  The law of allometric growth is one of basic rules for understanding urban
evolution. The general form of this law is allometric scaling law. However, the
deep meaning and underlying rationale of the scaling exponents remain to be
brought to light. In this paper, the theories of linear algebra and regression
analysis are employed to reveal the mathematical and statistic essence of
allometric scaling exponents. Suppose that the geometric measure relations
between a set of elements in an urban system follow the allometric growth law.
An allometric scaling exponent is proved to equal in theory to the ratio of the
standard deviation of one logarithmic measure to the standard deviation of
another logarithmic measure. In empirical analyses based on observational data,
the scaling exponent is equal to the product between the standard deviation
ratio and the corresponding Pearson correlation coefficient. The mathematical
derivation results can be verified by empirical analysis: the scaling exponent
values based on the standard deviation ratios are completely identical to those
based on the conventional method. This finding can be generalized to city
fractals and city size distribution to explain fractal dimensions of urban
space and Zipf scaling exponent of urban hierarchy. A conclusion can be reached
that scaling exponents reflect the ratios of characteristic lengths. This study
may be helpful for comprehending scaling from a new perspective and the
connections and distinctions between scaling and characteristic scales.
