Practical Statistics for the Analytical  Scientist
A Bench Guide
2nd Edition
Chapter 1 Introduction – Choosing the Correct Statistics 1
1.1 Introduction 1
1.2 Choosing the Right Statistical Procedures 2
1.2.1 Planning Experiments 2
1.2.2 Representative Test Portions for Large Samples 2
1.2.3 Reviewing and Checking Data 2
1.2.4 Reporting Results – Summarising and Describing Data 3
1.2.5 Decisions About Differences and Limits 3
1.2.6 Calibrating Instruments 4
1.2.7 Describing Analytical Method Performance 4
1.2.8 Analytical Method Validation 5
1.2.9 Analytical Quality Control 5
1.2.10 Testing Laboratory Performance – Proficiency Testing 6
1.2.11 Measurement Uncertainty 6
Chapter 2 Graphical Methods 7
2.1 Some Example Data 7
2.2 Dot Plots 7
2.3 Stem-and-Leaf Plots 8
2.4 Tally Charts 9
2.5 Histograms 9
2.6 Frequency Polygon 10
2.7 Cumulative Distribution 10
2.8 Box Plots 11
2.9 Scatter Plots 12
2.10 Normal Probability Plots 13
Chapter 3 Distributions and Types of Data 16
3.1 Introduction 16
3.2 Describing Distributions 16
Practical Statistics for the Analytical Scientist: A Bench Guide, 2nd Edition
Stephen L R Ellison, Vicki J Barwick, Trevor J Duguid Farrant
r LGC Limited 2009
Published by the Royal Society of Chemistry, www.rsc.org
ix
3.3 Distributions of Analytical Data 17
3.3.1 The Normal Distribution 17
3.3.2 The Lognormal Distribution 20
3.3.3 Poisson and Binomial Distributions 20
3.4 Distributions Derived from the Normal Distribution 20
3.4.1 Distribution of Student’s t 20
3.4.2 The Chi-squared Distribution 21
3.4.3 The F Distribution 22
3.5 Other Distributions 22
3.5.1 Rectangular Distribution (or Uniform Distribution) 22
3.5.2 Triangular Distribution 22
3.6 Populations and Samples 23
3.7 Checking Normality 23
3.8 Types of Data 24
References 24
Chapter 4 Basic Statistical Techniques 25
4.1 Summarising Data: Descriptive Statistics 25
4.1.1 Introduction 25
4.1.2 Counts, Frequencies and Degrees of Freedom 25
4.1.3 Measures of Location 26
4.1.4 Measures of Dispersion 27
4.1.5 Skewness 29
4.1.6 Kurtosis 29
4.2 Significance Testing 29
4.2.1 Introduction 29
4.2.2 A Procedure for Significance Testing 30
4.2.3 Tests on One or Two Mean Values – Student’s t-Test 34
4.2.4 Comparing Two Observed Standard Deviations or Variances – the
F-Test 42
4.2.5 Comparing Observed Standard Deviation or Variance with an
Expected or Required Standard Deviation Using Tables for the F
Distribution 45
4.3 Confidence Intervals for Mean Values 46
Chapter 5 Outliers in Analytical Data 48
5.1 Introduction 48
5.2 Outlier Tests 48
5.2.1 The Purpose of Outlier Tests 48
5.2.2 Action on Detecting Outliers 49
5.2.3 The Dixon Tests 49
5.2.4 The Grubbs Tests 51
5.2.5 The Cochran Test 53
5.3 Robust Statistics 53
5.3.1 Introduction 53
5.3.2 Robust Estimators for Population Means 54
5.3.3 Robust Estimates of Standard Deviation 55
x Contents
5.4 When to Use Robust Estimators 57
References 58
Chapter 6 Analysis of Variance 59
6.1 Introduction 59
6.2 Interpretation of ANOVA Tables 60
6.2.1 Anatomy of an ANOVA Table 60
6.2.2 Interpretation of ANOVA Results 62
6.3 One-way ANOVA 63
6.3.1 Data for One-way ANOVA 63
6.3.2 Calculations for One-way ANOVA 63
6.4 Two-factor ANOVA 65
6.4.1 Applications of Two-factor ANOVA 65
6.5 Two-factor ANOVA With Cross-classification 66
6.5.1 Two-factor ANOVA for Cross-classification Without Replication 66
6.5.2 Two-factor ANOVA for Cross-classification With Replication 69
6.6 Two-factor ANOVA for Nested Designs (Hierarchical Classification) 76
6.6.1 Data for Two-factor ANOVA for Nested Designs 76
6.6.2 Results Table for Two-factor ANOVA for Nested Designs 76
6.6.3 Variance Components 77
6.6.4 F-Tests for Two-factor ANOVA on Nested Designs 77
6.7 Checking Assumptions for ANOVA 79
6.7.1 Checking Normality 79
6.7.2 Checking Homogeneity of Variance – Levene’s Test 80
6.8 Missing Data in ANOVA 81
Appendix: Manual Calculations for ANOVA 82
Chapter 7 Regression 92
7.1 Linear Regression 92
7.1.1 Introduction to Linear Regression 92
7.1.2 Assumptions in Linear Regression 92
7.1.3 Visual Examination of Regression Data 93
7.1.4 Calculating the Gradient and Intercept 94
7.1.5 Inspecting the Residuals 97
7.1.6 The Correlation Coefficient 98
7.1.7 Uncertainty in Predicted Values of x 99
7.1.8 Interpreting Regression Statistics from Software 100
7.1.9 Testing for Non-linearity 102
7.1.10 Designing Linear Calibration Experiments 103
7.1.11 Two Common Mistakes 107
7.2 Polynomial Regression 108
7.2.1 Polynomial Curves and Non-linearity 108
7.2.2 Fitting a Quadratic (Second-order Polynomial) 109
7.2.3 Using Polynomial Regression for Checking Linearity 109
Appendix: Calculations for Polynomial Regression 109
References 113
Contents xi
Chapter 8 Designing Effective Experiments 114
8.1 Some New Terminology 114
8.2 Planning for Statistical Analysis 115
8.2.1 Measuring the Right Effect 115
8.2.2 Single- Versus Multi-factor Experiments 115
8.3 General Principles 115
8.4 Basic Experimental Designs for Analytical Science 116
8.4.1 Simple Replication 116
8.4.2 Linear Calibration Designs 116
8.4.3 Nested Designs 118
8.4.4 Factorial Designs 118
8.5 Number of Samples 119
8.5.1 Number of Samples for a Desired Standard Deviation of
the Mean 119
8.5.2 Number of Samples for a Given Confidence Interval
Width 120
8.5.3 Number of Samples for a Desired t-Test Power 122
8.5.4 Number of Observations for Other Applications and Tests 124
8.6 Controlling Nuisance Effects 124
8.6.1 Randomisation 125
8.6.2 Pairing 128
8.6.3 Blocked Designs 129
8.6.4 Latin Square and Related Designs 131
8.6.5 Validating Experimental Designs 133
8.7 Advanced Experimental Designs 133
8.7.1 Fractional Factorial Designs 133
8.7.2 Optimisation Designs 135
8.7.3 Mixture Designs 137
8.7.4 D-optimal Designs 138
8.7.5 Advanced Blocking Strategies 139
Appendix: Calculations for a Simple Blocked Experiment 140
References 143
Chapter 9 Validation and Method Performance 144
9.1 Introduction 144
9.2 Assessing Precision 145
9.2.1 Types of Precision Estimate 146
9.2.2 Experimental Designs for Evaluating Precision 146
9.2.3 Precision Limits 149
9.2.4 Statistical Evaluation of Precision Estimates 149
9.3 Assessing Bias 150
9.3.1 Statistical Evaluation of Bias Data 151
9.4 Accuracy 152
9.5 Capability of Detection 153
9.5.1 Limit of Detection 153
9.5.2 Limit of Quantitation 155
9.6 Linearity and Working Range 156
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9.7 Ruggedness 157
9.7.1 Planning a Ruggedness Study 158
9.7.2 Evaluating Data from a Ruggedness Study 158
References 159
Chapter 10 Measurement Uncertainty 161
10.1 Definitions and Terminology 161
10.2 Principles of the ISO Guide to the Expression of Uncertainty in
Measurement 162
10.2.1 Steps in Uncertainty Assessment 162
10.2.2 Specifying the Measurand 162
10.2.3 Identifying Sources of Uncertainty – the Measurement Equation 162
10.2.4 Obtaining Standard Uncertainties for Each Source of
Uncertainty 164
10.2.5 Converting Uncertainties in Influence Quantities to
Uncertainties in the Analytical Result 165
10.2.6 Combining Standard Uncertainties and ‘Propagation of
Uncertainty’ 166
10.2.7 Reporting Measurement Uncertainty 168
10.3 Practical Implementation 168
10.3.1 Using a Spreadsheet to Calculate Combined Uncertainty 168
10.3.2 Alternative Approaches to Uncertainty Evaluation – Using
Reproducibility Data 170
10.4 A Basic Methodology for Uncertainty Estimation in Analytical Science 170
References 171
Chapter 11 Analytical Quality Control 173
11.1 Introduction 173
11.2 Shewhart Charts 173
11.2.1 Constructing a Shewhart Chart 173
11.2.2 Shewhart Decision Rules 175
11.3 CuSum Charts 175
11.3.1 Constructing a CuSum Chart 175
11.3.2 CuSum Decision Rules 177
References 178
Chapter 12 Proficiency Testing 180
12.1 Introduction 180
12.2 Calculation of Common Proficiency Testing Scores 180
12.2.1 Setting the Assigned Value and the Standard Deviation for
Proficiency Assessment 180
12.2.2 Scoring PT Results 184
12.3 Interpreting and Acting on Proficiency Test Results 185
12.4 Monitoring Laboratory Performance – Cumulative Scores 187
12.5 Ranking Laboratories in Proficiency Tests 188
References 188
Contents xiii
Chapter 13 Simple Sampling Strategies 189
13.1 Introduction 189
13.2 Nomenclature 189
13.3 Principles of Sampling 190
13.3.1 Randomisation 190
13.3.2 Representative Samples 190
13.3.3 Composite Samples 190
13.4 Sampling Strategies 191
13.4.1 Simple Random Sampling 191
13.4.2 Stratified Random Sampling 192
13.4.3 Systematic Sampling 194
13.4.4 Cluster and Multi-stage Sampling 195
13.4.5 Quota Sampling 197
13.4.6 Sequential Sampling 197
13.4.7 Judgement Sampling 198
13.4.8 Convenience Sampling 198
13.4.9 Sampling in Two Dimensions 199
13.5 Uncertainties Associated with Sampling 201
13.6 Conclusion 201
References 201
Appendices 203
Appendix A Statistical Tables 205
Appendix B Symbols, Abbreviations and Notation 216
Appendix C Questions and Solutions 220
Questions 220
Solutions 234
Subject Index 263