Quantitative Methods

Statistics

Statistical techniques can be divided in two basic types: descriptive and inferential. Descriptive statistics refer to measures that summarize data. Descriptive statistics assess the distribution, central tendency, and dispersion of data and can be used to identity a simple pattern of observed conditions. To describe the distribution of data, researchers can perform observation frequencies that detail how many observations share the same value. Measures of central tendency include the mean (or average), median, and mode. Dispersion refers to techniques that describe the overall similarity or dissimilarity of empirical observations based on an observed standard deviation, variance, or range. In addition to these measures, descriptive statistics can be expressed graphically as scatterplots, histograms, or boxplots. Finally, statistics can also refer to numerical quantities that describe either a sample or a population.

Inferential statistics are an extension of descriptive statistics and are used to make key inferences about an observed pattern within the data set. The objective of inferential statistics is to determine whether the relationship is statistically significant. To determine whether a pattern is significant, researchers may calculate estimates and/or test hypotheses. One example of hypothesis testing would be to determine whether the wage structures of two regional economies are significantly different. For example, consider that the mean wage of residents in Region X is 30% higher than the mean wage of residents in Region Y. In this case, the null hypothesis is tested using a simple z test to determine whether the difference is statistically significant. Beyond basic hypothesis testing, inferential statistics include a full suite of bivariate and multivariate techniques that determine whether statistically significant relationships can be observed within and between variables and/or explain observed variance within a parameter. These techniques include correlation, regression, and cluster analysis as well as principal components analysis and factor analysis. In addition to parametric techniques, nonparametric statistics, such as chi-square and Spearman's rank order correlation, can be used to determine whether relationships exist within or between nominal variables and within or between ordinal variables, respectively.

Models

Mathematical modeling refers to the integration of techniques from other disciplines that can be used to model spatial relationships. Neoclassical models such as central place theory were adapted to chart and explain the hierarchy of places and markets. Neoclassical modeling is closely associated with location theory. Geographers also socialized well-known models from the natural sciences, including the gravity model. These models explained observed spatial interactions between or among two or more locations. Today, geographers are using these models and many others to investigate the full range of spatial and topological relationships. Indeed, the growth and expansion of geographic information science (GIScience) has facilitated the rapid integration of new methods, such as artificial neural networks, into the practice of spatial modeling.

Quantitative Geography and the Quantitative Revolution

Geographers always have used quantitative data to describe the world around us. In particular, geographers have used statistics as numerical quantities to describe regions or social processes. Yet statistical methods are closely associated with the quantitative revolution of the 1950s and 1960s that sought to embed geographers and the discipline within a new theoretical framework that mirrored trends observed across the academy. Using statistics and models to chart urban change, migration patterns, and other behaviors, theoretical geographers began to explore how methods adopted from other social or natural[Page 394] sciences can or cannot be effectively integrated into the geographic research agenda. Likewise, geographers began to revisit and reconsider the implications of older—more established—explanatory frames with the aid of new statistical tools and models. In some cases, geographers were at the fore in establishing the existence of the difference geography makes or in explaining how geography complicates statistical analysis. One example of this would be the emergence of spatial statistics that examine spatial autocorrelation.

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