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Maybe this question has answer in medicine, but are there any statistical reasons why BMI index is calculated as $\text{weight}/\text{height}^2$? Why not for example just $\text{weight}/\text{height}$? My first idea is that it has something to do with quadratic regression.

Sample of real data (200 individuals with weight, height, age and gender):

structure(list(Age = c(18L, 21L, 17L, 20L, 19L, 53L, 27L, 22L, 
19L, 27L, 19L, 20L, 19L, 20L, 42L, 17L, 23L, 20L, 20L, 19L, 20L, 
19L, 19L, 18L, 19L, 15L, 19L, 15L, 19L, 21L, 60L, 19L, 17L, 23L, 
60L, 33L, 24L, 19L, 19L, 22L, 20L, 21L, 19L, 19L, 20L, 18L, 19L, 
20L, 22L, 20L, 20L, 27L, 19L, 22L, 19L, 20L, 20L, 21L, 16L, 19L, 
41L, 54L, 18L, 23L, 19L, 19L, 22L, 18L, 20L, 19L, 25L, 18L, 20L, 
15L, 61L, 19L, 34L, 15L, 19L, 16L, 19L, 18L, 15L, 20L, 20L, 20L, 
20L, 19L, 16L, 37L, 37L, 18L, 20L, 16L, 20L, 36L, 18L, 19L, 19L, 
20L, 18L, 17L, 22L, 17L, 22L, 16L, 24L, 17L, 33L, 17L, 17L, 15L, 
18L, 18L, 16L, 20L, 29L, 24L, 18L, 17L, 18L, 36L, 16L, 17L, 20L, 
16L, 43L, 19L, 18L, 20L, 19L, 18L, 21L, 19L, 20L, 23L, 19L, 19L, 
20L, 24L, 19L, 20L, 38L, 18L, 17L, 19L, 19L, 20L, 20L, 21L, 20L, 
20L, 42L, 17L, 20L, 25L, 20L, 21L, 21L, 22L, 19L, 25L, 19L, 40L, 
25L, 52L, 25L, 21L, 20L, 41L, 34L, 24L, 30L, 21L, 27L, 47L, 21L, 
16L, 31L, 21L, 37L, 20L, 22L, 19L, 20L, 25L, 23L, 20L, 20L, 21L, 
36L, 19L, 21L, 16L, 20L, 18L, 21L, 21L, 18L, 19L), Height = c(180L, 
175L, 178L, 160L, 172L, 172L, 180L, 165L, 160L, 187L, 165L, 176L, 
164L, 155L, 166L, 167L, 171L, 158L, 170L, 182L, 182L, 175L, 197L, 
170L, 165L, 176L, 167L, 170L, 168L, 163L, 155L, 152L, 158L, 165L, 
180L, 187L, 177L, 170L, 178L, 170L, 170L, NA, 188L, 180L, 161L, 
178L, 178L, 165L, 187L, 178L, 168L, 168L, 180L, 192L, 188L, 173L, 
193L, 184L, 167L, 177L, 177L, 160L, 167L, 190L, 187L, 163L, 173L, 
165L, 190L, 178L, 167L, 160L, 169L, 174L, 165L, 176L, 183L, 166L, 
178L, 158L, 180L, 167L, 170L, 170L, 180L, 184L, 170L, 180L, 169L, 
165L, 156L, 166L, 178L, 162L, 178L, 181L, 168L, 185L, 175L, 167L, 
193L, 160L, 171L, 182L, 165L, 174L, 169L, 185L, 173L, 170L, 182L, 
165L, 160L, 158L, 186L, 173L, 168L, 172L, 164L, 185L, 175L, 162L, 
182L, 170L, 187L, 169L, 178L, 189L, 166L, 161L, 180L, 185L, 179L, 
170L, 184L, 180L, 166L, 167L, 178L, 175L, 190L, 178L, 157L, 179L, 
180L, 168L, 164L, 187L, 174L, 176L, 170L, 170L, 168L, 158L, 175L, 
174L, 170L, 173L, 158L, 185L, 170L, 178L, 166L, 176L, 167L, 168L, 
169L, 168L, 178L, 183L, 166L, 165L, 160L, 176L, 186L, 162L, 172L, 
164L, 171L, 175L, 164L, 165L, 160L, 180L, 170L, 180L, 175L, 167L, 
165L, 168L, 176L, 166L, 164L, 165L, 180L, 173L, 168L, 177L, 167L, 
173L), Weight = c(60L, 63L, 70L, 46L, 60L, 68L, 80L, 68L, 55L, 
89L, 55L, 63L, 60L, 44L, 62L, 57L, 59L, 50L, 60L, 65L, 63L, 72L, 
96L, 50L, 55L, 53L, 54L, 49L, 72L, 49L, 75L, 47L, 57L, 70L, 105L, 
85L, 80L, 55L, 67L, 60L, 70L, NA, 76L, 85L, 53L, 69L, 74L, 50L, 
91L, 68L, 55L, 55L, 57L, 80L, 98L, 58L, 85L, 120L, 62L, 63L, 
88L, 80L, 57L, 90L, 83L, 51L, 52L, 65L, 92L, 58L, 76L, 53L, 64L, 
63L, 72L, 68L, 110L, 52L, 68L, 50L, 78L, 57L, 75L, 55L, 75L, 
68L, 60L, 65L, 48L, 56L, 65L, 65L, 88L, 55L, 68L, 74L, 65L, 62L, 
58L, 55L, 84L, 60L, 52L, 92L, 60L, 65L, 50L, 73L, 51L, 60L, 76L, 
48L, 50L, 53L, 63L, 68L, 56L, 68L, 60L, 70L, 65L, 52L, 75L, 65L, 
68L, 63L, 54L, 76L, 60L, 59L, 80L, 74L, 96L, 68L, 72L, 62L, 58L, 
50L, 75L, 70L, 85L, 67L, 65L, 55L, 78L, 58L, 53L, 56L, 72L, 62L, 
60L, 56L, 82L, 70L, 53L, 67L, 58L, 58L, 49L, 90L, 58L, 77L, 55L, 
70L, 64L, 98L, 60L, 60L, 65L, 74L, 99L, 49L, 47L, 75L, 77L, 74L, 
68L, 50L, 66L, 75L, 54L, 60L, 65L, 80L, 90L, 95L, 79L, 57L, 70L, 
60L, 85L, 44L, 58L, 50L, 88L, 60L, 54L, 68L, 56L, 69L), Gender = c(1L, 
1L, 1L, 1L, 1L, 1L, 2L, 1L, 1L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 2L, 1L, 2L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 2L, 2L, 2L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 1L, 1L, 2L, 1L, 2L, 
2L, 1L, 1L, 1L, 2L, 2L, 1L, 2L, 1L, 1L, 1L, 2L, 1L, 1L, 2L, 2L, 
1L, 1L, 1L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 1L, 1L, 1L, 2L, 
1L, 2L, 1L, 2L, 2L, 1L, 2L, 1L, 1L, 1L, 1L, 2L, 1L, 2L, 2L, 1L, 
2L, 1L, 1L, 2L, 1L, 1L, 2L, 1L, 1L, 1L, 2L, 1L, 2L, 2L, 1L, 1L, 
1L, 2L, 1L, 1L, 1L, 1L, 2L, 2L, 1L, 2L, 1L, 2L, 1L, 1L, 2L, 1L, 
1L, 2L, 2L, 2L, 2L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 1L, 1L, 2L, 
1L, 1L, 2L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 1L, 1L, 1L, 2L, 1L, 
2L, 1L, 1L, 1L, 2L, 1L, 1L, 2L, 2L, 1L, 1L, 1L, 2L, 2L, 1L, 2L, 
1L, 1L, 1L, 1L, 1L, 1L, 2L, 1L, 2L, 2L, 1L, 1L, 1L, 2L, 1L, 1L, 
1L, 2L, 1L, 1L, 2L, 2L, 1L)), .Names = c("Age", "Height", "Weight", 
"Gender"), row.names = 304:503, class = "data.frame")
amoeba
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sitems
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    Nowadays formulas like this would drop out of a *linear* regression of log(weight) against log(height), which (for biological and statistical reasons) is a more natural way to analyze these quantities. – whuber Jul 12 '13 at 18:26
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    I had hoped to illustrate this with real data. The first Google hit found on "weight height data" is a large [UCLA-hosted dataset](http://wiki.stat.ucla.edu/socr/index.php/SOCR_Data_Dinov_020108_HeightsWeights). *It clearly is faked!* The marginal distributions are *perfectly* normally distributed (SW tests with subsamples of 5000 almost always have p-values near 1/2): no outliers, no low kurtosis (from a mixture of genders), no skewness (from a mixture of ages). These data were allegedly "used to develop Hong Kong's ... growth charts for ... body mass index (BMI)." That's *extremely* fishy. – whuber Jul 12 '13 at 18:39
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    The dependency of weight on height is not only nonlinear (allometric, as biologist use to say), it changes along with age and, also, it is heteroscedastic. – ttnphns Jul 12 '13 at 18:44
  • @whuber, you can use my real data (I edited my question) - 100 individuals (with mixture of ages and genders). – sitems Jul 12 '13 at 18:55
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    Thanks, but those data may be too limited to give a good sense of how height and weight co-vary. At a minimum they need to be classified by gender and age. It is clear, though, that the logarithms of height and weight are better to analyze: they reduce the heteroscedasticity to which @ttnphns refers and they also help make the distributions of residuals more symmetric. It is interesting that a regression of log weight against log height gives a slope of $2.55\pm 0.28$. That compares almost *exactly* with Quetelet's estimate of $5/2=2.5$ quoted by AdamO. – whuber Jul 12 '13 at 19:04
  • I added 100 individuals, gender and age. – sitems Jul 12 '13 at 19:12
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    Compare `library(MASS); rlm(log(Weight) ~ log(Height) + cut(Age, 3) + as.factor(Gender), data=y)` to `rlm(Weight ~ Height + cut(Age, 3) + as.factor(Gender), data=y)` (and plot diagnostics for both fits) to see the salutary effect of using logarithms: they do indeed stabilize and symmetrize the residuals. In either model gender is significant and so is age; the relationship with age is nonlinear. It is very interesting that the coefficient of log(height) in the first model is now around $1.6$ instead of $2.5$. (`y` is your data with the missing values deleted.) I don't see any interactions. – whuber Jul 12 '13 at 19:56
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    @whuber, I tried your code with full sample size (n=1336) and coefficient of log(height) is around 1.77. – sitems Jul 12 '13 at 20:24
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    To bring this full circle: (1) because my standard error is $0.3$, the results for the first 100 records are perfectly consistent with your result of $1.77$ (and both differ significantly from $2.5$). (2) The richer, more complex structure we have unearthed in your data--especially for the *logs* of the weights--is practically conclusive proof that the Hong Kong dataset I mentioned is either bogus or has been so thoroughly doctored as to be irrelevant. I cannot believe that *any* natural subgroup of Hong Kong people has a perfectly Normal distribution of weights and heights. – whuber Jul 12 '13 at 20:29
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    See now also https://people.maths.ox.ac.uk/trefethen/bmi.html for a contribution from a very distinguished applied mathematician. – Nick Cox Mar 31 '16 at 10:42

3 Answers3

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This review, by Eknoyan (2007) has far more than your probably wanted to know about Quetelet and his invention of the body mass index.

The short version is that BMI looks approximately normally distributed, while weight alone, or weight/height doesn't, and Quetelet was interested in describing a "normal" man via normal distributions. There are some first-principles arguments too, based on how people grow, and some more recent work has attempted to relate that scaling back to some biomechanics.

It's worth noting that the value of the BMI is fairly hotly debated. It does correlate with fatness pretty well, but the cut-offs for underweight/overweight/obese don't quite match up with healthcare outcomes.

Matt Krause
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    More importantly, he considered `weight/height^3` which would be interpreted as a density (intuitively makes sense), but opted for the classic BMI because of its normal distribution as you said. – AdamO Jul 12 '13 at 18:25
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    @AdamO However, adults generally only grow in 2 of the 3 dimensions... – James Jul 13 '13 at 13:36
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From Adolphe Quetelet's "A Treatise on Man and the Development of his Faculties":

If man increased equally in all dimensions, his weight at different ages would be as the cube of his height. Now, this is not what we really observe. The increase of weight is slower, except during the first year after birth; then the proportion we have just pointed out is pretty regularly observed. But after this period, and until near the age of puberty, weight increases nearly as the square of the height. The development of weight again becomes very rapid at puberty, and almost stops after the twenty-fifth year. In general, we do not err much when we assume that during development the squares of the weight at different ages are as the fifth powers of the height; which naturally leads to this conclusion, in supporting the specific gravity constant, that the transverse growth of man is less than the vertical.

See here.

He wasn't interested in characterizing obesity but the relationship between weight and height as he was very interested in biometry and bell curves. Quetelet's findings indicated that BMI had an approximately normal distribution in the population. This signified to him that he had found the "correct" relationship. (interestingly, only a decade or two later Francis Galton would approach the issue of the "distribution of height" in populations and coin the term "Regression to the Mean").

It's worth noting that the BMI has been a scourge of biometry in modern days because of the Framingham's study's far reaching utilization of BMI as a way of identifying obesity. There is still a lack of any good predictor of obesity (and health related outcomes thereof). The waist to hip measurement ratio is a promising candidate. Hopefully as ultrasounds become cheaper and better, doctors will use them to identify not only obesity, but fatty deposits and calcification in organs and make recommendations for care based on those.

AdamO
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    The phrase "the squares of the weight at different ages are as the fifth powers of the height" suggests $2.5$ to me. In any case, the quote is not related to different adults with different heights but to an individual during development – Henry Jul 12 '13 at 22:13
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    Quetelet is inferring about the development of the individual from observing a population based sample. I think he additionally comments that, on average, one can do well with a 2.5 exponent related weight and height (over all or most age ranges), but specifically in adults the relationship is quadratic. – AdamO Jul 12 '13 at 22:37
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    I *think* that the waist-to-hip ratio was actually considered by Quetelet or his contemporaries, but also got rejected because it wasn't normally distributed either. How far we've come.... – Matt Krause Jul 12 '13 at 23:24
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BMI is primarily used nowadays because of its ability to approximate abdominal visceral fat volume, useful in studying cardiovascular risk. For a case study analyzing the adequacy of BMI in screening for diabetes see Chapter 15 of http://biostat.mc.vanderbilt.edu/CourseBios330 under Handouts. Several assessments are there. You will see that a better power of height is closer to 2.5 but you can do better than using height and weight.

Frank Harrell
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