ABSTRACT
Body Mass
Index (BMI), defined as the ratio of individual mass (in kilograms) to the
square of the associated height (in meters), is one of the most widely discussed and utilized risk
factors in medicine and public health, given the increasing obesity worldwide
and its relation to metabolic disease. Statistically, BMI is a composite random
variable, since human weight (converted to mass) and height are themselves
random variables. Much effort over the years has gone into attempts to model or
approximate the BMI distribution function. This paper derives the mathematically exact BMI probability density function (PDF), as well as the exact bivariate PDF for human weight and height. Taken together, weight and
height are shown to be correlated bivariate lognormal variables whose marginal
distributions are each lognormal in form. The mean and variance of each
marginal distribution, together with the linear correlation coefficient of the
two distributions, provide 5 nonadjustable parameters for a given population
that uniquely determine the corresponding BMI distribution, which is also shown
to be lognormal in form. The theoretical analysis is tested experimentally by gender against a large anthropometric
data base, and found to predict with near perfection the profile of the
empirical BMI distribution and, to great accuracy, individual statistics including mean, variance, skewness, kurtosis, and correlation.
Beyond solving a longstanding statistical problem, the significance of these
findings is that, with knowledge of the exact BMI distribution functions for
diverse populations, medical and public health professionals can then make
better informed statistical inferences regarding BMI and public health policies
to reduce obesity.