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Allometric Exponent and Randomness

An allometric height-mass exponent $γ$ gives an approximative power-law relation $< M> \propto H^γ$ between the average mass $< M>$ and the height $H$, for a sample of individuals. The individuals in the present study are humans but could be any biological organism. The sampling can be for a specific age of the individuals or for an age-interval. The body-mass index (BMI) is often used for practical purposes when characterizing humans and it is based on the allometric exponent $γ=2$. It is here shown that the actual value of $γ$ is to large extent determined by the degree of correlation between mass and height within the sample studied: no correlation between mass and height means $γ=0$, whereas if there was a precise relation between mass and height such that all individuals had the same shape and density then $γ=3$. The connection is demonstrated by showing that the value of $γ$ can be obtained directly from three numbers characterizing the spreads of the relevant random Gaussian statistical distributions: the spread of the height and mass distributions together with the spread of the mass distribution for the average height. Possible implications for allometric relations in general are discussed.