Quantitative Geography: The Basics
Publication Year: 2016
DOI: http://dx.doi.org/10.4135/9781473920446
Subject: Quantitative Methods in Geography, Research Methods for Geography, Quantitative/Statistical Research
 Chapters
 Front Matter
 Back Matter
 Subject Index

Part 1: ABOUT QUANTITATIVE GEOGRAPHY
Part 2: FOUNDATIONS OF QUANTITATIVE GEOGRAPHY
 Chapter 3: Principles of Statistics (or, How Statistics Work)
 Chapter 4: Some Maths and Notation
 Chapter 5: Descriptive and Inferential Statistics
 Chapter 6: Statistical Testing, Statistical Significance and Why They Are Contentious
Part 3: DOING QUANTITATIVE GEOGRAPHY

Copyright
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© Richard Harris 2016
First published 2016
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act, 1988, this publication may be reproduced, stored or transmitted in any form, or by any means, only with the prior permission in writing of the publishers, or in the case of reprographic reproduction, in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers.
Library of Congress Control Number: 2015958574
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ISBN 9781446296530
ISBN 9781446296547 (pbk)
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Acknowledgements
To my family, Rabbits included.
List of Figures
 1.1 Government gross debt as a percentage of GDP for selected European countries, 2000–13 4
 1.2 Some of the components that make up quantitative geography 10
 1.3 House prices by neighbourhood in London 11
 2.1 Showing differences between schools on a league table measure of ‘valueadded’ 19
 2.2 The number of suicides has increased at the same time as US spending on science, space and technology, but are they really directly related to each other? 21
 2.3 Exploring the relationship between foreign aid and economic growth in Africa 22
 2.4 Be wary of annotated graphs with attentiongrabbing headlines 23
 2.5 Two views of London and South East England (left). The decline of the White British population in the capital (right). Nearly all places have become more ethnically mixed 24
 2.6 A graph from Thomas Piketty’s Capital in the TwentyFirst Century 26
 2.7 A second graph from Thomas Piketty’s Capital in the TwentyFirst Century 27
 3.1 Estimated maternal mortality rate (maternal deaths per 100,000 live births) in Sweden, UK and USA 41
 3.2 Average selling price in London’s neighbourhoods, 2011–13 45
 4.1 A set of values, collectively represented by the notation X (left). The magnitudes of the same values, X (right) 50
 4.2 The same set of values shown in Figure 4.1 (left). The values squared (right) 55
 4.3 An example of an unpaid debt growing exponentially and at a constant rate from one month to the next 57
 4.4 A line of best fit showing that there is, on average, a greater percentage of households exhibiting deprivation in London’s neighbourhoods where a greater percentage of the population are from a nonWhite British background 59
 4.5 The quadratic relationship between Y and X produces a curve, the gradient of which changes at different points along it 61
 4.6 The route of my journey from home to work: the actual route and the shortest (straightline) distance 62
 4.7 Showing the greatcircle distance from Heathrow to JFK airports 63
 4.8 The Australian states and territories. The map shows which locations share a border and is used to generate matrix, W, in equation (4.43) 72
 5.1 The mean average is like a pivot point balancing the lower and higher numbers 77
 5.2 Frequency of responses to the World Values Survey question about whether the state making incomes equal is an essential characteristic of democracy (1 = not at all essential; 10 = definitely essential) 78
 5.3 Percentage of the population estimated to be living below the poverty line in southern Californian counties, 2013 81
 5.4 Distances travelled from home to school in London in 2011–12 82
 5.5 Box plots indicating and comparing the distances travelled from home to school in London and the rest of South East England 84
 5.6 For normally distributed data, 95 per cent of the observations are within 1.96 standard deviations of the mean, and 99 per cent are within 2.58 standard deviations 87
 5.7 Showing the number of ways n = 20 coins can land with k of them heads side up 88
 5.8 Examples of negatively skewed data (left, where the mean is less than the median) and positively skewed data (right, the mean is greater than the median) 89
 5.9 Applying a mathematical transformation has normalised the distance to school data 90
 5.10 The central limit theorem and the law of large numbers ensure that, with a sufficiently large sample, the sampling distribution of the mean is approximately normal and centred very close to the true or expected value 91
 5.11 The t distribution is ‘fatter’ than the normal distribution and leads to wider confidence intervals, although as the sample size and degrees of freedom increase, the two distributions become increasingly similar 94
 6.1 The probability of exceeding our test score of 6.35 is, in principle, p = 0.012. Since this is less than p = 0.05, the result conventionally is regarded as statistically significant, at a 95 per cent confidence level 102
 6.2 The cutoff for statistical significance is a function not only of the test score but also of the degrees of freedom (df) for the test. With one or two degrees of freedom, a test score of 6.38 is significant at a 95 per cent confidence (because p < 0.05) but not with three or more degrees of freedom 104
 6.3 The only difference between data set A and data set B is that all the values in B are 0.1 greater than their equivalent value in A 108
 7.1 Pie charts are a bad way of displaying information. In this case, the information is the number of passengers at the world’s five busiest airports in 2013 116
 7.2 Adding annotation can make the pie chart more interpretable 117
 7.3 A doughnut chart is just a pie chart with a hole in the middle so faces all the same criticisms that a pie chart does 118
 7.4 A 3D pie chart and an exploding 3D pie chart 118
 7.5 Bar plots of airport passenger numbers data. The righthand plot does not rely on the software’s default settings and is the better of the two in my opinion 119
 7.6 Starting the vertical axis at values other than zero gives a deceptive impression of the numbers of passengers in each airport relative to each other 120
 7.7 Bar plots can be easier to read when drawn horizontally, but a more effective way of presenting the same information is the dot chart (right) 120
 7.8 Stacked bar plots are hard to interpret because it is difficult to work out the length of each part of the stack 121
 7.9 A simple infographic of the airport passenger data 124
 7.10 An infographic produced by the Guardian newspaper in association with the children’s charity Barnado’s 125
 7.11 Word clouds showing the frequency of words appearing in tweets referencing various news channels: @BBCNews, @CNN, @FoxNews and @AJEnglish. Can you identify which is which? 127
 7.12 Population pyramids showing the age distributions of some of the ethnic groups recorded in the 2011 Census of England. The median age is indicated by the black line 128
 7.13 Showing various ways of portraying the spread of electricity prices across European countries 130
 7.14 Using scatter plots to explore the relationship between electricity and gas prices in European countries in 2014: (a) is the basic scatter plot; (b) adds a regression line and the 95 per cent confidence interval around it; (c) includes part of a bean plot; (d) employs hexagonal binning 132
 7.15 Methods to deal with overplotting, which occurs with larger data sets: (a) uses a smaller plotting symbol; (b) uses hexagonal binning; (c) uses transparency; (d) uses kernel density estimation 134
 7.16 A matrix of scatter plots, exploring the pairwise relationship between three different variables 135
 7.17 A threedimensional scatter plot. Although it indicates the relationship between deprivation, car/van nonownership and unemployment, it is not a straightforward graph to interpret 136
 7.18 Using colour and symbol size to represent a third variable on a scatter plot 137
 7.19 A coplot exploring the relationship between the percentage of households with neither a car nor a van and the percentage of the economically active population unemployed, conditional on the percentage of households exhibiting three or four of the dimensions of deprivation 139
 7.20 A coplot with two conditioning variables, which, in this example, are geographical coordinates producing what is, in effect, a map indicating the geographically varying relationship between car/van nonownership and unemployment 140
 7.21 A parallel coordinates plot 141
 7.22 Radar charts can be used to illustrate a multivariate profile of different neighbourhoods and how they compare or differ 142
 8.1 A map showing the ethnic composition of London’s boroughs in 2011 146
 8.2 A raster representation of the Census population of Northern Ireland in 2001 (top). A vector representation (bottom) 147
 8.3 Showing the areas in and around the city of Belfast: in raster format (top); in vector format (bottom) 148
 8.4 Digital elevation model showing the height of the Rocky Mountains through the United States 150
 8.5 Examples of raster methods of analysis 151
 8.6 Illustrating a moving window operation 152
 8.7 An example of map algebra. In this case, the number of persons per household is calculated by dividing the raster count of population by the raster count of households 153
 8.8 The results of a leastcost procedure, finding a flat route from A to B on the map 154
 8.9 Three layers of vector data showing the area around the Brandenburg Gate 156
 8.10 Examples of vector analysis, used to model the core catchment area of a school 158
 8.11 The union and intersection of two vector layers 160
 8.12 A map designed to support political campaigning 161
 8.13 Changing the map classes changes our understanding of how concentrated or dispersed the nonWhite British groups were in England according to the 2011 Census data 163
 8.14 Showing the map classes used in Figure 8.13 and how they relate to the distribution of the variable percentage of residents not White British 164
 8.15 Showing the local authorities ranked by the percentage of their population that is not White British (highest rank = highest percentage) 165
 8.16 Showing the use of cartograms as well as a map insert to map the Census data 166
 9.1 Showing the negative relationship between the neighbourhood average house price and the unemployment rate 170
 9.2 Showing various values of the Pearson correlation coefficient, r 172
 9.3 Examples of nonlinear relationships 175
 9.4 The relationship between the average neighbourhood house price and the unemployment rate appears not to be linear 176
 9.5 Transforming the data can work to straighten the relationship 177
 9.6 Showing positive and negative regression residuals 183
 9.7 Visual tests of the regression assumption, N(0, σ2) 188
 9.8 Illustrating two examples of an extreme residual. Only one is influential on the regression line and is a leverage point 190
 10.1 From a geographical perspective, regression analysis can be understood as trying to explain the map of the Y variable in relation to one or more X variables 194
 10.2 Visual diagnostics for the simple bivariate model 200
 10.3 Visual diagnostics for the multiple regression model 201
 10.4 Showing how the measure of spatial autocorrelation is scaledependent: patterns of similarity and difference can coexist in a study region 203
 10.5 A partial regression plot exploring the relationship between neighbourhood average house price and the crime rate 204
 10.6 Showing the spatial multiplier: the expected decrease in the neighbourhood average house price as a consequence of a oneunit increase in the unemployment rate at A or B 215
 11.1 Concentration profile showing the proportion of London pupils living within given distances of each other 220
 11.2 Using kernel density estimation to map the places where school children of each ethnic group are most prevalent in London 221
 11.3 A second concentration profile showing what proportion of each ethnic group’s school age population can be found in a given proportion of the areas shown on the left 221
 11.4 Incidents of crime in Baton Rouge, LA, in 2014 222
 11.5 Quadrat analysis of the homicides data 223
 11.6 An example of a Poisson distribution 224
 11.7 Applying the Gfunction to the crime data 225
 11.8 Showing the distance between consecutive crime events of the same type as each other 226
 11.9 Moran plot of the neighbourhood average house price in Birmingham, England 227
 11.10 Map of the neighbourhood average house price on a log scale 229
 11.11 The local Moran values 230
 11.12 Showing hot spots and cold spots in the neighbourhood average house price 231
 11.13 G*statistics, expressed as standardised zvalues, calculated for the homicide data and at varying distance thresholds 232
 11.14 A variogram, which can be used to help decide which is the best distance threshold to use with the homicide data and with the local statistics 233
 11.15 Calculating locally varying means from the neighbourhood house price data 235
 11.16 A connectivity graph showing which places are treated as neighbours in the calculation of the locally varying means 236
 11.17 Examples of different types of inverse distance weighting 237
 11.18 Using geographically weighted averaging for interpolation 239
 11.19 Simulated land price data for Beijing (log of renminbi per square metre) 240
 11.20 The distribution of the local beta values estimated by GWR for the distance to an elementary school variable 241
 11.21 Map of the local beta values for the distance to an elementary school variable 242
 11.22 Estimates of the districtlevel variations in land parcel prices, having controlled for the X variables 244
 11.23 A caterpillar plot of the districtlevel residuals 245
 11.24 The districtlevel variations in the effect of distance to an elementary school on land prices, estimated as a random slope model (left). The GWR estimates (right) 246
 12.1 Choropleth map of the neighbourhood average house prices in Birmingham 263
 12.2 Local Moran values 265
 12.3 The geographically weighted mean (left) and standard deviation (right) of the house price data 266
 12.4 GWR estimates of the effects of (left) the unemployment rate on the log of the average house price and (right) the proportion of properties sold that were detached 268
List of Tables
 3.1 The ten most populous countries in 2014 according to the US Census Bureau 32
 4.1 Example of writing numbers using scientific notation 55
 4.2 The first eight entries in the distance log 64
 4.3 An extract from the table of data used to produce Figure 4.4 66
 4.4 Average speed and time taken between points on my journey to work 68
 5.1 Percentage of the population estimated to be living below the poverty line in selected Californian counties, 2013 79
 5.2 A sixnumber summary of the distance to school data, comparing London and the rest of South East England 83
 6.1 In this hypothetical example, one categorical variable (disciplinary identity) fully predicts agreement or disagreement with the statement ‘learning quantitative methods is more important for scientists than for social scientists’ 98
 6.2 In this example, the two variables are independent of one another because the percentage of scientists who agree/disagree with the statement is the same as for social scientists 98
 6.3 The actually observed values: the numbers of students who identified as scientists or social scientists and whether they agreed with the statement about learning quantitative methods 99
 6.4 The expected values if the variables were independent of one another 100
 6.5 As soon as one value in the table is changed, all the other values must ‘fall into line’ if the column and row totals are fixed 103
 6.6 The original values multiplied by 10. These lead to a chisquare statistic of 63.5 (1 df; p < 0.001) 106
 7.1 An ugly table of the airport passenger data 121
 7.2 A better tabulation of the data, with the rows ordered to emphasise the busiest airport 122
 7.3 In this example the airport with greatest growth is emphasised 122
 9.1 Showing how the correlation between unemployment rate (X) and the percentage of households with neither a car nor a van (Y) changes with the scale of analysis 179
 9.2 Typical regression output, here from a regression of neighbourhood average house price against the unemployment rate for neighbourhoods in Birmingham, England 184
 10.1 Regression output for a multiple regression model of neighbourhood average house prices 197
 10.2 Comparing measures of model fit for the simple bivariate and multiple regression models of neighbourhood average house prices 199
 10.3 Regression output for the multiple regression model with meancentred X variables 206
 10.4 Regression output for the multiple regression model, now including the square of one of the X variables and also an interaction term. The X variables are meancentred 208
 10.5 Regression output for the multiple regression model with the logarithm of the original Y variable. The X variables are meancentred 210
 10.6 The results of a logit model looking at which groups of pupils in London attended one of their three nearest secondary schools 211
 11.1 Showing the proportion (and percentage) of each ethnic group living within 5 km and 10 km of each other 219
 12.1 The additional R libraries used in the production of this book 252
About the Author
Richard Harris is professor of quantitative social geography at the School of Geographical Sciences, University of Bristol. He is the lead author on two textbooks about quantitative methods in geography and related disciplines: Statistics for Geography and Environmental Science (Prentice Hall, 2011) and Geodemographics, GIS and Neighbourhood Targeting (Wiley, 2005). Richard’s research interests include the geographies of education and the education of geographers. He is currently Director of Bristol QStep Centre, part of a multimillionpound UK initiative to raise quantitative skills training amongst social science students, and has worked with both the Royal Geographical Society (with IBG) and Higher Education Academy to promote numeracy and to support the transition of students from schools to university. The University of Bristol QStep Centre is one of 15 Centres across the UK working to promote a stepchange in quantitative social science training for undergraduates. The QStep Centres are delivering specialist undergraduate programmes, including new courses, work placements, and pathways to postgraduate study. QStep is funded by the Nuffield Foundation, the Economic and Social Research Council (ESRC) and the Higher Education Funding Council for England, and was developed as a strategic response to the shortage of quantitatively skilled social science graduates. You can find out more at www.nuffieldfoundation.org/qstep.
Preface
The plan was simple. Basic, even. I’d hide myself away in Ireland and write as much of this book as possible in one long splurge. Easy. Or maybe not: I went on the trip but now, almost two years later, I am typing these words at the weekend and on the morning of my birthday. That’s a clue to how overdue the manuscript is. So what went wrong?
I could, I suppose, blame my hosts. I’m not yet persuaded that Guinness tastes better in the Emerald Isle, but I do know it tastes better in good company. And they definitely were good company. But, no, I can’t blame others for my tardiness, so there are no excuses there.
It might be that I lack focus. That’s definitely true. Running every day for a year in support of the mental health charity, Mind, is a bit of a distraction but a healthy one, I think. Today is day 318 of that challenge.1 So far I have run 3545 km. It takes time out from my day, but I do some of my best thinking on the move. Besides, I can outrun the bus into work.
What happened was a change of context. Specifically, a renewed interest in quantitative methods in geography and in the social sciences more generally, and a concerted effort to increase numeracy and statistical literacy amongst students. Given that environment, an overly simple introduction to quantitative methods would lack ambition. It would sell the reader short and fail to convey what is a vibrant area of research and activity within geography. Consequently, the book you are reading is longer than I had intended and covers a wider range of topics, from the basic to the more advanced. I hope it is more useful as a consequence and gives a better impression of what quantitative geography is about.
My first book thanked almost everyone I had ever met. There is probably a dedication to my neighbour’s brother’s younger son’s best mate’s gerbil in there. The second was a little more constrained and dedicated to Les Hepple, a professor of geography whom I regard to this day as a model scholar and outstanding teacher. For this book, I will simply thank my family (Siân, Rhys and Timothy), my colleagues (Ron, Kelvyn, Ed, Clive, David, Winnie, Malcolm and Sean) and all the members of the running community, especially those at Emersons Green Running Club and Pomphrey Hill Parkrun who keep me broadly sane when the black dog threatens to bark.
South Gloucestershire(aged 42, and feeling it) 
Postscript
It’s 7.15 in the evening and very windy outside. It’s also a few days since I wrote the preface, so no longer my birthday. It is, however, my sister’s. She is a geography graduate too.
Being the evening, it will soon be time to put my children to bed. However, before I do so, there’s another task more pressing. It’s time to put this book to bed too. If you started at the beginning and have read this far, then you deserve a rest as well.
At the University I help to teach a course called ‘Convincing Stories? Numbers as Evidence in the Social Sciences’. The idea is to provide an introduction to quantitative social science, looking at how numbers are used and abused to create ‘stories’ in the media, public policy, and in social and scientific debate. The aim of the unit is to prepare students for the sorts of methods and techniques they will encounter in their disciplines by discussing and debating the ideas and concepts that are used to create evidence in an uncertain world, and upon which decisions are made. The unit encourages students to engage critically with research and debate in their subject areas, placing them in a better position to learn quantitative skills, to conduct research and to enhance their studies. That’s what the blurb says anyway. Sometimes we succeed (see Milligan et al., 2014).
As part of that course, I ask the participants to read a research paper I cowrote called ‘The changing interaction of ethnic and socioeconomic segregation in England and Wales, 1991–2011’ (Harris et al., 2015). The students are asked to go through the paper and to pick out all the different statistical techniques it contains and the ways they are used to tell a story. They include graphs, averages, correlations, index values and indices of segregation. I invite the students to pick holes in the paper. I pretend to have thick skin. I also ask the students to selfrate the techniques from easy to hard. Unsurprisingly, what I consider easy some regard as difficult. That is not a criticism of the students in any way. What I still grapple with, other researchers find pretty trivial. We are all at different stages in our learning.
I mention this for two reasons. First, because I am conscious that for a book about quantitative basics, some readers won’t have found it basic at all, whereas others (probably longserving professors) will mutter into their beards and wish I had gone deeper into certain topics. Both are right. It’s simply a matter of perspective. However, I hope I have provided an introduction to some core ideas, methods and concepts in quantitative geography and that, in doing so, it will have helped ‘students to engage critically with research and debate in their subject areas’. Since that subject area is geography, I can express it more plainly: I hope I have helped geographers to engage with geography (or at least an important part of it).
The second reason is that I hope the engagement will indeed be critical but in the best sense of the word – informed, not merely negative, and with the aim of improving the research, not simply criticising it. Within the geographical literature right now there is a lot of innovation and creativity in the use of quantitative methods as we move from the small samples of the past to trying to make sense of complex surveys, administrative data, big data, volunteered data and other less traditional sources of information. Data are everywhere. Some of them are useful. The challenge is to find out which.
Dennis Prager is quoted as saying, ‘our scientific age demands that we provide definitions, measurements, and statistics in order to be taken seriously. Yet most of the important things in life cannot be precisely defined or measured. Can we define or measure love, beauty, friendship, or decency, for example?’ It’s not often that I agree with a politically conservative radio talk show host, but in this instance I do. Quantitative geography is not the only thing that matters in geography, let alone in life. In the greater scheme of things, it may not matter that much at all. But it remains important precisely because definitions, measurements and statistics are taken seriously. Sometimes it is important to debunk that authority and show where the evidence is lacking. Sometimes it is important to create the evidence to better understand the world in which we live or to challenge what takes place within it – to provide evidence about the process that generate spatial inequalities and injustices, for example, or their consequences on things that matter like educational outcomes or life expectancy.
It’s easy to find quotes about statistics. Apparently Jean Baudrillard said: ‘Like dreams, statistics are a form of wish fulfilment.’ Well, maybe, but then sometimes it’s good to dream. It’s now 11pm and I risk being flippant but I would appreciate that dreaming coming sooner rather than later.
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