Essential Mathematics and Statistics for Science

Preface xi

On-line Learning Support xv

1 Mathematics and Statistics in Science 1

1.1 Data and Information 2

1.2 Experimental Variation and Uncertainty 2

1.3 Mathematical Models in Science 4

2 Scientific Data 7

2.1 Scientific Numbers 8

2.2 Scientific Quantities 15

2.3 Chemical Quantities 20

2.4 Angular Measurements 31

3 Equations in Science 41

3.1 Basic Techniques 41

3.2 Rearranging Simple Equations 53

3.3 Symbols 63

3.4 Further Equations 68

3.5 Quadratic and Simultaneous Equations 78

4 Linear Relationships 87

4.1 Straight Line Graph 89

4.2 Linear Regression 99

4.3 Linearization 107

5 Logarithmic and Exponential Functions 113

5.1 Mathematics of e, ln and log 114

5.2 Exponential Growth and Decay 128

6 Rates of Change 145

6.1 Rate of Change 145

6.2 Differentiation 152

7 Statistics for Science 161

7.1 Analysing Replicate Data 162

7.2 Describing and Estimating 168

7.3 Frequency Statistics 176

7.4 Probability 190

7.5 Factorials, Permutations and Combinations 203

8 Distributions and Uncertainty 211

8.1 Normal Distribution 212

8.2 Uncertainties in Measurement 217

8.3 Presenting Uncertainty 224

8.4 Binomial and Poisson Distributions 230

9 Scientific Investigation 243

9.1 Scientific Systems 243

9.2 The ‘Scientific Method’ 245

9.3 Decision Making with Statistics 246

9.4 Hypothesis Testing 250

9.5 Selecting Analyses and Tests 256

10 t-tests and F-tests 261

10.1 One-sample t-tests 262

10.2 Two-sample t-tests 267

10.3 Paired t-tests 272

10.4 F-tests 274

11 ANOVA – Analysis of Variance 279

11.1 One-way ANOVA 279

11.2 Two-way ANOVA 286

11.3 Two-way ANOVA with Replication 290

11.4 ANOVA Post Hoc Testing 296

12 Non-parametric Tests for Medians 299

12.1 One-sample Wilcoxon Test 301

12.2 Two-sample Mann–Whitney U-test 305

12.3 Paired Wilcoxon Test 308

12.4 Kruskal–Wallis and Friedman Tests 311

13 Correlation and Regression 315

13.1 Linear Correlation 316

13.2 Statistics of Correlation and Regression 320

13.3 Uncertainty in Linear Calibration 324

14 Frequency and Proportion 331

14.1 Chi-squared Contingency Table 332

14.2 Goodness of Fit 340

14.3 Tests for Proportion 343

15 Experimental Design 349

15.1 Principal Techniques 349

15.2 Planning a Research Project 357

Appendix I: Microsoft Excel 359

Appendix II: Cumulative z-areas for Standard Normal Distribution 363

Appendix III: Critical Values: t-statistic and Chi-squared, χ² 365

Appendix IV: Critical F-values at 0.05 (95 %) Significance 367

Appendix V: Critical Values at 0.05 (95 %) Significance for: Pearson’s Correlation Coefficient, r, Spearman’s Rank Correlation Coefficient, r_S , and Wilcoxon Lower Limit, W_L 369

Appendix VI: Mann–Whitney Lower Limit, U_L, at 0.05 (95 %) Significance 371

Short Answers to ‘Q’ Questions 373

Index 379