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The RankSize Rule of City
Populations
Geography, Statistical Theory, Data Tables, Statistical Modelling Date posted: The ranksize rule (or ranksize distribution) of city populations, is a commonly observed statistical relationship between the population sizes and population ranks of a nation’s cities. For many years, the reasoning behind the ranksize rule was unknown (at least to geographers). where: x is the rank of the city’s population i.e. a 1 for the highest population, 2 for the second highest etc. is the population size of the city ranked x is the population size of the largest city The rule can be loosened to the following: where A and b are parameters that don’t necessarily conform to and 1 respectively. The ranksize rule is a multiplicative relationship (hyperbola / curved) and when graphed, is convex to the origin. The relationship isn’t linear, nor is it concave to the origin. The loosened form of the equation fits better to real data, than its original restricted form. The loosened form of the equation usually fits very well to real ranksize data of a nation’s cities. The loosened form of the equation can be rewritten as follows: The equation in the final row can be fitted easily using regression (doublelog model). The RankSize
Distribution of US Cities
The following data set is of the 2009 population sizes and ranks of USAcities2009.csv When plotting the population sizes and ranks, the two variables appear to have a strong multiplicative relationship. When plotting the logs of the two variables, it makes the multiplicative relationship even more obvious. A doublelog model (as described earlier) was fitted to the data via regression. The following regression output was obtained: Call: lm(formula = log(Population) ~ log(Rank)) Residuals: Min 1Q Median 3Q Max 0.21820 0.01990 0.00368 0.02279 0.26833 Coefficients: Estimate Std. Error t value Pr(>t) (Intercept) 15.674447 0.013896 1128.0 <2e16 *** log(Rank) 0.742429 0.002936 252.9 <2e16 ***  Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.04691 on 274 degrees of freedom Multiple Rsquared: 0.9957, Adjusted Rsquared: 0.9957 Fstatistic: 6.394e+04 on 1 and 274 DF, pvalue: < 2.2e16 For those who aren’t familiar with regression output: The very high (close to one) Rsquared statistic shows that the model fits very strongly to the data: Multiple Rsquared: 0.9957 The very low (close to zero) pvalues of the tvalues, show that the parameters of the model are highly significant: Pr(>t) <2e16 *** <2e16 *** The coefficient estimates replace the unknown parameters, and the equation can then be rearranged to be in the form of the ranksize rule formula shown earlier. Estimate (Intercept) 15.674447 log(Rank) 0.742429 Reasons for the
RankSize Rule
According to The Dictionary of Human Geography published in 1986 (ISBN 0 631 14656 3): For many years, the reasoning behind the ranksize rule was unknown (at least to geographers). Some geographers gave some plausible but weak reasons (based around economic and geographical theories) for why it might occur. The reason why the ranksize rule is so common and strong is due to statistical/mathematical reasons. In short, the ranksize rule occurs because:
Gibrat’s law and the unequal growth rates, lead to a strengthening of the multiplicative relationship over time. In the long run, a multiplicative convex (to the origin) relationship would develop even if the initial relationship was linear or multiplicative but concave to the origin, or if the initial population sizes were equal. The following animated gifs show simulated ranksize distributions changing over time. Each example has a different starting distribution (linear, constant or concave to the origin), and each observation within an example is randomly assigned a growth rate. At each step of the simulation, a unit of time passes, and the populations of each city grow proportionately by their assigned growth rate. The rankings change if one city’s population overtakes another. Example 1: Initial Relationship is Linear Example 2: Initial Population Sizes are Equal Example 3: Initial Relationship is Concave See
also:

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