The concept of BMI was introduced as long ago as 1835 by Quetelet [5]. Initially assumed to be a normal (i.e. Gaussian) distribution by early developers of modern statistics, such as Galton and Pearson, the assumption was largely accepted by statisticians and scholars concerned with human growth throughout the 20th Century [5]. With the recognition that empirical BMI distributions appeared skewed to the right (i.e. to higher values), various non-symmetric distributions such as lognormal, gamma, beta, and power-law have been suggested, but none to our knowledge was rigorously demonstrated and tested. See, for example [6] [7] [8] [9].

The principal objective of this paper is to establish the distribution of BMI on a more rigorous foundation and to test our findings experimentally against a large data base of personal weights and heights compiled in the 2012 Anthropometric Survey of US Army Personnel (ANSUR) [10]. Our analysis, discussed in the following sections, shows that, statistically, weight and height are correlated lognormal variables from which it rigorously follows that BMI is also a lognormal variable whose probability density function (PDF) is predictable from the 5 parameters that uniquely characterize the joint distribution of weight and height. The finding of lognormality is consistent with one of the authors (MPS) previous investigations [2] [11] of the distribution of composite random variables, which showed that such variables are ordinarily well represented by lognormal distributions irrespective of the distributions of the composite factors. A novel feature of BMI, however, not encountered in References [1] and [11] is the correlation of the individual factors of weight and height.

It has long been the practice in clinical medicine to use mean values of selected ratios, such as BMI to assess fat, LDL/HDL (low and high density lipoprotein)to assess cardiovascular risk, A/G (albumin and globulin) to assess liver function, BUN (blood urea nitrogen)/creatinine to assess kidney function, and many others. However, as emphasized by the authors in regard to statistical inferences [12] [11], the mean values alone may be uninformative and even misinformative. What is really required for valid interpretation and practical application of a biomedical index is its statistical distribution. By knowing the distribution of a random variable an analyst can determine with quantifiable confidence all population statistics (for comparison with empirical sample statistics) such as moments (mean, variance, skewness, kurtosis, etc.), cumulants, percentiles (such as median, quartiles, etc.), and, especially in the case of biomedical ratios, the cut-off values that determine degrees of health and risk.

In the analysis to follow, we calculate the exact distribution of BMI from its defining relation Equation (1), knowledge of the joint distribution of weight and height, and use of mathematical transformation relations for products and quotients of random variables [13] [14] [15]. The merit of this approach is that the form of the calculated distribution function is unique, apart from the empirical parameters that define the distributions of weight and height in a given population. Past attempts, such as cited above, to obtain mathematical expressions for the BMI distribution by curve-fitting data to assumed BMI profiles lack rigor and can actually lead to mathematically untenable results. For example, as cited above, statisticians have assumed throughout much of the 20^th century that height, weight (or mass), and BMI all followed normal (Gaussian) distributions. It is mathematically demonstrable [2], however, that if mass and height are normal variables (which they are not), then the distribution of BMI cannot possibly also be normal, although the distribution profile may suggest such an appearance. Moreover, if mass and height are lognormal variables, then the distribution of BMI is rigorously lognormal too.

The virtue of having the exact (in contrast to an assumed or approximate) distribution is that it is expected to be valid for all allowed values of its parameters and variables. Thus, since mass and height must take real, positive values, the BMI distribution must rigorously vanish at B = 0 . A normal, or other approximate, distribution for BMI may appear to vanish at B = 0 if the mean of the distribution is sufficiently larger than the width (i.e. standard deviation); in other words, if the distribution is sharply defined. Rigorously, it does not vanish at B = 0 . More problematic, however, is that a broad normal distribution can overlap the negative real axis leading to impossible values of BMI. Under such conditions, it may be thought that one could preserve normality simply by adopting a truncated normal distribution defined over the positive real axis. The outcome of truncation, although it may meet boundary conditions, will not fit data as well as the true distribution. Examples investigated by one of the authors (MPS) by means of the Principle of Maximum Entropy [16], have shown that a truncated Gaussian is astronomically less likely to be correct than the true distribution [11].

Before examining the statistical distribution of BMI, it is worth summarizing briefly how the distribution of BMI can influence the assessment of individual health and creation of public health policy.

The measurement of an individual BMI value requires only a person’s weight and height. BMI therefore provides an inexpensive screening method for determining whether a person is underweight, healthy, overweight, or obese—the four general weight categories used by physicians and epidemiologists². BMI is correlated with, and therefore seen as a proxy for, measurement of body fat [17], and is strongly correlated with metabolic disease [18]. Although there are other more accurate ways to measure body fat, such as bioimpedance analysis, dual-energy x-ray absorptiometry, computed tomography, and magnetic resonance imaging, such methods are expensive, not readily available to most patients or medical personnel, and require specially trained staff [17].

The statistical distribution of BMI provides the basis for setting the principal cut-off points that characterize various weight categories from severe thinness to severe obesity. Standard BMI cut-offs are independent of age and gender, although it is recognized that the same numerical value of BMI may correspond to different amounts of body fat in different populations, partly as a result of different body proportions [19]. More than 2 decades ago the World Health Organization (WHO) convened a working group of experts to study cut-offs in regard to the BMI of Asian populations, but the cut-offs remained largely unchanged [20] despite the conclusion that the proportion of Asians at high risk for type 2 diabetes and cardiovascular disease is substantial at BMIs lower than the existing WHO cut-off points for overweight. A recent report, based on a US population of adults and young adults reached a similar conclusion that a higher BMI only moderately increased the risks for diabetes among the healthy obese, and that unhealthy thin people were more likely than the aforementioned group to get diabetes [21]. Clearly, rigorous statistical distributions of BMI are needed for specific population groups. Otherwise, the matter of setting and interpreting BMI cut-off points will be remain controversial, not just in articles in the medical literature, but also in reports to the general public [22] [23].

In reference [22], questions were raised as to the accuracy and utility of using BMI to describe individual health. In our opinion, BMI was, and is, intended to be a statistical quantity. As such, it describes populations and not specific individuals. Nevertheless, given the exact distribution function specific to a well-defined demographic, cut-offs can be set more appropriately and less arbitrarily so as to be medically useful to clinicians in evaluating individual patients. To achieve this, a major first step would be to have an accurate BMI distribution function covering the full range of a sufficiently large and well-defined population of healthy individuals. We believe our analysis of the ANSUR data, which separately includes male and female members of the US military, provides such baseline information.

With regard to the establishment of public health policies to reduce adult obesity, knowledge of the exact BMI distribution can help resolve a debate over the optimal strategy for disease prevention. One approach, the “population strategy”, proposed by Rose [24] [25] and widely adopted by epidemiologists, public health practitioners and policy makers, is to shift the distribution of a risk factor in a desired direction by applying interventions to an entire population [26]. An alternative approach, the “high-risk strategy”, aims to lower the risk of disease within a population by detecting and treating the subgroup of people who manifest extreme values of the designated risk factor, and therefore appear to be at the highest risk.

Statistically, Rose and others found strong correlations between the mean value of a risk factor (e.g. BMI) and the prevalence of extreme values of that risk factor. In other words, in a specified population of people at risk for a particular disease over time, Rose expected the lower and middle sections of the distribution curve of the risk factor to move proportionally in the same direction as the high-end extreme section, thereby displacing the entire distribution curve to the right. He concluded from this, in regard to public health strategy, to implement a policy of intervention to all members of the population and not just those with risk factors in the upper tail of the distribution. The idea, as summarized in a review [26] of Rose’s work, is that “More clinical cases result from small but widespread risks than large but rare risks.”

Supporters of the “high-risk strategy” have pointed out problems to Rose’s proposal. One such problem in regard to BMI in particular is the assumption of a Gaussian, or at least symmetric, distribution of the risk factor. This assumption, as we indicated in the previous section, is almost certainly invalid for any realistic population distribution of human height and weight (mass). Speculations based on biological considerations have been made for a lognormal distribution [7], but such reasoning, while suggestive, is likewise not rigorous and leaves open the possibility of other skewed distribution functions.

Ultimately, which public health strategy is superior must be validated empirically. The efficacy of the “high-risk strategy” can be tested experimentally by randomized control trials. By contrast, it is more difficult to test the efficacy of the “population strategy”. According to reference [26], the determination of whether a benefit results from lowering the risk of a whole population would require implementation and monitoring of lifestyle changes starting from birth and extending over decades.

Nevertheless, in order that an experimental test of strategy yield useful information, the results must be interpretable, and a valid interpretation requires the exact BMI probability density function. This function provides the most reliable statistical tool to study the evolution over time of the population statistics required for the “population strategy”. Likewise, it helps the analyst decide quantitatively who falls within the category of high risk (i.e. proportion of a given population under the right tail of the distribution) as required for the “high risk strategy”. From a broader perspective, the exact distribution function allows public health specialists to define meaningfully the cutoff points by which degrees of fatness and risk are classified.

Moreover, drawing upon the methodology of physics, we expect that, where direct experimental tests may be impractical to implement, the use of computer-based modeling can play a constructive role. Knowledge of the exact theoretical BMI distribution derived here, combined with well-designed mathematical models representing proposed public health interventions, can lead to insights and solutions in time intervals short compared to decades of observation.

Over much of the period of modern statistics human attributes such as height and weight have been assumed or approximated to follow a Gaussian distribution. Justification for this may be attributed in part to empirical inferences drawn from coarse-graded statistical sampling, theoretical inferences based on the Central Limit Theorem, and a need for mathematical convenience [28]. Statisticians were certainly aware, however, that the tails of a Gaussian did not fit observed frequency data closely [29] [30], but this problem was generally regarded as minor since the number of events were few compared with the bulk of the observed frequency distribution. With regard to BMI, however, the tail of the distribution is important since it represents the subgroup of people with extreme risk factors. Nevertheless, because normal distributions serve as a kind of baseline model in the statistics of public health, we examine the case of independent normally distributed weight (mass) X and height Y,

X = N X ( m 1 , s 1 2 ) Y = N Y ( m 2 , s 2 2 ) (16)

represented statistically by the symbol N ( m , s 2 ) in which m is the mean of the variable, s 2 is the variance, and the PDF of a normal distribution (indicated by superscript N)takes the general form

p X ( N ) ( x ) = 1 2 π s 2 e − ( x − m ) 2 / 2 s 2 . (17)

Substitution into Equation (15) of PDF (17) with the parameters of distributions (16) leads to the explicit function

p Z ( N , N ) ( z ) = 1 2 π s 1 s 2 ∫ 0 ∞ y 2 e − ( z y 2 − m 1 ) 2 / 2 s 1 2 ( e − ( y − m 2 ) 2 / 2 s 2 2 + e − ( y + m 2 ) 2 / 2 s 2 2 ) d y . (18)

where the superscript ( N , N ) signifies that both component factors (mass and height) are normally distributed. It is clear from the form of Equation (18), in which the leading term in the exponent is z to the 4^th power (rather than quadratic), that normal distributions of mass and height, as expressed in relations (16), result in a non-normal and non-symmetrical distribution of body mass index.

To our knowledge, the integral (18) cannot be performed analytically, but must be evaluated numerically. A plot of p Z ( N , N ) ( z ) (solid curve) as a function of z for a hypothetical sample set of parameters is shown in Figure 1. The profile is skewed to the right and appears very much like the lognormal profile (dashed curve), superposed for comparison. As seen in the figure, the two profiles are distinguishable, but in close agreement, especially in the vicinity of the maximum, the origin, and along the right tail. If the range of the figure were extended to show the tail out to z = 200, the two profiles would appear to overlap apart from a slightly lower maximum value of the lognormal distribution.

The lognormal distribution of BMI shown in Figure 1, discussed more fully in the following sections, is the profile that would result if individual distributions of mass and height were both lognormal with parameters corresponding to the parameters of the normal distributions in the example. The PDF of the variable Z = X / Y 2 in which X and Y are lognormal variables takes the general form

p Z ( Λ , Λ ) ( z ) = 1 2 π s 2 z e − ( ln ( z ) − m ) 2 2 s 2 (19)

to be derived in Section 2.3. The superscript ( Λ , Λ ) signifies that both component factors are lognormal. Determination of the lognormal parameters corresponding to the parameters of the normal distributions that form Figure 1 is explained in the following section.

The exact PDF (18) and the corresponding lognormal PDF (19) yield, respectively, the following means, dispersions (standard deviation about the mean), and asymmetries (skewness, defined in the next section)

Mean                     μ Z ( N , N ) = 27.25                       μ Z ( Λ , Λ ) = 27.00 StdDev                       σ Z ( N , N ) = 21.93                     σ Z ( Λ , Λ ) = 18.23 Skewness S k Z ( N , N ) = 2.68                       S k Z ( Λ , Λ ) = 2.33 (20)

which are seen to be numerically close for the exact and lognormal distributions of BMI.

The statistics exhibited in relation (20) raise a cautionary issue in regard to skewed distribution functions. Ordinarily—i.e. primarily for symmetric distributions—the standard deviation is interpreted as a measure of the uncertainty of the mean, which, itself, is usually adopted in statistical physics and medicine as the experimental value of a distributed quantity. However, as seen in Figure 1, the modes (i.e. maxima), in contrast to the means, of the two profiles are actually fairly narrowly located and serve better than the mean for purposes of monitoring the evolution of the BMI distribution in a specified population over time. The large dispersions about the means are due to the long high-end tails. The skewness of a distribution, which is proportional to the 3^rd central moment, provides a quantitative measure of the asymmetry about the mean, and therefore a measure of the fraction of a population at greatest risk of metabolic disease.

In summary, for purposes of defining appropriate cutoff points for the various weight categories and demographics and to investigate evolving trends in BMI within a population, a lognormal distribution would serve equally satisfactory to the exact BMI distribution derived on the assumption of normally distributed weight and height. Our analysis indicates, however, that this assumption is not valid, and that the true distributions of weight and height are, themselves, lognormal, from which it follows that a lognormal BMI distribution is not an approximation, but rigorously exact. We discuss this in the following section.

If the natural logarithm of a set of variates { x i } , represented by the random variable X, gives rise to a normal distribution, represented by the variable Y, then X is said to be a lognormal random variable. The relation is expressed symbolically as

Y = ln ( X ) = N Y ( m , s 2 )         ⇒         X = exp ( Y ) = Λ X ( m , s 2 ) . (21)

From the transformation relations (21) and normal PDF (17), there follows the lognormal PDF

p X ( Λ ) ( x ) = 1 2 π s 2 e − ( ln ( x ) − m ) 2 / 2 s 2 x . (22)

Note that the parameters defining the lognormal distribution X are the mean and variance of the normal variable Y. In other words, m and s 2 are not moments of the lognormal distribution. The r ^th order moments M 0 ( r ) ≡ 〈 X r 〉 , r = 1 , 2 , ⋯ , of a lognormal distribution can be calculated straightforwardly as expectation values by using PDF (22)

〈 X r 〉 = ∫ 0 ∞ x r p X ( Λ ) ( x ) d x . (23)

However, it is simpler to calculate M 0 ( r ) from the moment generating function [13]

g Y ( t ) ≡ 〈 exp ( Y t ) 〉 = e m t + 1 2 s 2 t 2 = 〈 X t 〉 (24)

of the normal distribution Y by use of relation (21) and substitution of the discrete index r for the continuous dummy variable t. This leads to

M 0 ( r ) = e m r + 1 2 s 2 r 2 . (25)

The first few moments M 0 ( r ) and the principal combination statistics

Variance σ X 2 ≡ 〈 ( X − μ ) 2 〉 (26)

Skewness S k ≡ 〈 ( X − μ ) 3 〉 / σ X 3 (27)

Kurtosis K X ≡ 〈 ( X − μ ) 4 〉 / σ X 4 (28)

of the lognormal distribution are summarized in Table 1. The subscript 0 in M 0 ( r ) signifies that the moments are taken with respect to the origin. The preceding combination statistics are central moments, designated M μ ( r ) in Table 1, where the subscript μ signifies that the moments are taken with respect to the lognormal mean μ ≡ M 0 ( 1 )

M μ ( r ) = ∫ 0 ∞ p X ( x ) ( x − μ ) r d x = ∑ j = 0 r ( − 1 ) r − j C ( r , j ) μ r − j M 0 ( j )   . (29)

The symbol

C ( r , j ) ≡ r ! j ! ( r − j ) ! (30)

is a binomial coefficient.

The mean μ and variance σ 2 of the lognormal variable X is given in terms of the mean m and variance s 2 of the normal variable Y by the following relations from Table 1.

μ = e m + 1 2 s 2 σ 2 = e 2 m ( e 2 s 2 − e s 2 ) (31)

from which follow the inverse relations

m = ln ( μ 2 μ 2 + σ 2 ) (32)

s 2 = ln ( μ 2 + σ 2 μ 2 ) . (33)

Relations (31), (32), (33) will be applied shortly to the BMI distribution in Figure 1.

Consider next the case of independent lognormal factors for mass and height respectively

X = Λ X ( m 1 , s 1 2 ) Y = Λ Y ( m 2 , s 2 2 ) (34)

resulting in the BMI³

Z = X / Y 2 . (35)

To find the distribution of Z, take the natural logarithm of both sides of expression (35) to obtain

ln ( Z ) = ln ( X ) − 2 ln ( Y ) = N ( m 1 , s 1 2 ) − 2 N ( m 2 , s 2 2 ) = N ( m 1 − 2 m 2 , s 1 2 + 4 s 2 2 ) (36)

where the first equality of the second line of Equation (36) follows from the definition of a lognormal variable, and the second equality is the result of combining two independent normal distributions, which follows from the equivalence relation [13],

N ( m , s 2 ) = m + s N ( 0 , 1 ) . (37)

Thus, since the log of Z is a normal variable, then Z must be a lognormal variable Z = Λ Z ( m , s 2 ) with parameters

m = m 1 − 2 m 2 s 2 = s 1 2 + 4 s 2 2 (38)

From relations (31), the parameters (38) correspond to a mean BMI of

μ = M 0 ( 1 ) = e ( m 1 − 2 m 2 ) + 1 2 ( s 1 2 + 4 s 2 2 ) (39)

with standard deviation

σ = M 0 ( 2 ) − M 0 ( 1 ) 2 = e ( m 1 − 2 m 2 ) + 1 2 ( s 1 2 + 4 s 2 2 ) e 2 ( s 1 2 + 4 s 2 2 ) − e ( s 1 2 + 4 s 2 2 ) . (40)

In the example illustrated in Figure 1, a hypothetical sample population was characterized statistically by mass μ M ± σ M = 70 ± 20 kg and height μ H ± σ H = 1.8 ± 0.5 m. If mass and height independently follow the respective lognormal distributions Λ X ( m 1 , s 1 2 ) and Λ Y ( m 2 , s 2 2 ) , the four parameters of the distributions are rigorously determined from Equations (31)-(33) (to four decimal places)

m 1 = 4.2093 ,                     s 1 = 0.2801 m 2 = 0.5506 ,                   s 2 = 0.2726 (41)

The corresponding parameters of the BMI distribution Λ Z ( m , s 2 ) in Figure 1 (dashed curve) are then given by Equation (38)

m = 3.1080 ,             s = 0.6130 . (42)

From the inverse relations (39) and (40) one calculates the mean and standard deviation of the BMI distribution to be

μ B M I = 27.0018 ,             σ B M I = 18.2349 , (43)

which agree with the corresponding values in Equation (20) obtained by integration over the PDF as in Equation (23).

The exact BMI distribution expressed by Equation (15) contains in the integrand products of the PDFs of the variables X and Y. However, if the weight of an individual is influenced by his/her height (or vice versa), then the joint distribution function of mass and height, expressed as p X Y ( x , y ) , does not factorize into separate functions of x and y. In that case, the antecedent Equation (14) with W = Y 2 takes the form

p Z ( z ) = ∫ 0 ∞ d x ∫ 0 ∞ d w   p X   W ( x , w ) δ ( z − x w ) (44)

leading to the result

p Z ( z ) = ∫ 0 ∞ y 2 ( p X Y ( z ​ ​ ​ y 2 , y ) + p X Y ( z ​ ​ ​ y 2 , − y ) ) d y . (45)

In the following section we provide strong evidence that height and weight are significantly correlated and that the marginal distributions of both variables are lognormal in form. The joint distribution function of bivariate lognormal variables is derivable from the distribution function of bivariate normal variables [31]

p Y 1 Y 2 ( N ) ( y 1 , y 2 ) = 1 2 π s 1 s 2 1 − r 2 e − q Y / 2 q Y = 1 1 − r 2 [ ( y 1 − m 1 s 1 ) 2 − 2 r ( y 1 − m 1 s 1 ) ( y 2 − m 2 s 2 ) + ( y 2 − m 2 s 2 ) 2 ] (46)

In the preceding equation, m 1 , s 1 are the mean and standard deviation of a normal variable Y 1 , and likewise m 2 , s 2 are the mean and standard deviation of a normal variable Y 2 . The Pearson correlation coefficient r is defined as the expectation value [31]

r ≡ 〈 ( Y 1 − m 1 ) ( Y 2 − m 2 ) 〉 s 1 s 2 = cov ( Y 1 , Y 2 ) s 1 s 2 , (47)

which falls within the range 1 ≥ r ≥ − 1 . The expectation value in the numerator of Equation (47) is the covariance. A correlation coefficient r = 1 signifies that the two variables are perfectly correlated linearly; likewise, r = − 1 signifies perfect linear anticorrelation. An arbitrary value of r within the stated range is interpreted to mean that r 2 is the fraction of the variance of one variable attributable to the other [32].

The probability density function p R ( r ) of the Pearson r is a complicated mathematical expression involving gamma functions and a hypergeometric function of the type 2 F 1 . The exact form of the PDF and resulting statistical moments can be found in Ref. [33]. Of particular utility in this paper is the standard deviation σ r and standard error (SE)

S E r ≡ σ r n = 1 − r 2 n (48)

truncated at the first term of an expansion in inverse powers of the sample size n. Plots of p R ( r ) for different mean values 〈 r 〉 and two sample sizes n are displayed in Figure 2. The profiles are strongly skewed to the left for small sample size and rapidly approach Gaussian form as n increases. For the ANSUR data used in this paper, n > 1000 and the exact profile of p R ( r ) is indistinguishable from a normal distribution about 〈 r 〉 with width S E r given by (48).

If the normally distributed variates of Y 1 and Y 2 are obtained by taking the natural logarithm of the variates of X 1 and X 2 , then X 1 and X 2 are lognormal variables. In analogy to Equation (4), the transformation of PDF (46) to a PDF of X 1 and X 2 is implemented as follows

p X 1 X 2 ( Λ , Λ ) ( x 1 , x 2 ) = p Y 1 Y 2 ( N , N ) ( y 1 ( x 1 ) , y 2 ( x 2 ) ) | d y 1 d x 1 d y 2 d x 2 | (49)

where y ( x ) = ln ( x ) and leads to

p X 1 X 2 ( Λ , Λ ) ( x 1 , x 2 ) = 1 2 π s 1 ​ ​ ​ s 2 1 − r 2 e − q X / 2 x 1 x 2 q X = 1 1 − r 2 [ ( ln ( x 1 ) − m 1 s 1 ) 2 − 2 r ( ln ( x 1 ) − m 1 s 1 ) ( ln ( x 2 ) − m 2 s 2 ) + ( ln ( x 2 ) − m 2 s 2 ) 2 ] (50)

which generalizes Equation (22).

The marginal distribution of one variable is obtained by integrating the PDF (50) over the other variable as follows

∫ 0 ∞ p X 1 X 2 ( Λ , Λ ) ( x 1 , x 2 ) d x 2 = p X 1 ( Λ ) ( x 1 ) ∫ 0 ∞ p X 1 X 2 ( Λ , Λ ) ( x 1 , x 2 ) d x 1 = p X 2 ( Λ ) ( x 2 ) (51)

As one would expect, the correlation coefficient r vanishes from the marginal distributions, since both variables must be present if there is to be a correlation between them.

It is to be borne in mind that the Pearson coefficient r is a measure of the correlation between normal variables Y 1 and Y 2 . The Pearson coefficient of the lognormal variables X 1 and X 2 , which represent respectively mass and height in the context of BMI, is obtained from the relation corresponding to (47)

ρ = cov ( ( X 1 − μ 1 ) ( X 2 − μ 2 ) ) σ 1 σ 2 (52)

which can be reduced to

ρ = 〈 ( X 1 − μ 1 ) ( X 2 − μ 2 ) 〉 σ 1 σ 2 = 〈 X 1 X 2 〉 − μ 1 μ 2 σ 1 σ 2 . (53)

Equation (53) requires the integral

〈 X 1 X 2 〉 = ∫ 0 ∞ x 1 d x 1 ∫ 0 ∞ x 2 p X 1 X 2 ( x 1 , x 2 ) d x 2 = μ 1 μ 2 e r s 1 s 2 (54)

where mean value μ of a lognormal variable is given by Equation (31). From Equations (54), (53), and (31), it follows that the correlation coefficient ρ of a bivariate lognormal distribution takes the form

ρ = e r s 1 s 2 − 1 ( e s 1 2 − 1 ) ( e s 2 2 − 1 ) . (55)

It is worth noting that the moments, including all correlation statistics of a bivariate, or more generally a multivariate, distribution can in principle be obtained from a moment generating function [13] without having to perform integrals like the one in Equation (54), which can be difficult. This method, however, lies outside the scope of this paper. Nevertheless, integrations over the bivariate PDF (50) can be greatly simplified by transforming from the space of ( x 1 , x 2 ) back to the space of ( y 1 , y 2 ) and then transforming to variables ( u , v ) defined by

u = y 1 − m 1 1 − r 2 s 1 v = y 2 − m 2 1 − r 2 s 2 (56)

which generates the probability density

f ( u , v ) = 1 − r 2 2 π e − 1 2 ( u 2 − 2 r u v + v 2 ) . (57)

The range of variables u, v is ( − ∞ , ∞ ) . To calculate joint expectations of powers of X 1 and X 2 , substitute

x 1 = e y 1 = e 1 − r 2 s 1 u + m 1 x 2 = e y 2 = e 1 − r 2 s 2 v + m 2 (58)

in the integral with PDF f ( u , v ) .

Even with the preceding transformations to facilitate calculation, we have been unable to derive in closed form an expression for the variance of Equation (55). We approximate, therefore, the variance of ρ by using error propagation theory [34]

σ ρ 2 = ( ∂ ρ ∂ r ) 2 σ r 2 + ( ∂ ρ ∂ s 1 2 ) 2 σ s 1 2 + ( ∂ ρ ∂ s 2 2 ) 2 σ s 2 2 (59)

in which σ r 2 is given by Equation (48), and the variance of the variance s 2 of a normal random variable N ( m , s 2 ) is known to be [13]

σ s 2 2 = 2 s 4 . (60)

Standard errors are obtained by dividing the variances σ r 2 , σ s 1 2 2 , σ s 2 2 2 by the sample size n. The analytical evaluation of Equation (59) leads to a long, and not particularly illuminating expression and will not be given explicitly, since, when its evaluation is needed later, both the partial derivatives and numerical substitutions are carried out by computer.

Calculation of the probability density function of the ratio Z = X 1 / X 2 2 using PDF (50) with lognormal factors X 1 = Λ X 1 ( m 1 , s 1 2 ) for mass and X 2 = Λ X 2 ( m 2 , s 2 2 ) for height proceeds most readily from the defining transformation

p Z ( z ) = ∫ 0 ∞ ∫ 0 ∞ p X 1 X 2 ( x 1 , x 2 ) δ ( z − x 1 x 2 2 ) d x 1 d x 2 = ∫ 0 ∞ x 2 2 p X 1 X 2 ( z x 2 2 , x 2 ) d x 2 (61)

where the second line of relation (61) results immediately from property (10) of the delta function. The remaining integration can be performed by transforming to the integration variable y 2 = ln ( x 2 ) and leads to the exact PDF

p Z ( Λ ) ( z ) = e − ( ln ( z ) − ( m 1 − 2 m 2 ) ) 2 2 ( s 1 2 + 4 s 2 2 − 4 r s 1 s 2 ) 2 π ( s 1 2 + 4 s 2 2 − 4 r s 1 s 2 )   z (62)

for BMI of a population with correlated weight and height.

From the form of PDF (62), it is seen that the variable Z is exactly lognormal

Z = Λ Z ( m , s 2 ) , (63)

with parameters

m = m 1 − 2 m 2 s 2 = s 1 2 + 4 s 2 2 − 4 r s 1 s 2 (64)

Comparison with Equation (38) shows that the mean m is the same as for independent lognormal factors, but the variance s 2 is a function of the correlation coefficient r. The influence of correlation on the probability density (and therefore also on statistical moments)can be quite strong, as illustrated in Figure 3 which shows plots of PDF (62) as a function of the BMI variate z for values of r ranging from −1 to +1 in intervals of 0.25. The plots are color coded such that

profiles of the same | r | have the same color, but are distinguished by their widths ranging from a maximum for r = − 1 (dashed blue curve) to a minimum of r = 1 (solid blue curve). The solid black profile corresponds to uncorrelated weight and height, r = 0 . As shown in the figure, increasing the correlation of weight and height displaces the maximum of the BMI distribution to the right and narrows the spread. For perfect linear correlation r = 1 , the variance takes its minimum value, s 2 | min = ( s 1 − 2 s 2 ) 2 , which, as expected, can never be negative. As a corollary of the narrower spread, the tail of the BMI distribution with positive correlation drops off more rapidly than if weight and height were uncorrelated or anticorrelated.

BMI population statistics, of which the most important are the mean, dispersion about the mean (standard deviation), and asymmetry about the mean (skewness)

μ Z ≡ 〈 Z 〉 = exp ( ( m 1 − 2 m 2 ) + 1 2 ( s 1 2 + 4 s 2 2 − 4 r s 1 s 2 ) ) (65)

σ Z ≡ 〈 Z 2 〉 − 〈 Z 〉 2 = e ( m 1 − 2 m 2 ) e ( 2 s 1 2 + 8 s 2 2 − 8 r s 1 s 2 ) − e ( s 1 2 + 4 s 2 2 − 4   r s 1 s 2 ) (66)

S k Z ≡ 〈 ( Z − μ ) 3 〉 σ Z 3 = e 3 ( s 1 2 + 4 s 2 2 − 4 r s 1 s 2 ) − 3 e s 1 2 + 4 s 2 2 − 4 r s 1 s 2 + 2 ( e s 1 2 + 4 s 2 2 − 4 r s 1 s 2 − 1 ) 3 / 2 (67)

are also markedly affected by the correlation of weight and height, as plotted in Figure 4 as a function of correlation coefficient r. As shown in the figure, the higher the correlation, the lower are the BMI mean, standard deviation, and skewness.

Figure 5 shows a histogram (gray bars) of the distribution of heights of the male subgroup (left panel) and female subgroup (right panel) in the ANSUR population. Corresponding histograms of the natural logarithm of the heights are shown in Figure 6. Table 2 summarizes the sample statistics obtained from

analysis of the two sets of data.

The histograms of log-height in Figure 6 appear symmetric about the mean and can be well fitted by Gaussian profiles with sample means and standard deviations

Male Subgroup: m H M = 0.5624 s H M = 0.0390 (68)

Female Subgroup: m H F = 0.4869 s H F = 0.0394 (69)

calculated directly from the unpartitioned data (in contrast to partitioning the data into categories and applying a maximum likelihood or Bayesian estimation procedure).

Chi-square tests of the goodness of fit of the log-height histograms to Gaussian profiles are summarized in Table 3. For ν = 24 degrees of freedom (data partitioned into 25 categories), the tests yielded respective p-values of 35.73% (male) and 58.77% (female). It is to be recalled that the p-value is the probability that a subsequent random sample from the same total population would result in a chi-square value equal to or greater than the observed value, assuming the null hypothesis is correct [35]. The null hypothesis in testing the histograms of Figure 6 is that they are samples from Gaussian distributions with parameters given by Equations (68) and (69). The critical statistic of a chi-square test is the chi-square value beyond which the p-value is below 5%. The p-values in Table 3 are all well above 5%.

The significance of a chi-square test is not that it proves the null hypothesis to be true, but that the null hypothesis cannot be rejected on the basis of the test. Nevertheless, the test supports the inference that, if the histograms of log-height are Gaussian, then the height, itself, is distributed lognormally for both male and female subgroups. This is evidenced in Figure 5 by the superposed lognormal profiles corresponding to the distributions Λ ( m H M , s H M 2 ) for males and Λ ( m H F , s H F 2 ) for females. Chi-square tests of the lognormal fits, reported in Table 3, show p-values of 41.67% for males and 59.08% for females, which again support the null hypothesis.

Although the visual appearance of the histograms of height in Figure 5, for both male and female subgroups, may suggest that these data are distributed normally, this appearance is deceptive and incorrect, given that the natural logarithm of the set of variates yield normal distributions. By contrast, the natural logarithm of a normal variable is not distributed normally, as shown in Figure 7. The blue profile is a normal (Gaussian) distribution based on the same height parameters ( m = 1.8 , s = 0.5 ) as the example used in Figure 1. The red profile is the distribution of the natural logarithm of the Gaussian variates.

To examine this issue analytically, consider a normal variable Y = N ( m , s 2 ) and the log-of-normal variable X = ln ( Y ) . To avoid confusing the term “log-of-normal” with the entrenched designation “lognormal” for a variable whose natural logarithm is normal, we will call X in this example a logGauss random variable. Employing the transformation methods of previous sections, one can readily show that the PDF of a logGauss variable takes the form

p X ( x ) = 1 2 π s 2 exp ( − ( e x − m ) 2 2 s 2 + x ) , (70)

which is not equivalent to the PDF of a normal (Gaussian) distribution. For variates x in the vicinity of the maximum point at ln ( m ) , one can truncate at first order a Taylor series expansion of the numerator ( e x − m ) in Equation (70) to obtain an approximate PDF of Gaussian form. However, the expansion is not valid at the wings, which descend more quickly than a Gaussian on the right side and extend more slowly and into the nonphysical negative range on the left side.

It is clear, then, that the distribution of heights of males and females in the ANSUR data is not a normal distribution, but, in conformity with our applied statistical tests and the theoretical analyses of [2] [11], is consistent with a lognormal distribution. Moreover, given that the same biological processes are likely to determine height in any population of healthy males or females with access to adequate nutrition, we believe it reasonable to infer that human height in all such populations is distributed lognormally.What distinguishes one population from another would be the parameters, not the form, of the distribution.

Figure 8 shows a histogram (gray bars) of the distribution of weight (converted to mass) of the male subgroup (left panel) and female subgroup (right panel) in the ANSUR population. The mass histograms in Figure 8 are skewed to the right and are clearly non-Gaussian. Corresponding histograms of the natural logarithm of the masses are shown in Figure 9. Table 2 summarizes the sample statistics obtained from analysis of the two sets of data.

As with the attribute of height in the previous section, the histograms of log-mass in Figure 9 appear symmetric about the mean and are well fitted by Gaussian profiles with the following sample means and standard deviations

Male Subgroup: m W M = 4.4351 s W M = 0.1654 (71)

Female Subgroup: m W F = 4.2030 s W F = 0.1604 (72)

calculated directly from the unpartitioned data. (Note: We use the subscript W for weight in relations (71) and (72), even though the distribution function and associated moments are for mass, since weight was the attribute actually measured. Also, we reserve the subscript M to represent “Male”.)

Chi-square tests of the goodness of fit of the log-mass histograms in Figure 9 to Gaussian profiles are summarized in Table 3. For ν = 24 degrees of freedom, the tests yielded respective p-values of 61.71% (male) and 59.08% (female). Likewise, chi-square tests of the fit of the mass histograms to lognormal profiles in Figure 8 yielded p-values of 27.89% (male) and 54.97% (female). Altogether, the chi-square tests of the histograms in Figure 8 and Figure 9 well support the null hypothesis that weight (mass) is distributed lognormally in both male and female subgroups of the ANSUR population. As with height, there is reason to infer that the attribute of weight in healthy human populations accessible to adequate nutrition will follow a lognormal distribution.

Figure 10 shows a scatter plot of the weight (converted to mass) against height for males (left panel) and females (right panel) of the ANSUR sample. Each point in a scatter plot is the mass and height of a single individual. The elongated shapes of the scatter plots clearly demonstrate that the data are linearly correlated. There may also be higher order correlations, but in this paper we are concerned exclusively with linear correlation as quantified by the Pearson correlation coefficients r and ρ defined by Equations (47) and (53), respectively, and predicted by Equation (55) for lognormal distributions.

Figure 11 displays the scatter plots of Figure 10 rescaled by dividing the variates of the two random variables by their sample standard deviations. The resulting variate is a pure number without units or dimensions. Superposed on the

dimensionless scatter plots is the line of regression obtained from a least-squares fit to the scaled data. The respective slopes of the lines in the left and right panels accurately yielded the correlation coefficient ρ for males and females, respectively, as recorded in Table 2 and Table 4. For purposes of comparison, Figure 12 shows a simulated scatter plot of uncorrelated weight and height, obtained from 10,000 samples drawn independently from lognormal random number generators (RNGs) with the same parameters as given in relations (69) and (72) for the female subgroup in the ANSUR data. The overall shape is circular, apart from fluctuations at the periphery.

It is an important point worth clarifying why the slope of the line of regression to the scaled scatter plot is an exact geometric representation of the Pearson correlation coefficient. We have not seen this point discussed elsewhere although Galton seems to have understood this point empirically in 1888 [36]. A linear least-squares (LLS) fit with slope a and intercept b

y = a x + b (73)

to the raw data (i.e. the scatter plot of variates y against variates x) leads to the standard LLS slope [32]

a ^ = 1 n ∑ x ​ y − ( 1 n ∑ x ) ( 1 n ∑ y ) 1 n ∑ x 2 − ( 1 n ∑ x ) 2 (74)

which is the sample statistic corresponding to the population statistic

a = 〈 X Y 〉 − 〈 X 〉 〈 Y 〉 σ X 2 . (75)

Relation (75) is not the Pearson correlation coefficient expressed by Equation (53). However, if one substitutes into Equation (74) the scaled variables

x ′ ≡ x / σ X y ′ ≡ y / σ Y (76)

then it follows straightforwardly that the resulting sample statistic corresponds to the population statistic a ′

a ′ = 〈 X Y 〉 − 〈 X 〉 〈 Y 〉 σ X σ Y , (77)

which is the Pearson correlation coefficient.

The quantity ρ in Table 4 is especially revealing, for it shows the agreement to three or four decimal places of the values of the empirical weight-height correlation coefficient for male and female subgroups obtained in 3 different ways: 1) direct calculation of the covariance of unpartitioned data displayed in Figure 10; 2) calculation of the slope of the scaled data in Figure 11; and 3) prediction of ρ by lognormal theory from the Pearson correlation coefficient r of the bivariate normal distribution N W H ( m W , s W 2 ; m H , s H 2 ; r ) . Thus, analysis of the Pearson correlation coefficient ρ reinforces the conclusion that weight and height comprise correlated bivariate lognormal random variables symbolized by Λ W H ( m W , s W 2 ; m H , s H 2 ; r ) .

We also note here, in anticipation of the next section, the very close agreement in Table 4 of the sample mean and standard deviation of the log-BMI data with the corresponding values predicted from Equation (64), which again depend on the Pearson correlation r.

Figure 13 shows histograms of the BMI for males (left panel) and females (right panel) calculated from the weight and height data of the ANSUR sample and normalized to unit area. The dashed red-blue envelope curve in each panel is actually a superposition of two theoretical curves: a) a lognormal profile (red) with mean and variance obtained directly from the unpartitioned set of natural logarithms of the empirical BMI variates; and b) the lognormal profile (blue) from Equation (62) with parameters predicted by Equation (64) and Gaussian statistics (68), (69), (71), (72). The perfect superposition of the two theoretical profiles is strong evidence that human weight and height are described by a correlated bivariate lognormal distribution, and that BMI is likewise distributed lognormally with theoretically determined, nonadjustable parameters.

Chi-square tests of the hypothesis that BMI is a lognormal variable is summarized in Table 3. For ν = 24 degrees of freedom, the tests yielded respective p-values of 44.37% for the male subgroup and 12.21% for the female subgroup.

Corresponding histograms of the natural logarithm of BMI are shown in Figure 14, superposed by Gaussian envelope curves computed with the parameters used in Figure 13. Chi-square tests of the goodness of fit, given in Table 3, yielded

p-values of 52.12% for the male subgroup and 12.66% for the female subgroup.

It is to be emphasized that the excellent fit of the theoretical probability density to the normalized histograms of BMI depends crucially on the correlation of the two variables, weight and height. Recall that the 4 parameters (2 pairs of means and variances) that separately characterize the lognormal distributions of weight (mass) and height are obtained experimentally from the marginal distributions of what is actually a bivariate normal distribution N W H ( m W , s W 2 ; m H , s H 2 ; r ) . However, the marginal distributions are independent of the Pearson correlation parameter r. If one is ignorant of, or intentionally disregards, the correlation of weight and height, the resulting theoretical probability density function may then fit the observed BMI distribution very poorly, as illustrated in Figure 15.

The gray translucent bars in Figure 15 comprise the BMI histogram of the ANSUR female subgroup displayed in Figure 13. The black enveloping curve is the theoretical PDF, Equation (62), with empirical correlation r = 0.5387, also shown in Figure 13. By contrast, the orange bars comprise a normalized histogram of BMI simulated by 10,000 samples drawn independently from RNGs for mass and height. The magenta envelope is the theoretical PDF, Equation (62), with r = 0. The histogram composed of uncorrelated samples of weight and height is much wider that the true (i.e. empirically obtained) histogram, and of lower maximum (since the total area under a normalized histogram is unity). As shown in Figure 15, the tails of the two histograms, which characterize the subpopulations at greatest risk of obesity and metabolic disease, differ significantly. To disregard positive (negative) correlation of weight and height is to significantly overcount (undercount) the population at greatest risk.