MHz ΓS ΓX Mismatch at 18 GHz ΓS ΓX Repeatabilit y uc(Kx) Expanded uncertainty 0.0 0.0 1.0 0.0008 0.0134 0.04.0 1.00056 ∞ ∞ ∞ ∞ 4 0.0 0.97) National Accreditation Board for Testing and Calibration Laboratories Doc.

C.8 Calibration of 4 ½ digital multimeter for its 100 V range
Introduction We discuss the method of calibration of a 4 ½ digital multimeter for its 100 V range with the application of 10 V from calibrated direct volt calibrator. Mathematical model The mathematical model used for the evaluation: Vx = Vs + ∆Vx Where Vx = Vs = Vs = (C.98)

Voltage indicated in the DMM Voltage applied from the calibrator Error of the multimeter

Some simplifying assumptions have been made: (i) (ii) errors due to loading and wire leads or connections are considered negligible, and all input quantities are uncorrelated.

We have the following inputs: (A) The calibrator is calibrated regularly at the interval of six months. For the range of 10 V, the specifications are: Resolution = 10 µV with the uncertainty at 99 % level of confidence as (4.5 × 10-6 of output + 100 µV), Vx = Vs + ∆Vx (B) (C.99)

4½ digital multimeter specification are: for range 100 V full display is 99.99 with resolution = 10 mV and the uncertainty at 99 % level of confidence as ± (10-5 of reading + 0.2 × 10-5 of full scale).

Observation: Applied voltage from calibrator 10.00000 V Indicated voltage in DMM 10.01 V

Even after repetition of the readings the multimeter reading indicating the same value or ± 1 due to digitizing process. This is due to better accuracy of the reference standard (calibrator).

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Combined standard uncertainty For uncorrelated input quantities, the combined standard uncertainty is

⎡⎛ δf u = ∑ ⎢⎜ ⎜ i =1 ⎣⎝ δxi
N
2 c

⎞⎤ 2 ⎟⎥ u ( xi ) ⎟ ⎠⎦

2

(C.100)

The components of the total measurement uncertainty consist of u1 (V) = the calibrator’s applied voltage uncertainty u2 (V) = multimeter’s random effect uncertainty Corresponding sensitivity coefficients are:

c1 =

∂V x ∂V x = 1 and c2 = =1 ∂Vs ∂∆Vx

(C.101)

Evaluation of uncertainty components:

Type A evaluation: Even after repetition of the readings the multimeter reading indicating the same value or ±1 due to digitizing process. This is due to better accuracy of the reference standard (calibrator). In this case, type A uncertainty can be assumed as negligible and the repeatability uncertainty can be treated as type B uncertainty using the resolution error of the multimeter.

Type B evaluation: (I) The uncertainty in applied voltage from the calibrator is a1 = 4.5 × 10-6 OF OUTPUT + 100 µV = (4.5 × 10 + 100) µV = 145 µV (C. 102)

At 99 % confidence level assuming normal distribution, coverage factor k = 2.58, the standard uncertainty in applied voltage is u1 (V) = a1/2.58 = (145 / 2.58) µV = 56.20 µV (C.103) Degree of freedom is = v1 = ∞ (C.104)

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(II)

From multimeter specification in the 100 V range, resolution is 10 mV (i.e. 1 count). Since the reading was unchanged and assuming the limit to be half the counts.

a2 =

10 mV = 5mV 2

(C.105)

For rectangular distribution, the standard uncertainty due to the resolution uncertainty of the multimeter is:

u 2V =

a2 3

=

5 3

mV = 2886.75 V

(C.106)

Degree of freedom is = v2 = ∞ Combined standard uncertainty
2 uc (V ) = u12 (V ) + u2 (V ) = 2.89mV

(C.107)

(C.108)

Effective degree of freedom veff Expanded uncertainty (U)

= ∞

as v1 = ∞ and

v2 = ∞

For 95.45 % level of confidence the coverage factor, k = 2, thus U = kuc (V) = 2 x 2.89 mV = 5.78 mV Result: The measured average voltage of the unknown cell is 10.01 V ± 5.78 mV. 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 95.45 %

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Doc. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.04.00 Amend No: 00 Amend Date: Page No: 68/ 70

04.2.25 µV 1.00 V Normal – Type B . No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02.5 56.89 mV 5.Type A 0 0 0 uc(Vx) Expanded Uncertainty k=2 2.75 µV ∞ Repeatabilit y Normal .2 µV 1.type A or B Standard Uncertainty u(xi) Sensitivity coefficient ci Uncertainty contribution ui (y) Degree of Freedom vi Vs 10.0 2886.01 V 5 mV Rectangular .0 56.00 Amend No: 00 Amend Date: Page No: 69/ 70 .Type B -√3 2886. 16: Uncertainty Budget: Source of Uncertainty Xi Estimates xi Limits ± ∆ xi Probability Distribution .Table C.78 mV ∞ ∞ National Accreditation Board for Testing and Calibration Laboratories Doc.2 µV ∞ ∆ Vx 0.

V. Gupta (NPL. Kandpal (NPL. M. No: NABL 141 Issue No: 02 Guidelines for Estimation and Expression of Uncertainty in Measurement Issue Date: 02. New Delhi) Dr.04. New Delhi) Dr. Chakrabarty (NABL.00 Amend No: 00 Amend Date: Page No: 70/ 70 .C.C. Ojha (NPL.K.CORE GROUP Dr. New Delhi) Dr.P. New Delhi) Dr. New Delhi) .N. P.Convenor National Accreditation Board for Testing and Calibration Laboratories Doc. New Delhi) Dr.K. H. New Delhi) Dr. Rustagi (NPL. Calcutta) – Chairman Dr. S. Rina Sharma (NPL. Bandyopadhyay (NPL. Mukherjee (University Of Calcutta. A. A. Gambhir (BIS. V.K. New Delhi) Dr.K.

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