f(x, y, z) = (ax, ay, az)

The exponent of 2 in the denominator of the formula for BMI is arbitrary. It is meant to reduce variability in the BMI associated only with a difference in size, rather than with differences in weight relative to one's ideal weight. If taller people were simply scaled-up versions of shorter people, the appropriate exponent would be 3, as weight would increase with the cube of height. However, on average, taller people have a slimmer build relative to their height than do shorter people, and the exponent which matches the variation best is between 2 and 3. An analysis based on data gathered in the USA suggested an exponent of 2.6 would yield the best fit for children aged 2 to 19 years old. The exponent 2 is used instead by convention and for simplicity.