# Quetelet's Equation, Upper Weight Limits and BMI Prime

## 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.

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• #### 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`

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• ##### 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.

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• ###### A Brief History of Body Mass Index (BMI)

Quetelet's Equation and 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.

In the metric system, this is expressed as the equation:

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

where mass is in kilograms, and height is in meters.

In the English (Avoirdupois) system, the equation is:

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

where weight is in pounds, and height is in inches.

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.

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• What Parameter Does BMI Represent?

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.

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BMI Units:

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.

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• BMI Tables:

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)`

where mass is rounded off to the nearest kilogram.

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`

where weight is rounded off to the nearest pound.

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

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• The Prognostic Value of BMI

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.

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• Weight Categories and BMI

Federal BMI Guidelines

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

TABLE 1:

Weight Categories

 Weight Category BMI Range Underweight < 18.5 Healthy Weight 18.5 - 24.9 Overweight 25.0 - 29.9 Obese 30.0 - 39.9 Morbidly Obese > 39.9

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• How Much Should I Weigh?

”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.

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• BMI Values in Laymen's Terms

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)`

where mass is in kilograms and height is in meters.

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

where weight is in pounds and height is in inches.

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• Upper Weight Limit - The Gold Standard

What Is My Upper Weight Limit and How Is It Calculated?

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)`

where mass is in kilograms and height is in meters.

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

where weight is in pounds and height is in inches.

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• Upper Weight Limits and Ease of Calculation

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:

Upper Weight Limit = [(24.9 lbs/sq in)(73 in)(73 in)] / 703

Upper Weight Limit = [(24.9 lbs/sq in)(5329 sq in)] / 703

Upper Weight Limit = (132692 lbs) / 703

Upper Weight Limit = 189 lbs

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

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• A New Linear Equation for Computing Upper Weight Limits

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`

where:

A = number of inches the patient stands above 4 feet

B = number of inches the patient stands above 5 feet

C = number of inches the patient stands above 6 feet

D = number of inches the patient stands above 7 feet

and

the coefficients 4 and 1 are in "pounds per inch"

the coefficient 2 is in "inches per pound"

For example, to calculate the Upper Weight Limit of a person standing 6 feet one inch, use the following algorithm:

 Calculation Value 1. Start with 80 lbs. = 80 lbs. 2. The patient is 25 inches over 4 feet. Multiply 25 in. by 4 lbs/in. = 100 lbs. 3. The patient is 13 inches over 5 feet. Divide 13 in. by 2 lbs/in. (round off) = 7 lbs. 4. The patient is 1 inch over 6 feet. Multiply 1 inch by 1 lb/in. = 1 lb. 5. Add the four values = 188 lbs.

In general, the Upper Weight Limits derived from the linear equation fall within 1 percent of those derived from Quetelet’s formula, and encompass heights from 4 feet to 8 feet.

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• A New Linear Equation for Computing Upper Mass Limits

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

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

where:

A = (Patient height, up to and including 180 cm) – 140 cm

B = (Patient height, exceeding 180 cm) – 180 cm

and

the numerical coefficients 4/5 and 1 are in kilograms per cm

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).

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• Computing Upper Mass Limits (Expanded Equation)

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`

where:

C = (Patient height up to and including 140 cm) – 120 cm

and

the numerical coefficients 3/5, 4/5 and 1 are in kilograms per cm

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• Using Upper Weight Limit to Determine Ancillary Weight Limits

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.

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• Lower Weight Limit as a Function of Upper Weight Limit

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

Lower Weight Limit / 18.5 = Upper Weight Limit / 25

Lower Weight Limit = (18.5 / 25) Upper Weight Limit

 `Lower Weight Limit = (0.75) Upper Weight Limit`

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

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• Obese Weight Limit as a Function of Upper Weight Limit

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

Obese Weight Limit / 30 = Upper Weight Limit / 25

Obese Weight Limit = (30/25) Upper Weight Limit

 `Obese Weight Limit = (1.2) Upper Weight Limit`

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• Morbidly Obese Weight Limit as a Function of Upper Weight Limit

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

Morbidly Obese Weight Limit / 40 = Upper Weight Limit / 25

Morbidly Obese Weight Limit = (40/25) Upper Weight Limit

 `Morbidly Obese Weight Limit = (1.6) Upper Weight Limit`

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• BMI as a Function of Upper Weight Limit

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

BMI / Actual Weight = 25 / Upper Weight Limit

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

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

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• Utility of the BMI Prime System

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:

BMI’ = Actual BMI / Upper Limit BMI

BMI' = 34 lbs per sq in / 25 lbs per sq in

BMI' = 1.36

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• Shortcomings of the Current BMI System

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?

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• BMI Prime Eliminates Confusing Units

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:

BMI’ = Actual Weight / Upper Weight Limit

BMI' = 250 lbs / 199 lbs

BMI' = 1.26

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• BMI Prime Gives Meaningful Quantitative Information

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:

If your BMI' = 0.74, you are 26 % below Upper Weight Limit

If your BMI' = 1.00, you have reached Upper Weight Limit

If your BMI' = 1.08, you are 8 % above Upper Weight Limit

If your BMI' = 1.36, you are 36 % above Upper Weight Limit

If your BMI' = 2.00, you are 100 % above Upper Weight Limit

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

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NIH Guidelines Expressed in Terms of BMI Prime

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

TABLE 2:

Weight Categories

 Weight Category BMI' Range Underweight < 0.74 Healthy Weight 0.74 - 0.99 Overweight 1.00 - 1.19 Obese 1.20 - 1.59 Morbidly Obese > 1.60

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• References and Communications

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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