Table of Contents:

BMI Prime: Introduction

What is BMI Prime?

BMI Prime (abbreviated BMI’) is an index of total body fat content. It is a simple modification of the familiar Body Mass Index (BMI) system.

How is BMI Prime Defined?

BMI Prime is the ratio of a person’s Actual Weight (or Mass) to his Upper Weight (or Mass) Limit.

BMI' = Actual Weight / Upper Weight Limit

It is also the ratio of a person’s Actual BMI to her Upper Limit BMI.

BMI' = Actual BMI / Upper Limit BMI

Why was BMI Prime Proposed?

If your doctor says your BMI is 34, what does that mean? Are you overweight? If so, to what degree?

Most people find BMI values difficult to interpret.

BMI Prime makes the BMI system more understandable and relevant. To appreciate why it was proposed, we must first consider how the older BMI system evolved.

A Brief History of Body Mass Index (BMI)

Lambert Adolphe Quetelet (1796 – 1874), a Belgian statistician, established that in adults, 20 years or older, body mass increases in direct proportion to the height squared.

Mass / [(Height) (Height)] = Constant

(703)(Weight) / [(Height)(Height)] = Constant

Originally, the numerical constant was called Quetelet’s Index. With time, it has become known, more generally, as the Body Mass Index, (abbreviated BMI). The terms Quetelet’s Index, Body Mass Index and BMI are interchangeable.

BMI correlates with total body fat content, and is used because total body fat is difficult to measure directly.

In general, for any given height, the greater the BMI, the greater the amount of fat stored in the body.

If two people are the same height, but have different BMI values, then the person with the higher BMI value is more likely to carry a greater amount of fat.

In the metric system, BMI is expressed in kilograms per square meter. In the English system, BMI is expressed in pounds per square inch.

Note that the English-system units are problematic. In physics, pounds per square inch is equivalent to force per unit area, which, in turn, is equivalent to pressure. Thus, BMI, a measure of total body fat content, is expressed in confusing units of pressure.

Convenient BMI tables have been constructed for both the metric and English systems.

Most metric BMI tables record body mass (in kilograms) as a function of both height (in meters) and BMI. Each value is generated by rearranging Quetelet’s metric system formula:

Mass = (BMI)(Height)(Height)

English-system BMI tables list weight (in pounds) as a function of both height (in inches) and BMI according to the formula:

Weight = [(BMI)(Height)(Height)] / 703

To see an example of a BMI Table, click on the following website:

According to the National Institute of Health (NIH), a person's BMI correlates with morbidity and mortality.

A person with too low a BMI has increased problems with malnutrition and immunocompetence.

A person with too high a BMI has an increased risk of sudden death, certain cancers (breast, colon), heart disease, osteoporosis, gallbladder disease, hypertension, and diabetes mellitus.

Federal BMI Guidelines

The NIH has used BMI to define both healthy and unhealthy weight categories:

”How much should I weigh?” is a frequently asked question in clinical practice.

The answer, based on the above NIH guidelines, is that a person’s weight should correspond to a BMI value between 18.5 and 24.9.

Unfortunately, such an answer is foreign to most people, who speak in terms of pounds or kilograms, rather than BMI values.

To most of us, BMI values become meaningful only when they are expressed in pounds or kilograms.

BMI values are converted either by using a standard BMI Table, or by direct calculation:

Mass = (BMI)(Height)(Height)

Weight = [(BMI)(Height)(Height)] / 703

The Upper Weight (or Mass) Limit is the weight (or mass) above which a person would be considered “fat” or unhealthy.

In food-abundant countries, people are more concerned with their upper weight limits, and less concerned with their lower weight limits.

Given the NIH guideline that a maximum healthy BMI is 24.9, and given a person’s height, an Upper Weight Limit or Upper Mass Limit for that individual is straightforward to calculate:

Upper Mass Limit = (24.9)(Height)(Height)

Upper Weight Limit = [(24.9)(Height)(Height)] / 703

Upper Weight Limits are straightforward to derive, but cumbersome to compute.

For example, to find the Upper Weight Limit of a person standing six feet, one inch (73 inches) tall:

The calculation of Upper Mass Limit is equally tedious, best accomplished with an electronic calculator.

A new linear equation facilitates computation of Upper Weight Limits for most adults in clinical practice.

Upper Weight Limit = 80 lb + 4A + B/2 + 1C + 1D

A new linear equation facilitates calculation of Upper Mass Limits in adult clinical practice.

Upper Mass Limit = 48 kg + (4/5)A + 1 B

In general, the Upper Mass Limits derived from the linear equation fall within 1 to 2 percent of those derived from Quetelet’s formula, and encompass heights over 139 cm. (4 feet, 7 inches).

For shorter adults, the linear metric equation can be expanded to encompass heights from 120 to 139 cm:

Upper Mass Limit = 36kg + (3/5)C + (4/5)A + 1B

The BMI to Weight Ratio is Constant, for any Given Height

Sadly, in adulthood, we continue growing outward long after we have ceased growing upward.

After you have stopped growing in height (ie once your height is constant), the Quetelet Equation states that your BMI increases linearly with your weight.

Expressed mathematically;

BMI 1 / Weight 1 = BMI 2 / Weight 2 = Constant

for any given height.

If you know your Upper Weight Limit, and pair it with the upper limit BMI value of 25, it is possible to compute the weight which corresponds to any arbitrary BMI value, using simple ratios.

25 / Upper Weight Limit = BMI 2 / Weight 2 = Constant

It is possible to generate an entire BMI table for any given height, independent of Quetelet’s equation, using simple ratios.

Simple ratios permit calculation of Lower Weight Limits, Obese Weight Limits, Morbidly Obese Weight Limits, and BMI values as a function of Upper Weight Limit.

Lower Weight Limit, defined at a BMI of 18.5, is derived as follows:

Lower Weight Limit = (0.75) Upper Weight Limit

Thus, the Lower Weight Limit is three-quarters the Upper Weight Limit.

Obese Weight Limit, defined at a BMI of 30, is derived as follows:

Obese Weight Limit = (1.2) Upper Weight Limit

Morbidly Obese Weight Limit, defined at a BMI of 40, is derived as follows:

Morbidly Obese Weight Limit = (1.6) Upper Weight Limit

A patient’s BMI can be computed, given his actual weight or mass, as follows:

BMI = (25) (Actual Weight / Upper Weight Limit)

Basis of the BMI Prime System

The BMI System arose from the observation that most patients tend to compare their Actual Weights to their Upper Weight Limits:

BMI' = Actual Weight / Upper Weight Limit

BMI' = Actual BMI / Upper Limit BMI = Actual BMI / 25

For example, a person with a BMI of 34 would be assigned a BMI’ of 1.36 as follows:

Current BMI values are associated with confusing units. The English system BMI has units of pressure, even though total body fat content, and not pressure, is measured indirectly.

An isolated BMI value does little to convey a quantitative sense of total body fat content. A BMI of 34 implies obesity, but to what degree?

Since BMI’ is a ratio, any confusing units in the numerator and denominator cancel out, yielding a pure, dimensionless number.

For example, a person 6 feet 3 inches tall who weighs 250 pounds would be assigned a BMI’ of 1.26 as follows:

The Upper Limit BMI Prime is 1.00.

Any person with a BMI Prime 1.00 or greater is overweight.

BMI Prime tells us immediately what percentage an individual exceeds, or falls below, his Upper Weight Limit, according to the formula:

(BMI'- 1.00)(100%) = % above or below Upper Weight Limit

For example:

Unlike BMI, BMI Prime gives quantitative information regarding total body fat content at a glance.

If the BMI Prime system were to be implemented, the NIH Guidelines would become:

Literature References

Quetelet A: Physique sociale: Ou essay sur le development des faculties de l’homme. Brussels. C Muquardt; 1869.

Keys A, Fidanza F, Karvonen M, et al: Indices of relative weight and obesity. J Chronic Dis. 1972; 25: 329 – 343.

Gadzik J: “How Much Should I Weigh?” – Quetelet’s Equation, Upper Weight Limits and BMI Prime. Connecticut Medicine Feb 2006; 70: 81 – 88.

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  © 2009 James P. Gadzik M.D.