Note: Intro -- Measurement Uncertainties -- Preface -- Contents -- Chapter 1: Some Important Definitions -- 1.1 Introduction -- 1.2 Terms Pertaining to Quantity -- 1.2.1 Quantity -- 1.2.2 System of Base Quantities -- 1.2.3 Derived Quantity -- 1.2.4 Quantity Equation -- 1.2.5 Dimension of a Quantity -- 1.2.6 Measurand -- 1.2.7 True Value of a Quantity -- 1.2.8 Conventional True Value of a Quantity -- 1.2.9 Measured Value -- 1.2.10 Relation in Between Measured Valueand True or Conventional True Value -- 1.3 Terms Pertaining to Measurement -- 1.3.1 Measurement -- 1.3.2 Method of Measurement -- 1.3.3 Substitution Method -- 1.3.4 Differential Method -- 1.3.5 Null Method -- 1.3.6 Measurement Procedure -- 1.3.7 Result of Measurement -- 1.3.8 Error -- 1.3.9 Spurious Error -- 1.3.10 Relative Error -- 1.3.11 Random Error -- 1.3.12 Systematic Error -- 1.3.13 Accuracy of Measurement -- 1.3.14 Precision of Measurement Result -- 1.3.15 Repeatability -- 1.3.16 Reproducibility (of Measurement Results) -- 1.3.17 Correction -- 1.4 Terms Pertaining to Statistics -- 1.4.1 Observation -- 1.4.2 Independent Observations -- 1.4.3 Population -- 1.4.4 Sample -- 1.4.5 Measurement -- 1.4.6 Population of Measurement -- 1.4.7 Sample of Measurements -- 1.4.8 Frequency/Relative Frequency -- 1.4.9 Mean -- 1.4.10 Sample Mean -- 1.4.11 Population Mean -- 1.4.12 Merits and Demerits of Arithmetic Mean [3] -- 1.4.12.1 Merits -- 1.4.12.2 Demerits -- 1.4.13 Median -- 1.4.14 Quartiles -- 1.4.15 Dispersion -- 1.4.16 Standard Deviation -- 1.4.17 Variance -- 1.4.18 Sample Standard Deviation -- 1.4.19 Population Standard Deviation -- 1.4.20 Estimate of Population Standard Deviation -- 1.4.21 Estimate of Population and Sample Standard Deviations-Relation -- 1.4.22 Independent Variable -- 1.4.23 Dependent Variable or Response Variable -- 1.4.24 Correlation -- 1.4.25 Correlation Coefficient. , 1.4.26 Covariance -- 1.4.27 Random Variable -- 1.4.28 Discrete Random Variable -- 1.4.29 Continuous Random Variable -- 1.4.30 Probability -- 1.4.31 Probability Distribution -- 1.4.32 Normal Distribution -- 1.4.32.1 Alternative Definition -- 1.4.33 Properties of Normal Distribution -- 1.4.34 Probable Error -- 1.4.35 Range -- 1.4.36 Confidence Level -- 1.4.37 Confidence Interval -- 1.4.38 Outlier -- 1.4.39 Parameter -- 1.4.40 Random Selection -- 1.4.41 Sample Statistic -- 1.4.42 Error -- 1.4.43 Standard Error, or Standard Deviationof the Mean -- 1.4.44 Uncertainty -- 1.4.45 Evaluations of Uncertainty -- 1.4.46 Random Uncertainty(er) -- 1.4.47 Systematic Uncertainty(Us) -- 1.4.48 Standard Uncertainty -- 1.4.49 Expanded Uncertainty -- 1.4.50 Expressing Uncertainty of Measurement -- 1.4.51 Coverage Interval -- 1.4.52 Coverage Probability -- 1.4.53 Central Limit Theorem -- 1.5 Influence Quantity -- 1.6 Instruments and Standards -- 1.6.1 Repeatability of an Instrument -- 1.6.2 Precision of the Instrument -- 1.6.3 Accuracy of an Instrument -- 1.6.4 Accuracy of a Standard -- 1.6.5 Difference Between Uncertaintyand Accuracy -- 1.6.6 Difference Between the Correction, Errorand Uncertainty -- 1.6.7 Correction Factor -- 1.6.8 Discrimination Threshold -- 1.7 Some Special Integrals and Functions -- 1.7.1 Gamma Function -- 1.7.1.1 Gamma Probability Density Function -- 1.7.2 Beta Function of First Kind B(m,n) -- 1.7.2.1 Beta Probability Functions of First Kind -- 1.7.3 Alternative Form of Beta Function -- 1.7.4 Beta Function of Second Kind B (m,n) -- 1.7.4.1 Beta Probability Functions of Second Kind -- 1.7.5 Cauchy Distribution -- 1.7.6 Arc Sine(U-Shaped) Distribution -- References -- Chapter 2: Distribution Functions -- 2.1 Introduction -- 2.2 Random Variable -- 2.3 Discrete and Continuous Variables -- 2.4 Discrete Functions. , 2.4.1 Probability Distribution of a Random Variable -- 2.4.2 Discrete Probability Function -- 2.5 Distribution Function -- 2.5.1 Continuous Distribution Function -- 2.5.2 Discrete Distribution -- 2.6 Probability Density Function -- 2.6.1 Discrete Probability Function -- 2.7 Discrete Probability Functions -- 2.7.1 Binomial Probability Distribution -- 2.7.1.1 Probability of the Binomial Distribution -- 2.7.1.2 Moments -- 2.7.1.3 Arithmetic Mean -- 2.7.1.4 Standard Deviation -- 2.7.2 Poisson's Distribution -- 2.7.2.1 Mean of the Poisson's Distribution -- 2.7.2.2 Standard Deviation of the Poisson's Distribution -- 2.8 Continuous Probability Distributions -- 2.8.1 Normal Probability Function -- 2.8.2 Cumulative Distribution of the NormalProbability Function -- 2.8.3 Normal Distribution and Probability Tables -- 2.8.4 Mean and Variance of a Linear Combinationof Normal Variates -- 2.8.5 Standard Deviation of Mean -- 2.8.6 Deviation from the Mean -- 2.8.7 Standard Deviation of Standard Deviation -- 2.8.8 Nomenclature for Normal Distribution -- 2.8.9 Probability Function of the Ratio of TwoNormal Variates [1] -- 2.8.10 Importance of Normal Distribution -- 2.8.11 Collation of Data from VariousLaboratories [2] -- 2.8.11.1 Most Probable Mean of the Data -- 2.8.11.2 Standard Deviation of the Most Probable Mean -- References -- Chapter 3: Other Probability Functions -- 3.1 Introduction -- 3.2 Important Distributions -- 3.2.1 Rectangular Distribution -- 3.2.1.1 Mean of the Rectangular Function -- 3.2.1.2 Variance of Rectangular Function -- 3.2.2 Triangular Probability Function -- 3.2.2.1 Mean of the Triangular Probability Function Is Given as -- 3.2.2.2 Variance 2 of Triangular Distribution -- 3.2.3 Trapezoidal Probability Function -- 3.2.3.1 Mean of the Trapezoidal Distribution -- 3.2.3.2 Variance of the Trapezoidal Distribution -- 3.3 Small Sample Distributions. , 3.3.1 The Student's t Distribution -- 3.3.1.1 Mean and Variance of Student's t Function -- 3.3.1.2 Comparison of Normal and t Distributions -- 3.3.1.3 Applications of t-Statistic -- 3.3.1.4 t-Test for a Sample Mean -- 3.3.1.5 Numerical Example -- 3.3.1.6 t-Test for Difference of Two Means -- 3.3.1.7 Numerical Example -- 3.3.1.8 Assumption Made for Student's t Test -- 3.3.1.9 Paired t-Test for Difference of Means -- 3.3.1.10 Numerical Example -- 3.3.2 The 2 Distribution -- 3.3.2.1 Use of 2 Distribution to Find a Range of Standard Deviationfor Given Probability -- 3.3.3 The F-Distribution -- 3.3.3.1 Parameters of F Distribution -- 3.3.4 Upper and Lower Percentage Points -- 3.3.4.1 Notation -- 3.3.5 Application of F-Test -- 3.3.5.1 Testing for Equality of Population Variances -- 3.3.5.2 Numerical Example -- 3.3.6 For Equality of Several Means -- 3.4 Combining of Variances of Variables Following Different Probability Distribution Functions -- References -- Chapter 4: Evaluation of Measurement Data -- 4.1 Introduction -- 4.2 Evaluation of Validity of Extreme Valuesof Measurement Results -- 4.2.1 Outline (Dixon)Test -- 4.3 Evaluation of the Means Obtained from Two Setsof Measurement Results -- 4.3.1 Two Means Coming from the Same Source -- 4.3.1.1 Standard Deviation of the Two Means -- 4.3.1.2 Test for Two Means of Samples of Smaller Size -- 4.3.1.3 SD and Mean Value of Two Means -- 4.3.1.4 Numerical Example -- 4.3.2 Test for Two Means Coming from Different Sources -- 4.3.2.1 Numerical Example -- 4.4 Comparison of Variances of Two Setsof Measurement Results -- 4.4.1 Numerical Example -- 4.5 Measurements Concerning Travelling Standards -- 4.5.1 Mean and Standard Deviation for each Laboratory -- 4.5.2 Inter-Laboratories Standard Deviation -- 4.5.3 Intra-Laboratory Standard Deviation -- 4.6 F-test for Internal and External Consistency. , 4.6.1 F-test for Inter- and Intra-Laboratory Variances -- 4.6.2 Weight Factors -- 4.6.2.1 Case 1 Weight Factor Is Unity -- 4.6.2.2 Case 2 Weight Factor Other than 1 -- 4.6.3 F-test for Variances -- 4.7 Standard Error of the Overall Mean -- 4.7.1 Results Inconsistent -- 4.8 Analysis of Variance -- 4.8.1 One-Way Analysis of Variance -- 4.8.1.1 Testing the Null Hypothesis -- 4.8.1.2 Numerical Example -- 4.9 Tests for Uniformity of Variances -- 4.9.1 Bartlett's Test for Uniformity of Many Variances -- 4.9.1.1 Numerical Example -- 4.9.2 Cochran Test for Homogeneity of Variances -- 4.9.2.1 Numerical Example -- References -- Chapter 5: Propagation of Uncertainty -- 5.1 Mathematical Modelling -- 5.1.1 Mean of Measurand (Dependent Variable) -- 5.1.2 Functional Relationship and Input Quantities -- 5.1.3 Expansion of a Function -- 5.1.4 Combination of Arithmetic Means -- 5.1.5 Combination of Variances -- 5.1.6 Variance of the Mean -- 5.2 Uncertainty -- 5.2.1 Combined Standard Uncertainty -- 5.2.1.1 Measurand (Output Quantity) Is a Function of Single Input Quantity -- 5.2.1.2 Measurand (Output Quantity) Is a Function of Several Quantities -- 5.2.2 Expanded Uncertainty -- 5.3 Type A Evaluation of Uncertainty -- 5.3.1 Numerical Example for Calculation of Type A Evaluation of Standard Uncertainty -- 5.4 Pooled Variance -- 5.4.1 Validity -- 5.4.2 Applicable -- 5.4.3 Uses -- 5.4.4 Need -- 5.4.4.1 Calibration of Measuring Instruments -- 5.4.5 Calculation of Pooled Variance -- 5.4.5.1 Category I: The Variance Is Independent of the Input Quantity -- 5.4.5.2 Category II: The Variance Depends Upon Input Quantity -- 5.4.6 Uses of Pooled Variance -- 5.4.6.1 Estimation of Type A Uncertainty of an Instrument -- 5.4.6.2 Testing the Authenticity of Observations -- 5.4.6.3 Maintenance of Laboratory Instruments -- 5.4.6.4 Fixing Maximum Permissible Error of an Instrument. , 5.4.6.5 Rejection of an Instrument Received for Calibration.