1 Dealing with data 1

1.1 The role of statistics in geography 1

1.1.1 Why do geographers need to use statistics? 1

1.2 About this book 3

1.3 Data and measurement error 3

1.3.1 Types of geographical data: nominal, ordinal, interval, and ratio 3

1.3.2 Spatial data types 5

1.3.3 Measurement error, accuracy and precision 6

1.3.4 Reporting data and uncertainties 7

1.3.5 Significant figures 9

1.3.6 Scientific notation (standard form) 10

1.3.7 Calculations in scientific notation 11

Exercises 12

2 Collecting and summarizing data 13

2.1 Sampling methods 13

2.1.1 Research design 13

2.1.2 Random sampling 15

2.1.3 Systematic sampling 16

2.1.4 Stratified sampling 17

2.2 Graphical summaries 17

2.2.1 Frequency distributions and histograms 17

2.2.2 Time series plots 21

2.2.3 Scatter plots 22

2.3 Summarizing data numerically 24

2.3.1 Measures of central tendency: mean, median and mode 24

2.3.2 Mean 24

2.3.3 Median 25

2.3.4 Mode 25

2.3.5 Measures of dispersion 28

2.3.6 Variance 29

2.3.7 Standard deviation 30

2.3.8 Coefficient of variation 30

2.3.9 Skewness and kurtosis 33

Exercises 33

3 Probability and sampling distributions 37

3.1 Probability 37

3.1.1 Probability, statistics and random variables 37

3.1.2 The properties of the normal distribution 38

3.2 Probability and the normal distribution: z‐scores 39

3.3 Sampling distributions and the central limit theorem 43

Exercises 47

4 Estimating parameters with confidence intervals 49

4.1 Confidence intervals on the mean of a normal distribution: the basics 49

4.2 Confidence intervals in practice: the t‐distribution 50

4.3 Sample size 53

4.4 Confidence intervals for a proportion 53

Exercises 54

5 Comparing datasets 55

5.1 Hypothesis testing with one sample: general principles 55

5.1.1 Comparing means: one‐sample z‐test 56

5.1.2 p‐values 60

5.1.3 General procedure for hypothesis testing 61

5.2 Comparing means from small samples: one‐sample t‐test 61

5.3 Comparing proportions for one sample 63

5.4 Comparing two samples 64

5.4.1 Independent samples 64

5.4.2 Comparing means: t‐test with unknown population variances assumed equal 64

5.4.3 Comparing means: t‐test with unknown population variances assumed unequal 68

5.4.4 t‐test for use with paired samples (paired t‐test) 71

5.4.5 Comparing variances: F‐test 74

5.5 Non‐parametric hypothesis testing 75

5.5.1 Parametric and non‐parametric tests 75

5.5.2 Mann–whitney U‐test 75

Exercises 79

6 Comparing distributions: the Chi‐squared test 81

6.1 Chi‐squared test with one sample 81

6.2 Chi‐squared test for two samples 84

Exercises 87

7 Analysis of variance 89

7.1 One‐way analysis of variance 90

7.2 Assumptions and diagnostics 99

7.3 Multiple comparison tests after analysis of variance 101

7.4 Non‐parametric methods in the analysis of variance 105

7.5 Summary and further applications 106

Exercises 107

8 Correlation 109

8.1 Correlation analysis 109

8.2 Pearson’s product‐moment correlation coefficient 110

8.3 Significance tests of correlation coefficient 112

8.4 Spearman’s rank correlation coefficient 114

8.5 Correlation and causality 116

Exercises 117

9 Linear regression 121

9.1 Least‐squares linear regression 121

9.2 Scatter plots 122

9.3 Choosing the line of best fit: the ‘least‐squares’ procedure 124

9.4 Analysis of residuals 128

9.5 Assumptions and caveats with regression 130

9.6 Is the regression significant? 131

9.7 Coefficient of determination 135

9.8 Confidence intervals and hypothesis tests concerning regression parameters 137

9.8.1 Standard error of the regression parameters 137

9.8.2 Tests on the regression parameters 138

9.8.3 Confidence intervals on the regression parameters 139

9.8.4 Confidence interval about the regression line 140

9.9 Reduced major axis regression 140

9.10 Summary 142

Exercises 142

10 Spatial statistics 145

10.1 Spatial data 145

10.1.1 Types of spatial data 145

10.1.2 Spatial data structures 146

10.1.3 Map projections 149

10.2 Summarizing spatial data 157

10.2.1 Mean centre 157

10.2.2 Weighted mean centre 157

10.2.3 Density estimation 158

10.3 Identifying clusters 159

10.3.1 Quadrat test 159

10.3.2 Nearest neighbour statistics 162

10.4 Interpolation and plotting contour maps 162

10.5 Spatial relationships 163

10.5.1 Spatial autocorrelation 163

10.5.2 Join counts 164

Exercises 171

11 Time series analysis 173

11.1 Time series in geographical research 173

11.2 Analysing time series 174

11.2.1 Describing time series: definitions 174

11.2.2 Plotting time series 175

11.2.3 Decomposing time series: trends, seasonality and irregular fluctuations 179

11.2.4 Analysing trends 180

11.2.5 Removing trends (‘detrending’ data) 186

11.2.6 Quantifying seasonal variation 187

11.2.7 Autocorrelation 189

11.3 Summary 190

Exercises 190

Appendix A: Introduction to the R package 193

Appendix B: Statistical tables 205

References 241

Index 243