Allometric exponent and randomness

An allometric height–mass exponent γ gives an approximative power-law relation 〈 M 〉∝ 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 is often used for practical purposes when characterizing humans and it is based on the allometric exponent γ = 2. It is shown here that the actual value of γ is to a 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.