Two tape readings and a height are enough for a man: the neck, the abdomen at the navel, and how tall he stands. A woman adds a third tape, around the hips. The equations behind the calculator were fitted in 1984 by James Hodgdon and Marcie Beckett at the Naval Health Research Center in San Diego. They measured 602 Navy men and 214 Navy women with a tape and also weighed them underwater.
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A man at 19.4 percent and a woman at 28.4 percent
A man is 178 cm tall (5 ft 10 in), with a 39 cm neck (15.4 in) and a 90 cm waist at the navel (35.4 in). Waist minus neck is 51 cm. The base-10 logarithm of 51 is 1.7076 and that of 178 is 2.2504.
Density = 1.0324 − 0.19077 × 1.7076 + 0.15456 × 2.2504 = 1.0545 g per cubic centimeter.
Percent fat = 495 ÷ 1.0545 − 450 = 19.4.
The calculator shows 19.4 percent, to be read as 15.9 to 22.9 percent. At 82 kg that is about 15.9 kg of fat and 66.1 kg of everything else.
A woman is 165 cm tall, with a 33 cm neck, a 74 cm waist and 99 cm hips. Waist plus hips minus neck is 140 cm, and the same two steps give a density of 1.0346 and 28.4 percent. The range is 24.7 to 32.1 percent. At 62 kg that is about 17.6 kg of fat and 44.4 kg of everything else.
The Navy equations in centimeters and in inches
| For | Centimeter form, 1984 | Inch form, 1999 |
|---|---|---|
| Men | Density = 1.0324 − 0.19077 × log10(waist − neck) + 0.15456 × log10(height) | Percent fat = 86.010 × log10(waist − neck) − 70.041 × log10(height) + 36.76 |
| Women | Density = 1.29579 − 0.35004 × log10(waist + hips − neck) + 0.22100 × log10(height) | Percent fat = 163.205 × log10(waist + hips − neck) − 97.684 × log10(height) − 78.387 |
In plain words: take the waist, subtract the neck (and for women add the hips first), take the logarithm, and weigh it against the logarithm of height. A bigger waist raises the result. A bigger neck or a taller frame lowers it.
The centimeter form needs one more step: percent fat = 495 ÷ density − 450. That step is William Siri's formula. In a 1956 report Siri took fat at 0.900 g per cubic centimeter and the fat-free body at 1.100. Those two densities give 4.950 ÷ density − 4.500 as the share of the body that is fat.
The inch form is not the centimeter form rearranged. In 1999 Hodgdon and Karl Friedl refitted the equations on 594 of the men and 202 of the women, so that they give percent fat directly. The two forms differ by a few tenths of a point. The man in the example gets 19.4 percent from one and 19.5 from the other, and the woman gets 28.4 and 28.7. The calculator uses the centimeter form and converts inches first.
Weight and age are not in the men's equation. Hodgdon and Beckett offered both to the regression, and neither entered the final model.
Neck, waist and hip tape sites for men and women
| Tape | Men | Women |
|---|---|---|
| Neck | Just under the larynx (Adam's apple), tape sloping slightly down toward the front. | The same site as for men. |
| Waist | Around the abdomen, level with the navel. | At the narrowest level of the abdomen, about midway between the lower end of the breastbone and the navel. |
| Hips | Not used. | Around the buttocks, at the level where they stick out most. |
The waist site is the one place where the sexes differ. A man who measures at his narrowest point, or a woman who measures at her navel, gives the equation a number it was not fitted on.
The Navy team laid the tape so that it followed the skin without pressing into it. They read the abdomen at the end of a normal breath out. Each site was measured twice, and a third time when the two readings differed by more than 1 cm. Tape technique for every site is set out in how to take body measurements.
The Navy's own error figures
| Equation and test | People measured | Standard error, in points of body fat |
|---|---|---|
| Men, 1984 fit | 602 Navy men aged 18 to 56 | 3.52 |
| Men, 1984 check on a second sample | 100 Navy men measured by another laboratory | 2.70 |
| Women, 1984 fit | 214 Navy women aged 18 to 44 | 3.72 |
| Women, 1984 check on a second sample | 66 Canadian Forces women | 4.36 |
| Women, 1984 check on a third sample | 80 Navy women | 4.04 |
| Men, 1999 inch form | 594 of the 1984 men | 3.52 |
| Women, 1999 inch form | 202 of the 1984 women | 3.61 |
Every figure is a standard error against underwater weighing. If errors follow the usual bell curve, about two results in three land within one standard error of the underwater figure. About one in three lands further away. The calculator's range uses the 1984 fit: 3.5 points either way for men and 3.7 for women.
Underwater weighing carries its own uncertainty. Siri put the uncertainty in fat estimated from density at about 4 percent of body weight. Most of it comes from the makeup of the fat-free body, which varies from person to person.
In 1999 the Navy also compared the equations with a four-part model of body composition in a newer sample. The standard errors were 3.15 points for men and 3.12 for women. The tape results ran higher on average, by 0.8 points for men and 2.0 points for women.
Body fat by tape difference and height
| Waist minus neck | Height 165 cm (5 ft 5 in) | Height 175 cm (5 ft 9 in) | Height 185 cm (6 ft 1 in) |
|---|---|---|---|
| 35 cm (13.8 in) | 8.1% | 6.4% | 4.9% |
| 40 cm (15.7 in) | 12.8% | 11.1% | 9.5% |
| 45 cm (17.7 in) | 17.1% | 15.4% | 13.7% |
| 50 cm (19.7 in) | 21.0% | 19.2% | 17.6% |
| 55 cm (21.7 in) | 24.5% | 22.7% | 21.1% |
| 60 cm (23.6 in) | 27.8% | 26.0% | 24.3% |
| 65 cm (25.6 in) | 30.9% | 29.1% | 27.4% |
| 70 cm (27.6 in) | 33.8% | 31.9% | 30.2% |
The men's equation uses only the difference between waist and neck. A 90 cm waist with a 40 cm neck reads the same as a 95 cm waist with a 45 cm neck.
| Waist plus hips minus neck | Height 155 cm (5 ft 1 in) | Height 165 cm (5 ft 5 in) | Height 175 cm (5 ft 9 in) |
|---|---|---|---|
| 120 cm (47.2 in) | 20.5% | 17.8% | 15.4% |
| 130 cm (51.2 in) | 26.0% | 23.3% | 20.7% |
| 140 cm (55.1 in) | 31.2% | 28.4% | 25.8% |
| 150 cm (59.1 in) | 36.2% | 33.3% | 30.7% |
| 160 cm (63.0 in) | 40.9% | 38.0% | 35.3% |
| 170 cm (66.9 in) | 45.4% | 42.5% | 39.7% |
| 180 cm (70.9 in) | 49.8% | 46.8% | 44.0% |
These figures are the Navy equations' output and nothing more. Each carries the same standard error as a single result.
Tape method against the BMI formula
When a weight is entered, the calculator adds a second figure from the formula of Deurenberg, Weststrate and Seidell (1991):
Body fat percent = 1.20 × BMI + 0.23 × age − 10.8 × sex − 5.4, where sex is 1 for men and 0 for women.
That formula was fitted on 1,229 people aged 7 to 83, with BMIs from 13.9 to 40.9. Its standard error for adults is 4.1 points, against 3.5 and 3.7 for the Navy equations.
The two methods read different things. The BMI formula knows only weight, height, age and sex, and the CDC notes that BMI does not distinguish between fat, muscle and bone mass. The tape method leaves weight out and reads where girth sits.
So the two can disagree. The man in the example, at 82 kg and age 35, gets 19.4 percent from the tape and 22.9 percent from the BMI formula. The woman, at 62 kg and age 30, gets 28.4 and 28.8. Deurenberg and colleagues reported that their formula slightly overestimated body fat in people with obesity. The BMI visualizer shows the BMI side on its own.
What a Navy body fat figure leaves open
The result is an estimate from a regression, shown with its error range. It does not measure your body directly and it diagnoses nothing.
The equations were fitted on Navy personnel. The men were aged 18 to 56 and averaged 21.6 percent body fat. The women were aged 18 to 44 and averaged 27.0 percent. The men's report itself says equations of this kind tend to be specific to the population they were built on. The reports give no error figures for teenagers, older adults or people far from those averages.
No category is attached to the percentage, because no agreed set of ranges exists to attach. Gallagher and colleagues wrote in 2000 that no accepted published ranges were available. The ranges they went on to propose were linked to BMI limits and labeled provisional.
If the waist is not larger than the neck, the equation has no answer, because it takes the logarithm of the difference. The calculator says so in place of a number.
The CDC describes a DEXA scan as a precise measure of body composition, and adds that the scans are expensive and not widely available.