The New BMI Formula: Trefethen's Fix for a 200-Year-Old Equation

For nearly two centuries, the Body Mass Index (BMI) has served as the world's go-to screening measure for weight status. Doctors use it, insurance companies rely on it, and the World Health Organization builds population health guidelines around it. But the formula behind BMI has a fundamental mathematical flaw — one that systematically penalizes tall people and flatters short people. In 2013, Oxford mathematician Nick Trefethen proposed a new BMI formula designed to fix this problem once and for all.

In this comprehensive guide, we will break down exactly what changed, why it matters, how much the two formulas differ across different body types, and whether you should start using the new BMI formula instead of the old one. We also provide an interactive side-by-side calculator so you can compare your own results instantly.

What Is BMI? The Original Quetelet Formula (1832)

The Body Mass Index was invented by Adolphe Quetelet, a Belgian mathematician, astronomer, and statistician, in 1832. Quetelet was not a physician — he was interested in the statistics of the "average man" and sought a simple mathematical relationship between a person's weight and height. The formula he devised divides a person's weight in kilograms by the square of their height in meters:

For example, a person who weighs 75 kg and stands 1.78 m tall would have a BMI of 75 / (1.78 × 1.78) = 23.7. According to WHO classification, this falls in the "Normal" range (18.5 to 24.9).

Quetelet's formula was not originally intended as a health metric. It was a statistical tool for characterizing populations. It was not until the 1970s that physiologist Ancel Keys popularized the term "Body Mass Index" in a landmark 1972 paper and proposed it as a practical screening tool for obesity. Keys tested several weight-for-height indices and found that Quetelet's formula correlated reasonably well with body fat percentage in large population studies, despite its simplicity.

The formula's greatest advantage has always been its simplicity. You need only two measurements — weight and height — and the calculation is trivial. This made it enormously appealing for public health screening, especially in the era before body composition scanners and DEXA machines.

How BMI categories were established

The WHO established the standard BMI classification thresholds that are used worldwide today:

• Underweight: BMI below 18.5
• Normal weight: BMI 18.5 to 24.9
• Overweight: BMI 25.0 to 29.9
• Obese Class I: BMI 30.0 to 34.9
• Obese Class II: BMI 35.0 to 39.9
• Obese Class III: BMI 40.0 and above

These thresholds were established based on large epidemiological studies linking BMI ranges to mortality and morbidity outcomes. They have remained essentially unchanged since their adoption in the late 1990s, and virtually all medical guidelines, research papers, and clinical protocols reference these categories.

Why the Original BMI Formula Is Flawed

Despite its widespread use, the traditional BMI formula contains a fundamental mathematical error in how it accounts for height. The core issue is the exponent of 2 applied to height. Squaring the height assumes that when a person grows taller, their body expands proportionally in only two dimensions (width and depth) while their mass increases accordingly. But human bodies are three-dimensional objects — when people are taller, they tend to be proportionally larger in all three spatial dimensions.

The height bias: penalizing the tall, flattering the short

Because the formula divides by height squared rather than a higher power, it systematically overestimates the BMI of tall people and underestimates the BMI of short people. Consider the practical implications:

• A 6'4" (193 cm) person with perfectly average body proportions for their height will get a higher BMI than they should, potentially pushing them from "Normal" into "Overweight" territory.
• A 5'0" (152 cm) person with the same proportional build will get a lower BMI than they should, potentially masking genuine overweight status.

This is not a minor technicality. Research has shown that the traditional BMI systematically misclassifies a meaningful percentage of very tall and very short individuals. A study published in the International Journal of Obesity found that approximately 18.5% of tall adults (over 6'0") who were classified as overweight by BMI actually had a body fat percentage in the normal range. Conversely, about 13% of shorter adults (under 5'3") classified as normal weight had a body fat percentage that placed them in the overweight or obese category.

Why Quetelet chose the square

Quetelet selected the exponent of 2 based on empirical observation of population data available in the early 19th century. His sample was limited to European men of relatively similar heights, and the square provided a reasonably good fit for that narrow dataset. He likely did not have the statistical tools or the diverse population data needed to detect the systematic height bias. Furthermore, Quetelet was creating a population-level statistical index, not an individual clinical measurement, so small per-person inaccuracies were less concerning for his purposes.

The dimensional analysis problem

From a physics and mathematics standpoint, the issue becomes clear through dimensional analysis. Mass is measured in kilograms, which relates to volume (a three-dimensional property), not area (a two-dimensional property). If we model the human body as a rough cylinder or ellipsoid that scales proportionally with height, then:

• Volume scales with height × width × depth
• If width and depth each scale with approximately height^0.5 to height^0.75 (based on allometric studies), then volume scales with approximately height² to height^2.5
• Since mass is proportional to volume, mass scales with approximately height² to height^2.5
• Dividing mass by height² therefore does not produce a height-independent index — the result still depends on height

This dimensional mismatch is the root cause of the height bias. A truly height-independent body mass index needs a higher exponent in the denominator — somewhere between 2 and 3.

Nick Trefethen's 2013 Revised BMI Formula

In January 2013, Nick Trefethen, a Professor of Numerical Analysis at the Mathematical Institute of the University of Oxford and a Fellow of the Royal Society, published a letter in The Economist and a more formal note referenced in the British Medical Journal (BMJ) proposing a revised BMI formula. His new formula is:

The two key changes from the traditional formula are:

1. The exponent changes from 2 to 2.5: This better accounts for how body mass scales with height in three-dimensional humans. The exponent of 2.5 sits between the 2-dimensional scaling (exponent 2) and the 3-dimensional scaling (exponent 3), reflecting the fact that not all body dimensions scale linearly with height.
2. A scaling factor of 1.3 is added: This normalizing constant ensures that the new BMI gives the same result as the traditional BMI for a person of average height (approximately 1.69 m or 5'6.5"). Without this constant, the change in exponent would shift all BMI values and render the existing WHO category thresholds meaningless.

"The BMI as currently defined is a bizarre measure. We live in a three-dimensional world, yet the BMI is defined as weight divided by height squared. If all three dimensions of a human being doubled, the weight would increase by a factor of eight, but the BMI would only increase by a factor of two." — Nick Trefethen, University of Oxford, 2013

What the 1.3 scaling factor does

The constant 1.3 is not arbitrary — it is calibrated so that for an "average" height person (around 170 cm / 5'7"), the old and new BMI produce essentially the same numerical result. This is a deliberate design choice that makes the new formula a drop-in replacement for the old one: the same WHO category thresholds (18.5, 25, 30, 35, 40) can still be applied. No recalibration of medical guidelines would be needed if the new formula were adopted, because the median-height person's BMI stays the same.

Trefethen's credentials and motivation

Nick Trefethen is not a fringe figure. He is one of the world's leading numerical analysts, an elected Fellow of the Royal Society (2005), and has held professorships at MIT, Cornell, and Oxford. He was motivated not by medical research per se, but by a mathematician's frustration with an equation that "everyone uses but which is clearly dimensionally unsatisfactory." His proposal gained significant attention in the mathematical, medical, and popular press, including coverage in the BBC, The Guardian, The New York Times, and Nature.

The Mathematical Explanation: Why the Exponent Matters

To understand why changing the exponent from 2 to 2.5 makes such a difference, let us work through the mathematics step by step.

Allometric scaling and the power law

In biology, allometric scaling describes how characteristics of organisms change with size. The relationship between body mass (M) and height (H) across individuals in a population follows a power law:

Where p is the scaling exponent. If humans were perfectly geometric shapes that scaled uniformly in all dimensions, p would equal exactly 3 (since volume, and therefore mass, scales as the cube of a linear dimension). In practice, research on human populations has found that the actual scaling exponent for mass versus height ranges from approximately 2.3 to 2.7, depending on the population studied, the age group, and the measurement methodology.

Why not exactly 3?

The scaling exponent is less than 3 because humans do not scale uniformly in all dimensions as they get taller. Taller people tend to be relatively more slender — they are taller, but not proportionally wider and deeper. Biomechanical constraints, growth hormone effects, and skeletal mechanics all contribute to this. The typical scaling exponent found across large population studies clusters around 2.5, which is exactly the value Trefethen chose.

The error at different heights

Let us quantify the error. Consider two people with identical body proportions — same relative amounts of muscle, fat, and bone — but different heights. Person A is 152 cm (5'0") and Person B is 193 cm (6'4"). If we calculate the traditional BMI and the new BMI for both:

For the traditional BMI (exponent 2), the index is not height-independent. Taller Person B will get a systematically higher value, and shorter Person A will get a systematically lower value, even though their body compositions are identical in proportional terms.

For the new BMI (exponent 2.5), the index is much closer to height-independent. Both persons get approximately the same BMI, which correctly reflects their identical proportional body composition.

The magnitude of the error depends on how far a person's height deviates from the calibration height (about 170 cm). At extreme heights, the discrepancy between old and new BMI can be 1 to 2 full BMI points — enough to shift someone between weight categories.

A concrete example

Consider a person who is 190 cm (6'3") and weighs 90 kg:

• Traditional BMI: 90 / (1.90)² = 90 / 3.61 = 24.9 (just barely Normal)
• New BMI: 1.3 × 90 / (1.90)^2.5 = 117 / 4.974 = 23.5 (comfortably Normal)

The difference of 1.4 BMI points is clinically meaningful: under the old formula, this person is teetering on the edge of "Overweight," while the new formula correctly identifies them as solidly in the "Normal" range.

Now consider a person who is 155 cm (5'1") and weighs 62 kg:

• Traditional BMI: 62 / (1.55)² = 62 / 2.4025 = 25.8 (Overweight)
• New BMI: 1.3 × 62 / (1.55)^2.5 = 80.6 / 2.991 = 26.9 (Overweight)

Here the new formula gives a higher BMI by 1.1 points, suggesting that the traditional BMI was actually flattering this shorter person by about a full BMI point.

Side-by-Side BMI Calculator: Old vs New Formula

Use the calculator below to compare your BMI under both the traditional (Quetelet) formula and the new (Trefethen) formula. Enter your height and weight, and see both results instantly with color-coded categories and the exact difference between them.

Comparison Table: Old vs New BMI at Different Heights and Weights

The table below shows how the traditional and new BMI compare across 12 different height/weight combinations. Notice how the difference grows as height deviates further from the average of approximately 170 cm. Negative differences mean the new formula gives a lower BMI (taller people), while positive differences mean the new formula gives a higher BMI (shorter people).

Why the WHO and CDC Haven't Adopted the New Formula

Despite the mathematical superiority of Trefethen's formula, neither the World Health Organization (WHO) nor the Centers for Disease Control and Prevention (CDC) have switched to the new BMI. There are several substantial reasons for this institutional inertia.

1. Decades of epidemiological data

Hundreds of thousands of research studies, spanning decades and involving millions of participants, have used the traditional BMI formula. Mortality risk curves, disease prevalence rates, treatment guidelines, and insurance tables are all built on the old BMI. Switching formulas would not invalidate these studies, but it would create confusion about how to interpret historical data alongside new data collected with the revised formula. The cost of this transition — in terms of recalculating baselines, updating software systems, retraining clinicians, and revising published guidelines — is enormous.

2. The difference is modest for average-height populations

For the majority of adults (those between about 5'3" and 5'11"), the difference between the old and new BMI is less than half a point. From a population health screening perspective, this difference is not large enough to justify the upheaval of changing the standard. Public health organizations focus on population-level screening, where small per-individual errors tend to average out, rather than individual-level precision.

3. BMI is already known to be a rough estimate

Medical professionals are already well aware that BMI is an imperfect measure. It does not distinguish between muscle and fat, does not account for body fat distribution (visceral vs. subcutaneous), varies in accuracy across ethnic groups, and is less useful for athletes, elderly individuals, and children. Given that BMI is already understood to be a crude screening tool, improving its height scaling by a fractional amount may not seem like a priority when the fundamental limitations of using only weight and height remain regardless of which formula is used.

4. Category thresholds would need validation

Even though the Trefethen formula was calibrated so that the numerical values match at average height, the question of whether the same category thresholds (18.5, 25, 30) produce the same health risk predictions under the new formula would need to be validated through new epidemiological studies. It is plausible that the thresholds would need adjustment for shorter and taller subpopulations, which would add further complexity.

5. Institutional inertia and consensus requirements

Changing a globally used health metric requires consensus among multiple international bodies, including the WHO, CDC, national health ministries, and professional medical organizations. This process is slow by design — it requires extensive review, multiple rounds of expert panels, public comment periods, and coordination across jurisdictions. Even well-supported changes can take years or decades to implement.

Other BMI Alternatives: Beyond Trefethen

The Trefethen formula is not the only attempt to improve upon or replace BMI. Several other indices and measures have been proposed over the years, each with its own strengths and weaknesses.

Ponderal Index (Rohrer Index)

The Ponderal Index (PI), also known as the Rohrer Index, uses height cubed instead of height squared:

By using an exponent of 3, the Ponderal Index better reflects the three-dimensional nature of the human body. Its normal range is approximately 11 to 15 kg/m³. However, it has gained less traction than BMI because its scale is less intuitive, its health-risk thresholds are less well-established, and research has suggested the scaling exponent of 3 actually overcorrects the height bias, producing a slight inverse bias where taller people now get BMI values that are too low.

Surface-Based Body Mass Index (SBMI)

The Surface-Based Body Mass Index normalizes body mass against body surface area (BSA) rather than height squared. It draws on the Du Bois formula for BSA estimation and maps the result onto a percentile-based scale from 0 to 70. SBMI provides a more physiologically grounded assessment because body surface area is directly relevant to metabolic rate, heat dissipation, and drug dosing. However, its complexity and the need for a lookup table have limited its adoption outside of specialized clinical and research settings.

Body Adiposity Index (BAI)

The Body Adiposity Index, proposed by Bergman et al. in 2011, attempts to estimate body fat percentage directly from hip circumference and height:

BAI was designed as a field-friendly alternative that does not require weighing the individual, making it useful in settings where scales are unavailable. However, subsequent validation studies have shown that BAI is not significantly more accurate than BMI for predicting body fat percentage and introduces its own biases across different populations, sexes, and activity levels.

Waist-to-Height Ratio (WHtR)

Perhaps the most promising simple alternative to BMI is the waist-to-height ratio, which divides waist circumference by height. A WHtR above 0.5 is associated with increased cardiometabolic risk regardless of sex, age, or ethnicity. Multiple systematic reviews and meta-analyses have found that WHtR is a better predictor of cardiovascular disease, type 2 diabetes, and all-cause mortality than BMI. It also captures central adiposity (belly fat), which is the most metabolically dangerous fat depot and which BMI completely ignores.

ABSI (A Body Shape Index)

The ABSI (A Body Shape Index), developed by Krakauer and Krakauer in 2012, combines waist circumference, height, and BMI into a single index designed to predict mortality risk. It explicitly accounts for the shape of the body (centrally obese vs. peripherally obese) rather than just overall mass. Research has shown that ABSI is a strong predictor of mortality risk independent of BMI, but its calculation is more complex and it has not yet been widely adopted in clinical practice.

Smart BMI and other proposals

Several researchers have proposed "Smart BMI" or "Corrected BMI" formulas that incorporate additional variables such as age, sex, ethnicity, and activity level. While these produce more accurate individual assessments, they sacrifice the simplicity that made BMI useful in the first place. The challenge of BMI alternatives is always the same: the more accurate you want to be, the more measurements you need, and at some point you might as well use DEXA, BIA, or hydrostatic weighing for a direct body composition measurement.

Should You Use the New or Old BMI Formula?

The answer depends on your situation and what you are using BMI for.

Use the traditional BMI if:

• You need to communicate with your doctor, insurer, or any institution that uses the standard BMI
• You are of roughly average height (160-180 cm / 5'3" to 5'11") and the difference between formulas is negligible
• You are tracking BMI over time and want consistency with your historical data
• You are participating in a study or program that requires the standard formula

Consider the new BMI if:

• You are notably tall (over 183 cm / 6'0") or short (under 160 cm / 5'3") and want a fairer assessment
• Your traditional BMI places you right at a category boundary (e.g., 24.8 or 25.2) and you want to check whether the height bias is affecting your classification
• You are mathematically inclined and want the more theoretically sound measure
• You are doing personal tracking and are not constrained by institutional standards

Best practice: use both and supplement with other measures

Regardless of which BMI formula you use, it is important to remember that BMI is just one data point. Neither the old nor the new formula can tell you about your body fat percentage, muscle mass, fat distribution, metabolic health, blood pressure, cholesterol, blood sugar, or fitness level. For a comprehensive health assessment, consider supplementing BMI with:

• Body fat percentage (measured via DEXA, BIA, calipers, or the US Navy method)
• Waist circumference or waist-to-height ratio (a better predictor of cardiometabolic risk)
• Blood biomarkers (fasting glucose, HbA1c, lipid panel, blood pressure)
• Fitness assessments (cardiovascular endurance, strength, flexibility)
• TDEE and activity level tracking

The new BMI formula is a meaningful improvement for individuals at the height extremes, but it does not solve the fundamental limitations of using only weight and height to assess health. Use it as one tool among many, and always consult a healthcare professional for personalized medical guidance.

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